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            <title><![CDATA[What Is a Schematic Diagram?]]></title>
            <description><![CDATA[<figure>
    <img src="https://www.thoughtco.com/thmb/zDnzif-Ssfn68ePLIyVEWdG-P_Y=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/MontyRakusen-5c4df34cc9e77c0001d760bd.jpg" referrerpolicy="no-referrer">
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<p id="mntl-sc-block_1-0" class="comp mntl-sc-block mntl-sc-block-html">
A schematic is defined as a picture that shows something in a simple way, using symbols. A <strong>schematic diagram</strong> is a picture that represents the components of a process, device, or other object using abstract, often standardized symbols and lines. Schematic diagrams only depict the significant components of a system, though some details in the diagram may also be exaggerated or introduced to facilitate the understanding of the system.
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<p id="mntl-sc-block_1-0-2" class="comp mntl-sc-block mntl-sc-block-html">
Schematic diagrams do not include details that are not necessary for comprehending the information that the diagram was intended to convey. For example, in a schematic diagram depicting an electrical circuit, you can see how the wires and components are connected together, but not photographs of the circuit itself.
</p>
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<h3 id="mntl-sc-block-callout-heading_1-0" class="comp mntl-sc-block-callout-heading mntl-text-block">
Key Takeaways: Schematic Diagram</h3>
<div id="mntl-sc-block-callout-body_1-0" class="comp mntl-sc-block-callout-body mntl-text-block">
<ul><li>A <strong>schematic diagram</strong> is a picture that represents the components of a process, device, or other object using abstract, often standardized symbols and lines.</li><li>Although schematic diagrams are commonly associated with electrical circuits, many examples can be found in other industries.</li></ul></div>
</div>
<span class="heading-toc" id="toc-what-is-a-schematic-diagram"></span> <h2 id="mntl-sc-block_1-0-5" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> What Is a Schematic Diagram? </span> </h2>
<p id="mntl-sc-block_1-0-6" class="comp mntl-sc-block mntl-sc-block-html">
Schematic diagrams can also differ in their level of abstraction. Although they are typically composed of only abstract symbols and lines, some diagrams can also be <strong>semi-schematic</strong> and contain more realistic elements. Some diagrams can also contain words, such as when a process contains multiple elements that have not been standardized.
</p>
<div id="mntl-sc-block_1-0-7" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-8" class="comp mntl-sc-block mntl-sc-block-html">
More simply, a schematic diagram is a simplified drawing that uses symbols and lines to convey important information. For example, if you are taking the subway you may see a “map” showing you all the stations along a subway line, but that map will not show all the roads and buildings you may pass along the way. In this case, the entire subway system can be represented as differently colored lines depicting the different subway routes, with dots indicating the stops along the lines.
</p>
<div id="mntl-sc-block_1-0-9" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<figure id="mntl-sc-block_1-0-10" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-square figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/VCc4N2wOqV7w-sIOmoj6mWFNWNE=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/Yuri_Arcurssubwaymap-5c4df0fd46e0fb0001f21f5e.jpg" referrerpolicy="no-referrer">
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<span class="figure-article-caption-text">An example of a public transportation map, using differnetly colored lines to represent the diffrent lines and dots to represent the stations along each line. Yuri_Arcurs / Getty Images.</span>
</figcaption>
</figure>
<p id="mntl-sc-block_1-0-11" class="comp mntl-sc-block mntl-sc-block-html">
Though schematic diagrams are most commonly associated with electronics, you have probably encountered many such diagrams, like the subway example above, even if you have never had to wire a circuit. Here are some examples of the many schematic diagrams you may encounter in your work or in your studies.
</p>
<div id="mntl-sc-block_1-0-12" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<span class="heading-toc" id="toc-examples-of-schematic-diagrams"></span> <h2 id="mntl-sc-block_1-0-13" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Examples of Schematic Diagrams </span> </h2>
<h3 id="mntl-sc-block_1-0-14" class="comp mntl-sc-block reference-sc-block-subheading mntl-sc-block-subheading"> <span class="mntl-sc-block-subheading__text"> Schematic Diagrams in Electronics </span> </h3>
<figure id="mntl-sc-block_1-0-15" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/VW0mjtseCeHeEFgvd8jr0hcSoFQ=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/kr7ysztofwiringdiagram-5c4df148c9e77c0001f32271.jpg" referrerpolicy="no-referrer">
  </figure>
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<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
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<span class="figure-article-caption-text">A circuit diagram behind a circuit board. kr7ysztof / Getty Images.</span>
</figcaption>
</figure>
<p id="mntl-sc-block_1-0-16" class="comp mntl-sc-block mntl-sc-block-html">
Schematic diagrams are typically associated with electrical circuits. Also called <strong>wiring diagrams</strong> or <strong>circuit diagrams</strong>, these diagrams show how the different components of a circuit are connected. In these diagrams, lines represent connecting wires, while other elements like resistors, lamps, and switches are represented by standardized symbols called <strong>electrical schematic symbols</strong>.
</p>
<div id="mntl-sc-block_1-0-17" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-18" class="comp mntl-sc-block mntl-sc-block-html">
In electronics, having a schematic diagram on hand may help a user design an entire circuit before building it, or troubleshoot an electronic that has stopped working.
</p>
<div id="mntl-sc-block_1-0-19" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-20" class="comp mntl-sc-block mntl-sc-block-html">
Schematic diagrams may also be used to explain the general way that an electronic functions without detailing the hardware or software used in the actual electronic. For example, to explain how a computer projects the words you type on a screen, you might use a schematic diagram that shows how information passes from the keys you press to a word processing program, and finally to the computer screen.
</p>
<div id="mntl-sc-block_1-0-21" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<h3 id="mntl-sc-block_1-0-22" class="comp mntl-sc-block reference-sc-block-subheading mntl-sc-block-subheading"> <span class="mntl-sc-block-subheading__text"> Schematic Diagrams in Manufacturing </span> </h3>
<figure id="mntl-sc-block_1-0-23" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/0QLbQ9-l9FFL1OhzIGTGeNJMYUU=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/EakachaiLeesin-EyeEm-manufacturingschematic-5c4df19b46e0fb0001a8e7df.jpg" referrerpolicy="no-referrer">
  </figure>
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<span class="figure-article-caption-text">Schematic diagrams of various parts. Eakachai Leesin / EyeEm / Getty Images.</span>
</figcaption>
</figure>
<p id="mntl-sc-block_1-0-24" class="comp mntl-sc-block mntl-sc-block-html">
Schematic diagrams can be used to depict machines as well. For example, in a textbook a car engine may be depicted as a set of shapes that show how the different parts are positioned relative to one another. A <strong>schematic drawing</strong> may also be created by an engineer while they are designing the machine so that they can properly understand how the parts work together, and make any necessary changes before building the actual system.
</p>
<div id="mntl-sc-block_1-0-25" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<h3 id="mntl-sc-block_1-0-26" class="comp mntl-sc-block reference-sc-block-subheading mntl-sc-block-subheading"> <span class="mntl-sc-block-subheading__text"> Schematic Diagrams in Chemistry </span> </h3>
<figure id="mntl-sc-block_1-0-27" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/SgrFNorGm300D4On0HLEbk86usI=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/chemistry-blackboard-186738050-5c50896546e0fb00018decd2.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-3" class="comp mntl-figure-caption figure-article-caption">
<span class="figure-article-caption-owner">traffic_analyzer / Getty Images</span>
</figcaption>
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<p id="mntl-sc-block_1-0-28" class="comp mntl-sc-block mntl-sc-block-html">
Many chemical products are often obtained by performing several different reactions at different steps of the process. A schematic diagram in chemistry can help someone understand all the reactions that were performed to yield a final product, without showing the actual products themselves. This may be portrayed, for example, as a series of boxes connected together with arrows, with words depicting the various elements and conditions that were used throughout the process.
</p>
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<p id="mntl-sc-block_1-0-30" class="comp mntl-sc-block mntl-sc-block-html">
Like for machines, a schematic diagram may also be used to depict the apparatus that was used for performing the reactions, especially if it is not typically used for reactions, or had been modified from an instrument that was already known.
</p>
<div id="mntl-sc-block_1-0-31" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<h3 id="mntl-sc-block_1-0-32" class="comp mntl-sc-block reference-sc-block-subheading mntl-sc-block-subheading"> <span class="mntl-sc-block-subheading__text"> Schematic Diagrams in Business </span> </h3>
<figure id="mntl-sc-block_1-0-33" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/BilEEwA3hhB8083zp-U8WOOrM8g=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/SeanGladwellbusinessmodel-5c4df1f7c9e77c00014afaee.jpg" referrerpolicy="no-referrer">
  </figure>
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<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
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<span class="figure-article-caption-text">A flowchart that could be used to depict a business process. Sean Gladwell / Getty Images.</span>
</figcaption>
</figure>
<p id="mntl-sc-block_1-0-34" class="comp mntl-sc-block mntl-sc-block-html">
Schematic diagrams are helpful in conveying the main parts of a complicated business model and showing how they all relate together. For example, a marketing plan might consist of many different elements, such as strategy, objectives, and an action plan. A schematic diagram would then be used to help organize all those elements, including elements within each category, in ways that would convey the main ideas in a clear and concise fashion.
</p>
</div>
</div>
]]></description>
            <pubDate>Tue, 29 Jan 2019 17:18:26 GMT</pubDate>
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            <link>https://www.thoughtco.com/what-is-a-schematic-diagram-4584811</link>
            <author><![CDATA[Alane Lim]]></author>
                <category>Science</category>
                <category>Tech</category>
                <category>Math</category>
        </item>
        <item>
            <title><![CDATA[Why You May Have Heard Humans Only Use 10% of the Brain and Why That's Wrong]]></title>
            <description><![CDATA[<figure>
    <img src="https://www.thoughtco.com/thmb/NkUaOtX5HTe8Pg-VKTJp-xzXzFA=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/GettyImages-1034250008-83ed606ca5c845cda1d05f21a8972609.jpg" referrerpolicy="no-referrer">
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<p id="mntl-sc-block_1-0" class="comp mntl-sc-block mntl-sc-block-html">
You may have heard that humans only use 10 percent of their brain power, and that if you could unlock the rest of your brainpower, you could do so much more. You could become a super genius, or acquire psychic powers like mind reading and telekinesis. However, there is a powerful body of evidence debunking the 10 percent myth. Scientists have consistently shown that humans use their entire brain throughout each day.
</p>
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<p id="mntl-sc-block_1-0-2" class="comp mntl-sc-block mntl-sc-block-html">
Despite the evidence, the 10 percent myth has inspired many references in the cultural imagination. Films like "Limitless" and "Lucy" depict protagonists who develop godlike powers thanks to drugs that unleash the previously inaccessible 90 percent of the brain. A 2013 study showed that about 65 percent of Americans believe the trope, and a 1998 study showed that a full third of psychology majors, who focus on the workings of the brain, fell for it.
</p>
<div id="mntl-sc-block_1-0-3" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<span class="heading-toc" id="toc-neuropsychology"></span> <h2 id="mntl-sc-block_1-0-4" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Neuropsychology </span> </h2>
<p id="mntl-sc-block_1-0-5" class="comp mntl-sc-block mntl-sc-block-html">
Neuropsychology studies how the anatomy of the brain affects someone’s behavior, emotion, and cognition. Over the years, brain scientists have shown that different parts of the brain are responsible for <a href="https://www.thoughtco.com/anatomy-of-the-brain-373479" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">specific functions</a>, whether it’s <a href="https://www.thoughtco.com/occipital-lobes-anatomy-373224" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">recognizing colors</a> or <a href="https://www.thoughtco.com/frontal-lobes-anatomy-373213" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="3">problem solving</a>. Contrary to the 10 percent myth, scientists have proven that every part of the brain is integral for our daily functioning, thanks to brain imaging techniques like positron emission tomography and functional magnetic resonance imaging.
</p>
<div id="mntl-sc-block_1-0-6" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-7" class="comp mntl-sc-block mntl-sc-block-html">
Research has yet to find a brain area that is completely inactive. Even studies that measure activity at the level of single neurons have not revealed any <a href="http://www.bmj.com/content/335/7633/1288.full" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">inactive areas of the brain</a>.&nbsp;Many brain imaging studies that measure brain activity when a person is doing a specific task show how different parts of the brain work together. For example, while you are reading this text on your smartphone, some parts of your brain, including those responsible for vision, reading comprehension, and holding your phone, will be more active.
</p>
<div id="mntl-sc-block_1-0-8" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-9" class="comp mntl-sc-block mntl-sc-block-html">
However, some brain images <a href="https://www.theatlantic.com/technology/archive/2014/07/you-already-use-way-way-more-than-10-percent-of-your-brain/374520/" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">unintentionally support the 10 percent myth</a>,&nbsp;because they often show small bright splotches on an otherwise gray brain. This may imply that only the bright spots have brain activity, but that isn’t the case. Rather, colored splotches represent brain areas that are<em> <a href="http://www.newsobserver.com/news/technology/article23721076.html" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="2">more</a></em><a href="http://www.newsobserver.com/news/technology/article23721076.html" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="3"> active </a>when someone’s doing a task compared to when they’re not. The gray spots are still active, just to a lesser degree.
</p>
<div id="mntl-sc-block_1-0-10" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-11" class="comp mntl-sc-block mntl-sc-block-html">
A more direct counter to the 10 percent myth lies in individuals who have suffered brain damage–through a stroke, head trauma, or carbon monoxide poisoning–and what they can no longer do as a result of that damage, or can still do just as well. If the 10 percent myth were true, damage to perhaps 90 percent of the brain wouldn’t affect daily functioning.
</p>
<div id="mntl-sc-block_1-0-12" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-13" class="comp mntl-sc-block mntl-sc-block-html">
Yet studies show that damaging even a very small part of the brain may have devastating consequences. For example, damage to <a href="https://www.thoughtco.com/brocas-area-anatomy-373215" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Broca’s area</a> hinders proper formation of words and fluent speech, though general language comprehension remains intact. In one highly publicized case, a Florida woman permanently lost her “capacity for thoughts, perceptions, memories, and emotions that are the very essence of being human” when a lack of oxygen destroyed half of her <a href="https://www.thoughtco.com/anatomy-of-the-brain-cerebrum-373218" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">cerebrum</a>, which makes up about 85 percent of the brain.
</p>
<div id="mntl-sc-block_1-0-14" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<span class="heading-toc" id="toc-evolutionary-arguments"></span> <h2 id="mntl-sc-block_1-0-15" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Evolutionary Arguments </span> </h2>
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Another line of evidence against the 10 percent myth comes from evolution. The adult brain only constitutes 2 percent of body mass, yet it consumes over 20 percent of the body’s energy. In comparison, the adult brains of many vertebrate species–including some fish, reptiles, birds, and mammals–consume <a href="https://www.physiology.org/doi/abs/10.1152/ajpregu.1981.241.3.R203%29" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">2 to 8 percent of their body’s energy</a>. The brain has been shaped by millions of years of <a href="https://www.thoughtco.com/survival-of-the-fittest-1224578" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">natural selection</a>, which passes down favorable traits to increase likelihood of survival. It is unlikely that the body would dedicate so much of its energy to keep an entire brain functioning if it only uses 10 percent of the brain.
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<span class="heading-toc" id="toc-the-origin-of-the-myth"></span> <h2 id="mntl-sc-block_1-0-18" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Origin of the Myth </span> </h2>
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The main allure of the 10 percent myth is the idea that you could do so much more <em>if only</em> you could unlock the rest of your brain. Even with ample evidence suggesting the contrary, why do many people still believe that humans only use 10 percent of their brains? It’s unclear how the myth spread in the first place, but it has been popularized by self-help books, and may even also be grounded in older, flawed, neuroscience studies.
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The myth could be aligned with messages espoused by self-improvement books, which show you ways to do better and live up to your "potential."&nbsp;For example, the preface to the notorious "How to Win Friends and Influence People" says that the average person “develops only 10 percent of his latent mental ability.” This statement, which is traced back to psychologist William James, refers to a person’s potential to achieve more rather than how much brain matter they used. Others have even said that Einstein explained his brilliance using the 10 percent myth, though these claims remain unfounded.
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Another possible source of the myth lies in “silent” brain areas from older neuroscience research. In the 1930s, for instance, neurosurgeon Wilder Penfield hooked electrodes to the exposed brains of his epilepsy patients while operating on them. He noticed that particular brain areas triggered the experience various sensations, but while others seemed to <a href="http://www.sci-con.org/2004/09/exploding-the-10-percent-myth/" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1" rel="nofollow">cause no reaction</a>. Still, as technology evolved, researchers found that these “silent” brain areas, which included the <a href="https://misuse.ncbi.nlm.nih.gov/error/abuse.shtml" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="2">prefrontal lobes</a>, had major functions after all.
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<span class="heading-toc" id="toc-resources-and-further-reading"></span> <h2 id="mntl-sc-block_1-0-25" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Resources and Further Reading </span> </h2>
<ul id="mntl-sc-block_1-0-26" class="comp mntl-sc-block mntl-sc-block-html">
<li><cite>Beyerstein, B.L. “Whence Cometh the Myth That We Only Use 10% of Our Brains?” <em>Mind Myths: Exploring Popular Assumptions about the Mind and Brain</em>, edited by Sergio Della Sala, Wiley, 1999, pp. 3-24.</cite></li>
<li><cite>Broadfoot, Marla Vacek. “<a href="https://www.newsobserver.com/news/technology/article23721076.html" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">How Do Brain Scans Work?</a>” <em>Raleigh News &amp; Observer</em>, 27 Jan. 2013.</cite></li>
<li><cite>“<a href="https://www.sci-con.org/2004/09/exploding-the-10-percent-myth/" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="2" rel="nofollow">Exploding the 10 Percent Myth</a>.” <em>Science &amp; Consciousness Review</em>.</cite></li>
<li><cite>Higbee, Kenneth L., and Samuel L. Clay. “<a href="https://doi.org/10.1080/00223989809599280" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="3">College Students' Beliefs in the Ten-Percent Myth</a>.” <em>The Journal of Psychology</em>, vol. 132, no. 5, 1998, pp. 469-476.</cite></li>
<li><cite>Jarrett, Christian. <em>Great Myths of the Brain</em>. Wiley Blackwell, 2014.</cite></li>
<li><cite>McDougle, Sam. “<a href="https://www.theatlantic.com/technology/archive/2014/07/you-already-use-way-way-more-than-10-percent-of-your-brain/374520/" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="4">You Already Use Way, Way More Than 10 Percent of Your Brain</a>.” <em>The Atlantic</em>, 7 Aug. 2014.</cite></li>
<li><cite>Mink, J. W., et al. “<a href="https://doi.org/10.1152/ajpregu.1981.241.3.R203" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="5">Ratio of Central Nervous System to Body Metabolism in Vertebrates: Its Constancy and Functional Basis</a>.” <em>American Journal of Physiology-Regulatory, Integrative and Comparative Physiology</em>, vol. 241, no. 3, 1 Sept. 1981, pp. R203-R212.</cite></li>
<li><cite>“<a href="https://www.michaeljfox.org/publication/new-survey-finds-americans-care-about-brain-health-misperceptions-abound" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="6">New Survey Finds Americans Care about Brain Health, but Misperceptions Abound</a>.” <em>The Michael J. Fox Foundation for Parkinson's Research</em>, 25 Sept. 2013.</cite></li><span class="mntl-sc-block-adslot mntl-sc-block-adslot-inline"></span>
<li><cite>Tandon, Prakashnarain. “<a href="https://doi.org/10.4103/0028-3886.125242" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="7">Not so ‘Silent’: The Human Prefrontal Cortex</a>.” <em>Neurology India</em>, vol. 61, no. 6, 2013, pp. 578-580.</cite></li>
<li><cite>Vreeman, Rachel C, and Aaron E Carroll. “<a href="https://doi.org/10.1136/bmj.39420.420370.25" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="8">Medical Myths</a>.” <em>BMJ</em>, vol. 335, no. 7633, 20 Dec. 2007, pp. 1288-1289.</cite></li>
<li><cite>Wanjek, Christopher. <em>Bad Medicine: Misconceptions and Misuses Revealed, from Distance Healing to Vitamin O</em>. Wiley, 2003.</cite></li>
</ul>
</div>
</div>
]]></description>
            <pubDate>Thu, 22 Feb 2018 15:31:24 GMT</pubDate>
            <guid isPermaLink="false">https://www.thoughtco.com/percentage-of-human-brain-used-4159438</guid>
            <link>https://www.thoughtco.com/percentage-of-human-brain-used-4159438</link>
            <author><![CDATA[Alane Lim]]></author>
                <category>Science</category>
                <category>Tech</category>
                <category>Math</category>
        </item>
        <item>
            <title><![CDATA[Déjà Vu: The Science Behind the Eerie Feeling of Familiarity]]></title>
            <description><![CDATA[<figure>
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If you’ve ever had the feeling that a situation feels very familiar even though you know it shouldn’t feel familiar at all, like if you’re traveling in a city for the very first time, then you’ve probably experienced <strong>déjà vu.</strong> Déjà vu, which means “already seen” in French, combines <em>objective</em> unfamiliarity – that you know, based on ample evidence, that something shouldn’t be familiar – with <em>subjective</em> familiarity – that feeling that it’s familiar anyway.
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Déjà vu is common. According to a paper published in 2004, more than 50 surveys on déjà vu suggested that about two-thirds of individuals have experienced it at least once in their lifetime, with many reporting multiple experiences. This reported number also appears to be growing as people become more aware of what déjà vu is.
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Most often, déjà vu is described in terms of what you see, but it’s not specific to vision and even people who were born blind can experience it.
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<span class="heading-toc" id="toc-measuring-dj-vu"></span> <h2 id="mntl-sc-block_1-0-6" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Measuring Déjà Vu </span> </h2>
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Déjà vu is difficult to study in the laboratory because it is a fleeting experience, and also because there is no clearly identifiable trigger for it. Nevertheless, researchers have used several tools to study the phenomenon, based on the hypotheses they’ve put forward. Researchers may survey participants; study possibly related processes, especially those involved in memory; or design other experiments to probe déjà vu.
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Because déjà vu is hard to measure, researchers have postulated many explanations for how it works. Below are several of the more prominent hypotheses.
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<p id="mntl-sc-block_1-0-11" class="comp mntl-sc-block mntl-sc-block-html">
<strong>Memory Explanations</strong>
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Memory explanations of déjà vu are based on the idea that you have previously experienced a situation, or something very much like it, but you don’t <em>consciously</em> remember that you have. Instead, you remember it <em>unconsciously</em>, which is why it feels familiar even though you don’t know why.
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<p id="mntl-sc-block_1-0-15" class="comp mntl-sc-block mntl-sc-block-html">
<em>Single element familiarity</em>
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<p id="mntl-sc-block_1-0-17" class="comp mntl-sc-block mntl-sc-block-html">
The single element familiarity hypothesis suggests you experience déjà vu if one element of the scene is familiar to you but you don’t consciously recognize it because it’s in a different setting, like if you see your barber out on the street.
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<p id="mntl-sc-block_1-0-19" class="comp mntl-sc-block mntl-sc-block-html">
Your brain still finds your barber familiar even if you don’t recognize them, and generalizes that feeling of familiarity to the entire scene. Other researchers have extended this hypothesis to multiple elements as well.
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<p id="mntl-sc-block_1-0-21" class="comp mntl-sc-block mntl-sc-block-html">
<em>Gestalt familiarity</em>
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The gestalt familiarity hypothesis focuses on how items are organized in a scene and how déjà vu occurs when you experience something with a similar layout. For example, you may not have seen your friend’s painting in their living room before, but maybe you’ve seen a room that’s laid out like your friend’s living room – a painting hanging over the sofa, across from a bookcase. Since you can’t recall the other room, you experience déjà vu.
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One advantage to the gestalt similarity hypothesis is that it can be more directly tested. In <a href="https://misuse.ncbi.nlm.nih.gov/error/abuse.shtml" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">one study</a>, participants looked at rooms in virtual reality, then were asked how familiar a new room was and whether they felt they were experiencing déjà vu.
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The researchers found that study participants who couldn’t recall the old rooms tended to think a new room was familiar, and that they were experiencing déjà vu, if the new room resembled old ones. Furthermore, the more similar the new room was to an old room, the higher these ratings were.
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<p id="mntl-sc-block_1-0-29" class="comp mntl-sc-block mntl-sc-block-html">
<strong>Neurological Explanations</strong>
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<p id="mntl-sc-block_1-0-31" class="comp mntl-sc-block mntl-sc-block-html">
<em>Spontaneous brain activity</em>
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<p id="mntl-sc-block_1-0-33" class="comp mntl-sc-block mntl-sc-block-html">
Some explanations posit that&nbsp;déjà vu is experienced when there is spontaneous brain activity unrelated to what you’re currently experiencing. When that happens in the part of your brain dealing with memory, you can have a false feeling of familiarity.
</p>
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Some evidence comes from individuals with <a href="https://www.thoughtco.com/temporal-lobes-anatomy-373228" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">temporal lobe </a>epilepsy, when abnormal electrical activity occurs in the part of the brain dealing with memory. When the brains of these patients are electrically stimulated as part of a pre-surgery evaluation, they may experience déjà vu.
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One <a href="https://misuse.ncbi.nlm.nih.gov/error/abuse.shtml" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">researcher</a>&nbsp;suggests that you experience déjà vu when the <a href="https://www.thoughtco.com/hippocampus-anatomy-373221" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">parahippocampal system</a>, which helps identify something as familiar, randomly misfires and makes you think something is familiar when it shouldn’t.&nbsp;
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<p id="mntl-sc-block_1-0-39" class="comp mntl-sc-block mntl-sc-block-html">
<a href="https://www.clinph-journal.com/article/S1388-2457%2811%2900561-X/fulltext" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1" rel="nofollow">Others have said</a> that déjà vu can’t be isolated to a single familiarity system, but rather involves multiple structures involved in memory and the connections between them.
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<p id="mntl-sc-block_1-0-41" class="comp mntl-sc-block mntl-sc-block-html">
<em>Neural transmission speed</em>
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Other hypotheses are based on how fast information travels through your brain. Different areas of your brain transmit information to “higher order” areas that combine the information together to help you make sense of the world. If this complex process is disrupted in any way – perhaps one part sends something more slowly or more quickly than it usually does – then your brain interprets your surroundings incorrectly.
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<span class="heading-toc" id="toc-which-explanation-is-correct"></span> <h2 id="mntl-sc-block_1-0-45" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Which Explanation is Correct? </span> </h2>
<p id="mntl-sc-block_1-0-46" class="comp mntl-sc-block mntl-sc-block-html">
An explanation for déjà vu remains elusive, though the hypotheses above appear to have one common thread: a temporary error in cognitive processing. For now, scientists can continue to design experiments that more directly probe the nature of déjà vu, to be more certain of the correct explanation.
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<span class="heading-toc" id="toc-sources"></span> <h2 id="mntl-sc-block_1-0-48" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Sources </span> </h2>
<ul id="mntl-sc-block_1-0-49" class="comp mntl-sc-block mntl-sc-block-html">
<li><cite>Tip-of-the-tongue states and related phenomena. Ed. Bennett L. Schwartz and Alan S. Brown. Cambridge University Press. New York, NY 2014. http://www.cambridge.org/gb/academic/subjects/psychology/biological-psychology/tip-tongue-states-and-related-phenomena?format=HB</cite></li>
<li><cite>C. Moulin. The cognitive neuropsychology of déjà vu. Part of the Essays in Cognitive Psychology series. Psychology Press. New York, NY 2018.<a href="https://www.routledge.com/The-Cognitive-Neuropsychology-of-Deja-Vu/Moulin/p/book/9781138696266" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1"> https://www.routledge.com/The-Cognitive-Neuropsychology-of-Deja-Vu/Moulin/p/book/9781138696266</a></cite></li>
<li><cite>Bartolomei, F., Barbeau, E., Gavaret, M., Guye, M., McGonigal, A., Régis, J., and P. Chauvel. “<a href="http://n.neurology.org/content/63/5/858.short" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="2">Cortical stimulation study of the role of rhinal cortex in déjà vu and reminiscence of memories.</a>” <em>Neurology</em>, vol. 63, no. 5, Sept. 2004, pp. 858-864, </cite><cite>doi</cite><cite>:10.1212/01.wnl.0000137037.56916.3f.</cite></li>
<li><cite>J. Spatt. “<a href="https://misuse.ncbi.nlm.nih.gov/error/abuse.shtml" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="3">Déjà vu: possible parahippocampal mechanisms.</a>” <em>The Journal of Neuropsychiatry &amp; Clinical Neurosciences</em>, vol. 14, no. 1, 2002, pp. 6-10, </cite><cite>doi</cite><cite>:10.1176/jnp.14.1.6.</cite></li>
<li><cite>Cleary, A. M., Brown, A. S., Sawyer, B.D., Nomi, J.S., Ajoku, A.C., and A. J. Ryals. “<a href="https://misuse.ncbi.nlm.nih.gov/error/abuse.shtml" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="4">Familiarity from the configuration of objects in 3-dimensional space and its relation to déjà vu: a virtual reality investigation.</a>” <em>Consciousness and Cognition</em>, vol. 21, no. 2, 2012, pp. 969-975, </cite><cite>doi</cite><cite>:10.1016/j.concog.2011.12.010.</cite></li><span class="mntl-sc-block-adslot mntl-sc-block-adslot-inline"></span>
<li><cite>A. S. Brown. The déjà vu experience. Part of the Essays in Cognitive Psychology series. Psychology Press. New York, NY 2004.<a href="https://www.routledge.com/The-Deja-Vu-Experience/Brown/p/book/9780203485446" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="5"> https://www.routledge.com/The-Deja-Vu-Experience/Brown/p/book/9780203485446</a></cite></li>
<li><cite>A. S. Brown. “<a href="https://misuse.ncbi.nlm.nih.gov/error/abuse.shtml" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="6">A review of the déjà vu experience.</a>” <em>Psychology Bulletin</em>, vol. 129, no. 3, 2003, pp. 394-413. </cite><cite>doi</cite><cite>:10.1037/0033-2909.129.3.394.</cite></li>
<li><cite>Bartolomei, F., Barbeau, E. J., Nguyen, T., McGonigal, A., Régis, J., Chauvel, P., and F. Wendling. “<a href="http://www.clinph-journal.com/article/S1388-2457%2811%2900561-X/abstract" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="7" rel="nofollow">Rhinal-hippocampal interactions during déjà vu.</a>” <em>Clinical Neurophysiology</em>, vol. 123, no. 3, March 2012, pp. 489-495. </cite><cite>doi</cite><cite>:10.1016/j.clinph.2011.08.012</cite></li>
</ul>
</div>
</div>
]]></description>
            <pubDate>Thu, 22 Feb 2018 15:31:01 GMT</pubDate>
            <guid isPermaLink="false">https://www.thoughtco.com/causes-of-deja-vu-4159448</guid>
            <link>https://www.thoughtco.com/causes-of-deja-vu-4159448</link>
            <author><![CDATA[Alane Lim]]></author>
                <category>Science</category>
                <category>Tech</category>
                <category>Math</category>
        </item>
        <item>
            <title><![CDATA[7 Extinction Level Events That Could End Life as We Know It]]></title>
            <description><![CDATA[<figure>
    <img src="https://www.thoughtco.com/thmb/v9MJhtNFw_KTAyDdJ8FtqewdvzI=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/GettyImages-724233197-5a834dfc8023b90037be80d3.jpg" referrerpolicy="no-referrer">
  </figure>
<div id="list-sc_1-0" class="comp article-content list list-sc mntl-block">
<div id="article-intro_1-0" class="comp article-content__description article-intro mntl-block">
<p id="mntl-sc-block_1-0" class="comp mntl-sc-block mntl-sc-block-html">
If you've watched the movies "2012" or "Armageddon" or read "On the Beach," you know about some of the threats that could end life as we know it. The <a href="https://www.thoughtco.com/things-you-should-know-about-the-sun-3073449" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Sun</a> could do something nasty. A <a href="https://www.thoughtco.com/astronomy-basics-4133556" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">meteor</a> could strike. We could nuke ourselves out of existence. These are only a few well-known extinction level events. There are so many more ways to die!
</p>
<p id="mntl-sc-block_1-0-1" class="comp mntl-sc-block mntl-sc-block-html">
But first, what exactly is an extinction event? An <strong>extinction level event</strong> or ELE is a catastrophe resulting in the extinction of the majority of species on the planet. It's not the normal extinction of species that occurs every day. It isn't necessarily the sterilization of all living organisms. We can identify major extinction events by examining the deposition and chemical composition of rocks, the <a href="https://www.thoughtco.com/famous-fossil-discoveries-1092049" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">fossil record</a>, and evidence of major events on moons and other planets.
</p>
<p id="mntl-sc-block_1-0-2" class="comp mntl-sc-block mntl-sc-block-html">
There are dozens of phenomena capable of causing widespread extinctions, but they can be grouped into a few categories:
</p>
</div>
<div id="list-sc__content_1-0" class="comp list-sc__content mntl-sc-page mntl-block" data-sc-sticky-offset="85" data-sc-ad-label-height="24" data-sc-ad-track-spacing="100" data-sc-min-track-height="250" data-sc-max-track-height="600" data-sc-breakpoint="50em" data-sc-load-immediate="3" data-sc-content-positions="[1, 1050, 1250, 1550, 1950, 2350, 2750, 3150]" data-bind-scroll-on-start="true">
<div id="list-sc-item_1-0" class="comp mntl-sc-list-item list-sc-item mntl-block">
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<div class="content-list-number">
<div class="item-number">01</div>
<div class="total">of 09</div>
</div>
</div>
<span class="heading-toc" id="toc-the-sun-will-kill-us"></span> <h2 id="mntl-sc-block_2-0" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Sun Will Kill Us </span> </h2>
<figure id="mntl-sc-block_2-0-1" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
</svg>
<span class="figure-article-caption-text">If a strong solar flare hit the Earth, the results could be devastating.</span>
<span class="figure-article-caption-owner">VICTOR HABBICK VISIONS, Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-2" class="comp mntl-sc-block mntl-sc-block-html">
Life as we know it wouldn't exist without the Sun, but let's be honest. The Sun has it out for planet Earth. Even if none of the other catastrophes on this list ever happen, the Sun will end us. Stars like the Sun burn brighter over time as they fuse hydrogen into helium. In another billion years, it will be about 10 percent brighter. While this might not seem significant, it will cause more water to evaporate. <a href="https://www.thoughtco.com/worst-greenhouse-gases-606789" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Water is&nbsp;a greenhouse gas</a>, so it traps heat in the <a href="https://www.thoughtco.com/most-abundant-gases-in-earths-atmosphere-607594" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">atmosphere</a>, leading to more evaporation. Sunlight will break water into hydrogen and oxygen, so it can <a href="https://www.thoughtco.com/if-earths-atmosphere-vanished-607906" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="3">bleed away into space</a>. Should any life survive, it will meet a fiery fate when the Sun enters its <a href="https://www.thoughtco.com/what-are-stars-really-3073631" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="4">red giant</a> phase, expanding out to the orbit of Mars. It's not likely any life will survive <em>inside</em> the Sun.
</p>
<p id="mntl-sc-block_2-0-3" class="comp mntl-sc-block mntl-sc-block-html">
But, the Sun can kill us any old day it wants <a href="https://www.thoughtco.com/solar-flares-4137226" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">via a coronal mass ejection</a> (CME). As you can guess from the name, this is when our favorite star expels charged particles outward from its corona. Since a CME can sent matter any direction, it doesn't usually shoot directly toward Earth. Sometimes only a tiny fraction of particles reach us, granting us an aurora or a solar storm. However, it's possible for a CME to barbecue the planet.
</p>
<p id="mntl-sc-block_2-0-4" class="comp mntl-sc-block mntl-sc-block-html">
The Sun has pals (and they hate Earth too). A nearby (within 6000 light years) <a href="https://www.thoughtco.com/supernovae-deaths-of-massive-stars-3073301" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">supernova</a>, nova, or gamma ray burst could irradiate organisms and destroy the ozone layer, leaving life&nbsp;at the mercy of the Sun's <a href="https://www.thoughtco.com/definition-of-ultraviolet-radiation-604675" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">ultraviolet radiation</a>. Scientists think a <a href="https://www.thoughtco.com/gamma-ray-burst-destroy-life-earth-3072521" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="3">gamma burst</a> or supernova might have led to the End-Ordovician extinction.
</p>
</div>
<div id="list-sc-item_1-0-1" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-1" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">02</div>
<div class="total">of 09</div>
</div>
</div>
<span class="heading-toc" id="toc-geomagnetic-reversals-may-kill-us"></span> <h2 id="mntl-sc-block_2-0-5" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Geomagnetic Reversals May Kill Us </span> </h2>
<figure id="mntl-sc-block_2-0-6" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/7CfPJl1EDfXgckaZvnvIqPRDuDA=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/GettyImages-693943194-5a84493fd8fdd500375c2f46.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-1" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
</svg>
<span class="figure-article-caption-text">Scientists believe magnetic pole reversals were involved in some past mass extinctions.</span>
<span class="figure-article-caption-owner">siiixth, Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-7" class="comp mntl-sc-block mntl-sc-block-html">
The Earth is a giant magnet that has a love-hate relationship with life. The magnetic field protects us from the worst the Sun throws at us. Every so often, the positions of the <a href="https://www.thoughtco.com/what-is-magnetic-reversal-1435340" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">north and south magnetic poles flip</a>. How often the reversals occur and how long it takes the magnetic field to get settled&nbsp;is highly variable. Scientists aren't completely sure what will happen when the poles flip. Maybe nothing. Or maybe the weakened magnetic field will expose the Earth to the <a href="https://www.thoughtco.com/the-effects-of-solar-storms-3073703" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">solar wind</a>, letting the Sun steal a lot of our oxygen. You know, that gas humans breathe. Scientists say magnetic field reversals aren't always extinction level events. Just sometimes.
</p>
</div>
<div id="list-sc-item_1-0-2" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-2" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">03</div>
<div class="total">of 09</div>
</div>
</div>
<span class="heading-toc" id="toc-the-big-bad-meteor"></span> <h2 id="mntl-sc-block_2-0-8" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Big Bad Meteor </span> </h2>
<figure id="mntl-sc-block_2-0-9" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/fRhKUp4HvIhoOQA3RRt9eSsNA3w=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/GettyImages-532101243-5a844ef13418c6003624983e.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-2" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
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<span class="figure-article-caption-text">A big meteor impact could be an extinction level event.</span>
<span class="figure-article-caption-owner">Marc Ward/Stocktrek Images, Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-10" class="comp mntl-sc-block mntl-sc-block-html">
You may be surprised to learn the impact of an asteroid or meteor has only been connected with certainty to one mass extinction, the Cretaceous-Paleogene extinction event. Other impacts have been contributing factors to extinctions, but not the primary cause.
</p>
<p id="mntl-sc-block_2-0-11" class="comp mntl-sc-block mntl-sc-block-html">
The good news is that <a href="https://nypost.com/2016/12/14/nasa-scientist-warns-earth-is-due-for-extinction-level-event/" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">NASA claims</a> about 95 percent of comets and asteroids bigger than 1 kilometer in diameter have been identified. The other good news is that scientists estimate an object needs to be about 100 kilometers (60 miles) across to wipe out all life. The bad news is there are another 5 percent out there and not much we can do about a significant threat with our present technology (no, Bruce Willis cannot detonate a nuke and save us).
</p>
<p id="mntl-sc-block_2-0-12" class="comp mntl-sc-block mntl-sc-block-html">
Obviously, living things at ground zero for a meteor strike will die. Many more will die from the shock wave, earthquakes, tsunamis, and firestorms. Those that survive the initial impact would have a hard time finding food, as the debris thrown into the atmosphere would change the climate, leading to mass extinctions. You're probably better off at ground zero for this one.
</p>
</div>
<div id="list-sc-item_1-0-3" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-3" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">04</div>
<div class="total">of 09</div>
</div>
</div>
<span class="heading-toc" id="toc-the-sea"></span> <h2 id="mntl-sc-block_2-0-13" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Sea </span> </h2>
<figure id="mntl-sc-block_2-0-14" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/v_x1oZvDSXc0FtAdEggw2a_eenM=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/GettyImages-128139136-5a8467ef3418c6003645ddc5.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-3" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
</svg>
<span class="figure-article-caption-text">A tsunami is dangerous, but the sea has more lethal tricks.</span>
<span class="figure-article-caption-owner">Bill Romerhaus, Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-15" class="comp mntl-sc-block mntl-sc-block-html">
A day at the beach might seem idyllic, until you realize the blue part of the marble we call Earth is deadlier than all of the sharks in its depths. The ocean has various ways of causing ELEs.
</p>
<p id="mntl-sc-block_2-0-16" class="comp mntl-sc-block mntl-sc-block-html">
Methane clathrates (molecules made of water and methane) sometimes break from the continental shelves, producing a methane eruption called a clathrate gun. The "gun" shoots immense amounts of the greenhouse gas methane into the atmosphere. Such events are linked to the <a href="https://www.thoughtco.com/permian-period-300-250-million-years-1091430" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">end-Permian extinction</a> and Paleocene-Eocene Thermal Maximum.
</p>
<p id="mntl-sc-block_2-0-17" class="comp mntl-sc-block mntl-sc-block-html">
Prolonged sea level rise or fall also leads to extinctions. Falling sea levels are more insidious, as exposing the continental shelf kills off innumerable marine species. This, in turn, upsets the terrestrial ecosystem, leading to an ELE.
</p>
<p id="mntl-sc-block_2-0-18" class="comp mntl-sc-block mntl-sc-block-html">
Chemical imbalances in the sea also cause extinction events. When the middle or upper layers of the ocean become anoxic, a chain reaction of death occurs. The Ordovician-Silurian, late Devonian, Permian-Triassic, and Triassic-Jurassic extinctions all included anoxic events.
</p>
<p id="mntl-sc-block_2-0-19" class="comp mntl-sc-block mntl-sc-block-html">
Sometimes the levels of essential trace elements (e.g., <a href="https://www.thoughtco.com/selenium-facts-606594" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">selenium</a>) fall, leading to mass extinctions. Sometimes the sulfate-reducing bacteria in thermal vents get out of control, releasing an excess of hydrogen sulfide that weakens the ozone layer, exposing life to lethal UV. The ocean also undergoes a periodic overturn in which the high-salinity surface water sinks to the depths. Anoxic deep water rises, killing surface organisms. The late-Devonian and <a href="https://www.thoughtco.com/the-permian-triassic-extinction-event-1092136" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">Permian-Triassic extinctions</a> are associated oceanic overturn.
</p>
<p id="mntl-sc-block_2-0-20" class="comp mntl-sc-block mntl-sc-block-html">
The beach doesn't look so nice now, does it?
</p>
</div>
<div id="list-sc-item_1-0-4" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-4" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">05</div>
<div class="total">of 09</div>
</div>
</div>
<span class="heading-toc" id="toc-and-the-winner-is-volcanoes"></span> <h2 id="mntl-sc-block_2-0-21" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> And the "Winner" Is... Volcanoes </span> </h2>
<figure id="mntl-sc-block_2-0-22" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/B6IEDBoM4dSm5JDu9JNS8j8L2xQ=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/GettyImages-675488626-5a84688e312834003700b60f.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-4" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
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<span class="figure-article-caption-text">Historically, most extinction level events have been caused by volcanoes.</span>
<span class="figure-article-caption-owner">Mike Lyvers, Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-23" class="comp mntl-sc-block mntl-sc-block-html">
While falling sea level has been associated with 12 extinction events, only seven involved a significant loss of species. On the other hand, <a href="https://www.thoughtco.com/what-happens-when-a-volcano-erupts-4151722" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">volcanoes</a> have led to 11 ELEs, <em>all</em> of them significant. The End-Permian, End-Triassic, and <a href="https://www.thoughtco.com/the-k-t-extinction-event-1092141" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">End-Cretaceous extinctions</a> are&nbsp;associated with volcanic eruptions called flood basalt events. Volcanoes kill by releasing dust, sulfur oxides, and <a href="https://www.thoughtco.com/carbon-dioxide-molecular-formula-608475" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="3">carbon dioxide</a> that collapse food chains by inhibiting photosynthesis, poison the land and sea with acid rain, and produce global warming. The 

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Mathematics is called the language of science. Italian astronomer and physicist <a href="https://www.thoughtco.com/galileo-galilei-biography-1991864" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Galileo Galilei</a> is attributed with the quote, "<em>Mathematics is the language in which God has written the universe</em>." Most likely this quote is a summary of his statement in&nbsp;<em>Opere Il Saggiatore:</em>
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<blockquote id="mntl-sc-block_1-0-1" class="comp mntl-sc-block mntl-sc-block-html">
[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.
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<p id="mntl-sc-block_1-0-2" class="comp mntl-sc-block mntl-sc-block-html">
Yet, is mathematics truly a language, like English or Chinese? To answer the question, it helps to know what language is and how the vocabulary and grammar of mathematics are used to construct sentences.
</p>
<div id="mntl-sc-block_1-0-3" class="comp theme-keytakeaways mntl-sc-block reference-sc-block-callout mntl-sc-block-callout mntl-block" data-tracking-id="mntl-sc-block-callout" data-tracking-container="true">
<h3 id="mntl-sc-block-callout-heading_1-0" class="comp mntl-sc-block-callout-heading mntl-text-block">
Key Takeaways: Why Math is a Language</h3>
<div id="mntl-sc-block-callout-body_1-0" class="comp mntl-sc-block-callout-body mntl-text-block">
<ul><li>In order to be considered a language, a system of communication must have vocabulary, grammar, syntax, and people who use and understand it.</li><li>Mathematics meets this definition of a language. Linguists who don't consider math a language cite its use as a written rather than spoken form of communication.</li><li>Math is a universal language. The symbols and organization to form equations are the same in every country of the world.</li></ul></div>
</div>
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<span class="heading-toc" id="toc-what-is-a-language"></span> <h2 id="mntl-sc-block_2-0" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> What Is a Language? </span> </h2>
<p id="mntl-sc-block_2-0-1" class="comp mntl-sc-block mntl-sc-block-html">
There are multiple definitions of "<a href="https://www.thoughtco.com/what-is-a-language-1691218" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">language</a>." A language may be a system of words or codes used within a discipline. Language may refer to a system of communication using symbols or sounds. <a href="https://www.thoughtco.com/what-is-a-linguist-1691239" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">Linguist</a> Noam Chomsky defined language as a set of sentences constructed using a finite set of elements. Some linguists believe language should be able to represent events and abstract concepts.
</p>
<p id="mntl-sc-block_2-0-2" class="comp mntl-sc-block mntl-sc-block-html">
Whichever definition is used, a language contains the following components:
</p>
<ul id="mntl-sc-block_2-0-3" class="comp mntl-sc-block mntl-sc-block-html">
<li>There must be a <strong>vocabulary</strong> of words or symbols.</li><li><strong>Meaning</strong> must be attached to the words or symbols.</li><li>A language employs <strong>grammar</strong>, which is a set of rules that outline how vocabulary is used.</li><li>A <strong>syntax</strong> organizes symbols into linear structures or propositions.</li><li>A <strong>narrative</strong> or discourse consists of strings of syntactic propositions.</li><li>There must be (or have been) a group of people who use and understand the symbols.</li>
</ul>
<p id="mntl-sc-block_2-0-4" class="comp mntl-sc-block mntl-sc-block-html">
Mathematics meets all of these requirements. The symbols, their meanings, syntax, and grammar are the same throughout the world. Mathematicians, scientists, and others use math to communicate concepts. Mathematics describes itself (a field called meta-mathematics), real-world phenomena, and abstract concepts.
</p>
</div>
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<span class="heading-toc" id="toc-vocabulary-grammar-and-syntax-in-mathematics"></span> <h2 id="mntl-sc-block_2-0-5" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Vocabulary, Grammar, and Syntax in Mathematics </span> </h2>
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    <img src="https://www.thoughtco.com/thmb/N2hMLp5mKoTWbfT8wpxt1VZ25eM=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/student-writing-on-blackboard-695556138-5a6e40fdeb97de0037e8a0c6.jpg" referrerpolicy="no-referrer">
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<span class="figure-article-caption-owner">Emilija Manevska / Getty Images</span>
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The vocabulary of math draws from many different alphabets and includes symbols unique to math. A mathematical equation may be stated in words to form a sentence that has a noun and a verb, just like a sentence in a spoken language. For example:
</p>
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3 + 5 = 8
</p>
<p id="mntl-sc-block_2-0-9" class="comp mntl-sc-block mntl-sc-block-html">
could be stated as "Three added to five equals eight."
</p>
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Breaking this down, <a href="https://www.thoughtco.com/what-is-concrete-noun-1689904" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">nouns</a> in math include:
</p>
<ul id="mntl-sc-block_2-0-11" class="comp mntl-sc-block mntl-sc-block-html">
<li>Arabic numerals (0, 5, 123.7)</li><li>Fractions (1⁄4, 5⁄9, 2 1⁄3)</li><li>Variables (a, b, c, x, y, z)</li><li>Expressions (3x, x<sup>2</sup>, 4 + x)</li><li>Diagrams or visual elements (circle, angle, triangle, tensor, matrix)</li><li>Infinity (∞)</li><li>Pi (π)</li><li>Imaginary numbers (i, -i)</li><li>The speed of light (c)</li>
</ul>
<p id="mntl-sc-block_2-0-12" class="comp mntl-sc-block mntl-sc-block-html">
Verbs include symbols including:
</p>
<ul id="mntl-sc-block_2-0-13" class="comp mntl-sc-block mntl-sc-block-html">
<li>Equalities or inequalities (=, &lt;, &gt;)</li><li>Actions such as addition, subtraction, multiplication, and division (+, -, x or *, ÷ or /)</li><li>Other operations (sin, cos, tan, sec)</li>
</ul>
<p id="mntl-sc-block_2-0-14" class="comp mntl-sc-block mntl-sc-block-html">
If you try to perform a sentence diagram on a mathematical sentence, you'll find infinitives, conjunctions, adjectives, etc. As in other languages, the role played by a symbol depends on its context.
</p>
<span class="heading-toc" id="toc-international-rules"></span> <h2 id="mntl-sc-block_2-0-15" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> International Rules </span> </h2>
<p id="mntl-sc-block_2-0-16" class="comp mntl-sc-block mntl-sc-block-html">
Mathematics grammar and syntax, like vocabulary, are international. No matter what country you're from or what language you speak, the structure of the mathematical language is the same.
</p>
<ul id="mntl-sc-block_2-0-17" class="comp mntl-sc-block mntl-sc-block-html">
<li>Formulas are read from left to right.</li>
<li>The Latin alphabet is used for parameters and variables. To some extent, the Greek alphabet is also used. <a href="https://www.thoughtco.com/understanding-classification-of-numbers-2312407" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Integers</a> are usually drawn from <em>i</em>, <em>j</em>, <em>k</em>, <em>l</em>, <em>m</em>, <em>n</em>. <a href="https://www.thoughtco.com/what-is-a-real-number-3126307" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">Real numbers</a> are represented by&nbsp;<em>a</em>,&nbsp;<em>b</em>,&nbsp;<em>c</em>, α<em>, β</em>, γ. Complex numbers are indicated by <em>w</em> and <em>z</em>. Unknowns are <em>x</em>, <em>y</em>, <em>z</em>. Names of functions are usually <em>f</em>, <em>g</em>, <em>h</em>.</li>
<li>The Greek alphabet is used to represent specific concepts. For example, λ is used to indicate wavelength and ρ means density.</li>
<li>Parentheses and brackets indicate the <a href="https://www.thoughtco.com/definition-of-bedmas-2312372" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="3">order in which the symbols interact</a>.</li>
<li>The way functions, integrals, and derivatives are phrased is uniform.</li>
</ul>
</div>
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<span class="heading-toc" id="toc-language-as-a-teaching-tool"></span> <h2 id="mntl-sc-block_2-0-18" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Language as a Teaching Tool </span> </h2>
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    <img src="https://www.thoughtco.com/thmb/3FBkjxj7Vj_djIbgjhs5Asro2ho=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/question-marks-on-black-background-184837701-5a6df53904d1cf00378895cb.jpg" referrerpolicy="no-referrer">
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<span class="figure-article-caption-owner">StockFinland / Getty Images</span>
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Understanding how mathematical sentences work is helpful when teaching or learning math. Students often find numbers and symbols intimidating, so putting an equation into a familiar language makes the subject more approachable. Basically, it's like translating a foreign language into a known one.
</p>
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While students typically dislike word problems, extracting the nouns, verbs, and modifiers from a spoken/written language and translating them into a mathematical equation is a valuable skill to have. Word problems improve comprehension and increase problem-solving skills.
</p>
<p id="mntl-sc-block_2-0-22" class="comp mntl-sc-block mntl-sc-block-html">
Because mathematics is the same all over the world, math can act as a universal language. A phrase or formula has the same meaning, regardless of another language that accompanies it. In this way, math helps people learn and communicate, even if other communication barriers exist.
</p>
</div>
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<span class="heading-toc" id="toc-the-argument-against-math-as-a-language"></span> <h2 id="mntl-sc-block_2-0-23" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Argument Against Math as a Language </span> </h2>
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  </figure>
<figcaption id="mntl-figure-caption_1-0-2" class="comp mntl-figure-caption figure-article-caption">
<span class="figure-article-caption-owner">Anne Helmenstine</span>
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</figure>
<p id="mntl-sc-block_2-0-25" class="comp mntl-sc-block mntl-sc-block-html">
Not everyone agrees that mathematics is a language. Some definitions of "language" describe it as a spoken form of communication. Mathematics is a written form of communication. While it may be easy to read a simple addition statement aloud (e.g., 1 + 1 = 2), it's much harder to read other equations aloud (e.g., Maxwell's equations). Also, the spoken statements would be rendered in the speaker's native language, not a universal tongue.
</p>
<p id="mntl-sc-block_2-0-26" class="comp mntl-sc-block mntl-sc-block-html">
However, sign language would also be disqualified based on this criterion. Most linguists accept sign language as a true language.&nbsp;There are a handful of dead languages that no one alive knows how to pronounce or even read anymore.
</p>
<p id="mntl-sc-block_2-0-27" class="comp mntl-sc-block mntl-sc-block-html">
A strong case for mathematics as a language is that modern elementary-high school curricula uses techniques from language education for teaching mathematics. Educational psychologist Paul Riccomini and colleagues wrote that students learning mathematics require "a robust vocabulary knowledge base; flexibility; fluency and proficiency with numbers, symbols, words, and diagrams; and comprehension skills."
</p>
<span class="heading-toc" id="toc-sources"></span> <h2 id="mntl-sc-block_2-0-28" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Sources </span> </h2>
<ul id="mntl-sc-block_2-0-29" class="comp mntl-sc-block mntl-sc-block-html">
<li><cite>Ford, Alan, and F. David Peat. "<a href="https://doi.org/10.1007/BF01889434" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">The Role of Language in Science</a>." <em>Foundations of Physics</em> 18.12 (1988): 1233–42.&nbsp;</cite></li>
<li><cite>Galilei, Galileo. "'The Assayer' ('Il Saggiatore' in Italian) (Rome, 1623)." <em>The Controversy on the Comets of 1618</em>. Eds. Drake, Stillman and C. D. O'Malley. Philadelphia: University of Pennsylvania Press, 1960.&nbsp;</cite></li>
<li><cite>Klima, Edward S., and Ursula Bellugi. "The Signs of Language. "Cambridge, MA: Harvard University Press, 1979.&nbsp;</cite></li>
<li><cite>Riccomini, Paul J., et al. "<a href="https://doi.org/10.1080/10573569.2015.1030995" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="2">The Language of Mathematics: The Importance of Teaching and Learning Mathematical Vocabulary</a>." <em>Reading &amp; Writing Quarterly</em> 31.3 (2015): 235-52. Print.</cite></li>
</ul>
</div>
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]]></description>
            <pubDate>Mon, 29 Jan 2018 19:48:57 GMT</pubDate>
            <guid isPermaLink="false">https://www.thoughtco.com/why-mathematics-is-a-language-4158142</guid>
            <link>https://www.thoughtco.com/why-mathematics-is-a-language-4158142</link>
            <author><![CDATA[Anne Marie Helmenstine, Ph.D.]]></author>
                <category>Science</category>
                <category>Tech</category>
                <category>Math</category>
        </item>
        <item>
            <title><![CDATA[Here's How Equivalent Systems of Equations Work]]></title>
            <description><![CDATA[<figure>
    <img src="https://www.thoughtco.com/thmb/xr1EIfRZtWAHfml2WyLPPa0MWrI=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/GettyImages-595348759-5372ddd37cf44fdb957bdb81028ef7ed.jpg" referrerpolicy="no-referrer">
  </figure>
<div id="article-content_1-0" class="comp article-content mntl-block">
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<p id="mntl-sc-block_1-0" class="comp mntl-sc-block mntl-sc-block-html">
Equivalent equations are systems of equations that have the same solutions. Identifying and solving equivalent equations is a valuable skill, not only in <a href="https://www.thoughtco.com/what-is-algebra-why-take-algebra-2311937" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">algebra class</a> but also in everyday life. Take a look at examples of equivalent equations, how to solve them for one or more variables, and how you might use this skill outside a classroom.
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<h3 id="mntl-sc-block-callout-heading_1-0" class="comp mntl-sc-block-callout-heading mntl-text-block">
Key Takeaways</h3>
<div id="mntl-sc-block-callout-body_1-0" class="comp mntl-sc-block-callout-body mntl-text-block">
<ul><li>Equivalent equations are algebraic equations that have identical solutions or roots.</li><li>Adding or subtracting the same number or expression to both sides of an equation produces an equivalent equation.</li><li>Multiplying or dividing both sides of an equation by the same non-zero number produces an equivalent equation.</li></ul></div>
</div>
<span class="heading-toc" id="toc-linear-equations-with-one-variable"></span> <h2 id="mntl-sc-block_1-0-3" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Linear Equations With One Variable </span> </h2>
<p id="mntl-sc-block_1-0-4" class="comp mntl-sc-block mntl-sc-block-html">
The simplest examples of equivalent equations don't have any variables. For example, these three equations are equivalent to each other:
</p>
<div id="mntl-sc-block_1-0-5" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-6" class="comp mntl-sc-block mntl-sc-block-html">
<li>3 + 2 = 5</li><li>4 + 1 = 5</li><li>5 + 0 = 5</li>
</ul>
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<p id="mntl-sc-block_1-0-8" class="comp mntl-sc-block mntl-sc-block-html">
Recognizing these equations are equivalent is great, but not particularly useful. Usually, an equivalent equation problem asks you to solve for a variable to see if it is the same (the same <strong>root</strong>) as the one in another equation.
</p>
<div id="mntl-sc-block_1-0-9" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-10" class="comp mntl-sc-block mntl-sc-block-html">
For example, the following equations are equivalent:
</p>
<div id="mntl-sc-block_1-0-11" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-12" class="comp mntl-sc-block mntl-sc-block-html">
<li>x = 5</li><li>-2x = -10</li>
</ul>
<div id="mntl-sc-block_1-0-13" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-14" class="comp mntl-sc-block mntl-sc-block-html">
In both cases, x = 5. How do we know this? How do you solve this for the "-2x = -10" equation? The first step is to know the rules of equivalent equations:
</p>
<div id="mntl-sc-block_1-0-15" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-16" class="comp mntl-sc-block mntl-sc-block-html">
<li><a href="https://www.thoughtco.com/3-digit-addition-worksheets-with-regrouping-2311922" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Adding</a> or subtracting the same number or expression to both sides of an equation produces an equivalent equation.</li>
<li>Multiplying or dividing both sides of an equation by the same non-zero number produces an equivalent equation.</li>
<li>Raising both sides of the equation to the <a href="https://www.thoughtco.com/exponents-and-bases-2312002" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">same odd power</a> or taking the same odd root will produce an equivalent equation.</li>
<li>If both sides of an equation are non-<a href="https://www.thoughtco.com/calculations-with-negative-numbers-2312514" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="3">negative</a>, raising both sides of an equation to the same even power or taking the same even root will give an equivalent equation.</li>
</ul>
<div id="mntl-sc-block_1-0-17" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<span class="heading-toc" id="toc-example"></span> <h2 id="mntl-sc-block_1-0-18" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Example </span> </h2>
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Putting these rules into practice, determine whether these two equations are equivalent:
</p>
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<ul id="mntl-sc-block_1-0-21" class="comp mntl-sc-block mntl-sc-block-html">
<li>x + 2 = 7</li><li>2x + 1 = 11</li>
</ul>
<div id="mntl-sc-block_1-0-22" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-23" class="comp mntl-sc-block mntl-sc-block-html">
To solve this, you need to find "x" for each <a href="https://www.thoughtco.com/exercise-worksheets-using-foil-2312026" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">equation</a>. If "x" is the same for both equations, then they are equivalent. If "x" is different (i.e., the equations have different roots), then the equations are not equivalent. For the first equation:
</p>
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<ul id="mntl-sc-block_1-0-25" class="comp mntl-sc-block mntl-sc-block-html">
<li>x + 2 = 7</li><li>x + 2 - 2 = 7 - 2 (subtracting both sides by same number)</li><li>x = 5</li>
</ul>
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<p id="mntl-sc-block_1-0-27" class="comp mntl-sc-block mntl-sc-block-html">
For the second equation:
</p>
<div id="mntl-sc-block_1-0-28" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-29" class="comp mntl-sc-block mntl-sc-block-html">
<li>2x + 1 = 11</li><li>2x + 1 - 1 = 11 - 1 (subtracting both sides by the same number)</li><li>2x = 10</li><li>2x/2&nbsp;= 10/2 (dividing both sides of the equation by the same number)</li><li>x = 5</li>
</ul>
<div id="mntl-sc-block_1-0-30" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-31" class="comp mntl-sc-block mntl-sc-block-html">
So, yes, the two equations are equivalent because x = 5 in each case.
</p>
<div id="mntl-sc-block_1-0-32" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<span class="heading-toc" id="toc-practical-equivalent-equations"></span> <h2 id="mntl-sc-block_1-0-33" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Practical Equivalent Equations </span> </h2>
<p id="mntl-sc-block_1-0-34" class="comp mntl-sc-block mntl-sc-block-html">
You can use equivalent equations in daily life. It's particularly helpful when shopping. For example, you like a particular shirt. One company offers the shirt for $6 and has $12 shipping, while another company offers the shirt for $7.50 and has $9 shipping. Which shirt has the best price? How many shirts (maybe you want to get them for friends) would you have to buy for the price to be the same for both companies?
</p>
<div id="mntl-sc-block_1-0-35" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-36" class="comp mntl-sc-block mntl-sc-block-html">
To solve this problem, let "x" be the number of shirts. To start with, set x =1 for the purchase of one shirt. For company #1:
</p>
<div id="mntl-sc-block_1-0-37" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-38" class="comp mntl-sc-block mntl-sc-block-html">
<li>Price = 6x + 12 = (6)(1) + 12 = 6 + 12 = $18</li>
</ul>
<div id="mntl-sc-block_1-0-39" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-40" class="comp mntl-sc-block mntl-sc-block-html">
For company #2:
</p>
<div id="mntl-sc-block_1-0-41" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-42" class="comp mntl-sc-block mntl-sc-block-html">
<li>Price = 7.5x + 9 = (1)(7.5) + 9 = 7.5 + 9 = $16.50</li>
</ul>
<div id="mntl-sc-block_1-0-43" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-44" class="comp mntl-sc-block mntl-sc-block-html">
So, if you're buying one shirt, the second company offers a better deal.
</p>
<div id="mntl-sc-block_1-0-45" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-46" class="comp mntl-sc-block mntl-sc-block-html">
To find the point where prices are equal, let "x" remain the number of shirts, but set the two equations equal to each other. Solve for "x" to find how many shirts you'd have to buy:
</p>
<div id="mntl-sc-block_1-0-47" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-48" class="comp mntl-sc-block mntl-sc-block-html">
<li>6x + 12 = 7.5x + 9</li>
<li>6x - 7.5x = 9 - 12 (<a href="https://www.thoughtco.com/2-digit-subtraction-worksheets-with-regrouping-2311924" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">subtracting</a> the same numbers or expressions from each side)</li>
<li>-1.5x = -3</li>
<li>1.5x = 3 (dividing both sides by the same number, -1)</li>
<li>x = 3/1.5 (dividing both sides by 1.5)</li>
<li>x = 2</li>
</ul>
<div id="mntl-sc-block_1-0-49" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-50" class="comp mntl-sc-block mntl-sc-block-html">
If you buy two shirts, the price is the same, no matter where you get it. You can use the same math to determine which company gives you a better deal with larger orders and also to calculate how much you'll save using one company over the other. See, algebra is useful!
</p>
<div id="mntl-sc-block_1-0-51" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<span class="heading-toc" id="toc-equivalent-equations-with-two-variables"></span> <h2 id="mntl-sc-block_1-0-52" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Equivalent Equations With Two Variables </span> </h2>
<p id="mntl-sc-block_1-0-53" class="comp mntl-sc-block mntl-sc-block-html">
If you have two equations and two unknowns (x and y), you can determine whether two sets of linear equations are equivalent.
</p>
<div id="mntl-sc-block_1-0-54" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-55" class="comp mntl-sc-block mntl-sc-block-html">
For example, if you're given the equations:
</p>
<div id="mntl-sc-block_1-0-56" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-57" class="comp mntl-sc-block mntl-sc-block-html">
<li>-3x + 12y = 15</li><li>7x - 10y = -2</li>
</ul>
<div id="mntl-sc-block_1-0-58" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-59" class="comp mntl-sc-block mntl-sc-block-html">
You can determine whether the following system is equivalent:
</p>
<div id="mntl-sc-block_1-0-60" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-61" class="comp mntl-sc-block mntl-sc-block-html">
<li>-x + 4y = 5</li><li>7x -10y = -2</li>
</ul>
<div id="mntl-sc-block_1-0-62" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-63" class="comp mntl-sc-block mntl-sc-block-html">
To <a href="https://www.thoughtco.com/problem-solving-in-mathematics-2311775" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">solve this problem</a>, find "x" and "y" for each system of equations. If the values are the same, then the systems of equations are equivalent.
</p>
<div id="mntl-sc-block_1-0-64" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-65" class="comp mntl-sc-block mntl-sc-block-html">
Start with the first set. To solve two <a href="https://www.thoughtco.com/solve-algebra-problems-step-by-step-2311970" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">equations</a> with two <a href="https://www.thoughtco.com/what-is-a-scientific-variable-3975929" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">variables</a>, isolate one variable and plug its solution into the other equation. To isolate the "y" variable:
</p>
<div id="mntl-sc-block_1-0-66" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-67" class="comp mntl-sc-block mntl-sc-block-html">
<li>-3x + 12y = 15</li><li>-3x = 15 - 12y</li><li>x = -(15 - 12y)/3 = -5 + 4y (plug in for "x" in the second equation)</li><li>7x - 10y = -2</li><li>7(-5 + 4y) - 10y = -2</li><li>-35 + 28y - 10y = -2</li><li>18y = 33</li><li>y = 33/18 = 11/6</li>
</ul>
<div id="mntl-sc-block_1-0-68" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-69" class="comp mntl-sc-block mntl-sc-block-html">
Now, plug "y" back into either equation to solve for "x":
</p>
<div id="mntl-sc-block_1-0-70" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-71" class="comp mntl-sc-block mntl-sc-block-html">
<li>7x - 10y = -2</li><li>7x = -2 + 10(11/6)</li>
</ul>
<div id="mntl-sc-block_1-0-72" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-73" class="comp mntl-sc-block mntl-sc-block-html">
Working through this, you'll eventually&nbsp;get x = 7/3.
</p>
<div id="mntl-sc-block_1-0-74" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-75" class="comp mntl-sc-block mntl-sc-block-html">
To answer the question,&nbsp;you <em>could</em> apply the same principles to the second set of equations to solve for "x" and "y" to find that yes, they are indeed equivalent. It's easy to get bogged down in the algebra, so it's a good idea to check your work using an <a href="http://www.webmath.com/solver2.html" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">online equation solver</a>.
</p>
<div id="mntl-sc-block_1-0-76" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-77" class="comp mntl-sc-block mntl-sc-block-html">
However, the clever student will notice the two sets of equations are equivalent <em>without doing any difficult calculations at all.</em> The only difference between the first equation in each set is that the first one is three times the second one (equivalent). The second equation is exactly the same.
</p>
</div>
</div>
]]></description>
            <pubDate>Wed, 17 Jan 2018 18:49:52 GMT</pubDate>
            <guid isPermaLink="false">https://www.thoughtco.com/understanding-equivalent-equations-4157661</guid>
            <link>https://www.thoughtco.com/understanding-equivalent-equations-4157661</link>
            <author><![CDATA[Anne Marie Helmenstine, Ph.D.]]></author>
                <category>Science</category>
                <category>Tech</category>
                <category>Math</category>
        </item>
        <item>
            <title><![CDATA[10 Math Tricks That Will Blow Your Mind]]></title>
            <description><![CDATA[<figure>
    <img src="https://www.thoughtco.com/thmb/FGRV9FpTnS_WqfKU6xTvkI9vI28=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/Math_class_in_Da_Ji_Junior_High_School_2006-12-1-89e9cb05eb2a4d53a503a97e3f3b1ae1.jpg" referrerpolicy="no-referrer">
  </figure>
<div id="list-sc_1-0" class="comp article-content list list-sc mntl-block">
<div id="article-intro_1-0" class="comp article-content__description article-intro mntl-block">
<p id="mntl-sc-block_1-0" class="comp mntl-sc-block mntl-sc-block-html">
Are you ready to give your mathematics skills a boost? These simple math tricks can help you perform calculations more quickly and easily. They also come in handy if you want to impress your teacher, parents, or friends.
</p>
</div>
<div id="list-sc__content_1-0" class="comp list-sc__content mntl-sc-page mntl-block" data-sc-sticky-offset="85" data-sc-ad-label-height="24" data-sc-ad-track-spacing="100" data-sc-min-track-height="250" data-sc-max-track-height="600" data-sc-breakpoint="50em" data-sc-load-immediate="3" data-sc-content-positions="[1, 1050, 1250, 1550, 1950, 2350, 2750, 3150]" data-bind-scroll-on-start="true">
<div id="list-sc-item_1-0" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">01</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-multiplying-by-6"></span> <h2 id="mntl-sc-block_2-0" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Multiplying by 6 </span> </h2>
<p id="mntl-sc-block_2-0-1" class="comp mntl-sc-block mntl-sc-block-html">
If you multiply 6 by an even number, the answer will end with the same digit. The number in the ten's place will be half of the number in the one's place.
</p>
<p id="mntl-sc-block_2-0-2" class="comp mntl-sc-block mntl-sc-block-html">
Example: 6 x 4 = 24.
</p>
</div>
<div id="list-sc-item_1-0-1" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-1" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">02</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-the-answer-is-2"></span> <h2 id="mntl-sc-block_2-0-3" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Answer Is 2 </span> </h2>
<ol id="mntl-sc-block_2-0-4" class="comp mntl-sc-block mntl-sc-block-html">
<li>Think of a number.</li><li>Multiply it by 3.</li><li>Add 6.</li><li>Divide this number by 3.</li><li>Subtract the number from Step 1 from the answer in Step 4.</li>
</ol>
<p id="mntl-sc-block_2-0-5" class="comp mntl-sc-block mntl-sc-block-html">
The answer is 2.
</p>
</div>
<div id="list-sc-item_1-0-2" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-2" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">03</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-same-three-digit-number"></span> <h2 id="mntl-sc-block_2-0-6" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Same Three-Digit Number </span> </h2>
<ol id="mntl-sc-block_2-0-7" class="comp mntl-sc-block mntl-sc-block-html">
<li>Think of any three-digit number in which each of the digits is the same. Examples include 333, 666, 777, and 999.</li>
<li>Add up the digits.</li>
<li><a href="https://www.thoughtco.com/basic-division-facts-no-remainder-2312184" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Divide</a> the three-digit number by the answer in Step 2.</li>
</ol>
<p id="mntl-sc-block_2-0-8" class="comp mntl-sc-block mntl-sc-block-html">
The answer is 37.
</p>
</div>
<div id="list-sc-item_1-0-3" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-3" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">04</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-six-digits-become-three"></span> <h2 id="mntl-sc-block_2-0-9" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Six Digits Become Three </span> </h2>
<ol id="mntl-sc-block_2-0-10" class="comp mntl-sc-block mntl-sc-block-html">
<li>Take any three-digit number and write it twice to make a six-digit number. Examples include 371371 or 552552.</li><li>Divide the number by 7.</li><li>Divide it by 11.</li><li>Divide it by 13.</li>
</ol>
<p id="mntl-sc-block_2-0-11" class="comp mntl-sc-block mntl-sc-block-html">
The order in which you do the division is unimportant!
</p>
<p id="mntl-sc-block_2-0-12" class="comp mntl-sc-block mntl-sc-block-html">
The answer is the three-digit number.
</p>
<p id="mntl-sc-block_2-0-13" class="comp mntl-sc-block mntl-sc-block-html">
Examples: 371371 gives you 371 or 552552 gives you 552.
</p>
<ol id="mntl-sc-block_2-0-14" class="comp mntl-sc-block mntl-sc-block-html">
<li>A related trick is to take any three-digit number.</li>
<li><a href="https://www.thoughtco.com/magical-multiplication-tricks-2086556" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Multiply</a> it by 7, 11, and 13.</li>
</ol>
<p id="mntl-sc-block_2-0-15" class="comp mntl-sc-block mntl-sc-block-html">
The result will be a six-digit number that repeats the three-digit number.
</p>
<p id="mntl-sc-block_2-0-16" class="comp mntl-sc-block mntl-sc-block-html">
Example: 456 becomes 456456.
</p>
</div>
<div id="list-sc-item_1-0-4" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-4" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">05</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-the-11-rule"></span> <h2 id="mntl-sc-block_2-0-17" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The 11 Rule </span> </h2>
<p id="mntl-sc-block_2-0-18" class="comp mntl-sc-block mntl-sc-block-html">
This is a quick way to multiply two-digit numbers by 11 in your head.
</p>
<ol id="mntl-sc-block_2-0-19" class="comp mntl-sc-block mntl-sc-block-html">
<li>Separate the two digits in your mind.</li><li>Add the two digits together.</li><li>Place the number from Step 2 between the two digits. If the number from Step 2 is greater than 9, put the one's digit in the space and carry the ten's digit.</li>
</ol>
<p id="mntl-sc-block_2-0-20" class="comp mntl-sc-block mntl-sc-block-html">
Examples: 72 x 11 = 792.
</p>
<p id="mntl-sc-block_2-0-21" class="comp mntl-sc-block mntl-sc-block-html">
57 x 11 = 5 _ 7, but 5 + 7 = 12, so put 2 in the space and add the 1 to the 5 to get 627
</p>
</div>
<div id="list-sc-item_1-0-5" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-5" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">06</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-memorizing-pi"></span> <h2 id="mntl-sc-block_2-0-22" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Memorizing Pi </span> </h2>
<p id="mntl-sc-block_2-0-23" class="comp mntl-sc-block mntl-sc-block-html">
To remember the first seven <a href="https://www.thoughtco.com/the-number-pi-3-141592654-3126451" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">digits of pi</a>, count the number of letters in each word of the sentence:
</p>
<p id="mntl-sc-block_2-0-24" class="comp mntl-sc-block mntl-sc-block-html">
"How I wish I could calculate pi."
</p>
<p id="mntl-sc-block_2-0-25" class="comp mntl-sc-block mntl-sc-block-html">
This becomes 3.141592.
</p>
</div>
<div id="list-sc-item_1-0-6" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-6" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">07</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-contains-the-digits-1-2-4-5-7-8"></span> <h2 id="mntl-sc-block_2-0-26" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Contains the Digits 1, 2, 4, 5, 7, 8 </span> </h2>
<ol id="mntl-sc-block_2-0-27" class="comp mntl-sc-block mntl-sc-block-html">
<li>Select a number from 1 to 6.</li><li>Multiply the number by 9.</li><li>Multiply it by 111.</li><li>Multiply it by 1001.</li><li>Divide the answer by 7.</li>
</ol>
<p id="mntl-sc-block_2-0-28" class="comp mntl-sc-block mntl-sc-block-html">
The number will contain the digits 1, 2, 4, 5, 7, and 8.&nbsp;
</p>
<p id="mntl-sc-block_2-0-29" class="comp mntl-sc-block mntl-sc-block-html">
Example: The number 6 yields the answer 714285.
</p>
</div>
<div id="list-sc-item_1-0-7" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-7" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">08</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-multiply-large-numbers-in-your-head"></span> <h2 id="mntl-sc-block_2-0-30" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Multiply Large Numbers in Your Head </span> </h2>
<p id="mntl-sc-block_2-0-31" class="comp mntl-sc-block mntl-sc-block-html">
To easily multiply two double-digit numbers, use their distance from 100 to simplify the math:
</p>
<ol id="mntl-sc-block_2-0-32" class="comp mntl-sc-block mntl-sc-block-html">
<li><a href="https://www.thoughtco.com/2-digit-subtraction-worksheets-with-regrouping-2311924" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Subtract</a> each number from 100.</li>
<li>Add these values together.</li>
<li>100 minus this number is the first part of the answer.</li>
<li>Multiply the digits from Step 1 to get the second part of the answer.</li>
</ol>
</div>
<div id="list-sc-item_1-0-8" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-8" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">09</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-super-simple-divisibility-rules"></span> <h2 id="mntl-sc-block_2-0-33" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Super Simple Divisibility Rules </span> </h2>
<p id="mntl-sc-block_2-0-34" class="comp mntl-sc-block mntl-sc-block-html">
You've got 210 pieces of pizza and want to know whether or not you can split them evenly within your group. Rather than whip out the calculator, use these simple shortcuts to do the math in your head:
</p>
<ul id="mntl-sc-block_2-0-35" class="comp mntl-sc-block mntl-sc-block-html">
<li>Divisible by 2 if the last digit is a multiple of 2 (210).</li>
<li>Divisible by 3 if&nbsp;the sum of the digits is divisible by 3&nbsp;(522 because the digits add up to 9, which is divisible by 3).</li>
<li>Divisible by 4 if&nbsp;the last two digits are divisible by 4 (2540 because 40 is divisible by 4).</li>
<li>Divisible by 5 if&nbsp;the last digit is 0 or 5 (9905).</li>
<li>Divisible by 6 if it passes the rules for both 2 and 3 (408).</li>
<li>Divisible by 9 if the <a href="https://www.thoughtco.com/3-digit-addition-worksheets-no-regrouping-2311907" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">sum</a> of the digits is divisible by 9 (6390 since 6 + 3 + 9 + 0 = 18, which is divisible&nbsp;by 9).</li>
<li>Divisible by 10 if the number ends in a 0 (8910).</li>
<li>Divisible by 12 if the rules for divisibility by 3 and 4 apply.</li>
</ul>
<p id="mntl-sc-block_2-0-36" class="comp mntl-sc-block mntl-sc-block-html">
Example: The 210 slices of pizza may be evenly distributed into groups of 2, 3, 5, 6, 10.
</p>
</div>
<div id="list-sc-item_1-0-9" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-9" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">10</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-finger-multiplication-tables"></span> <h2 id="mntl-sc-block_2-0-37" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Finger Multiplication Tables </span> </h2>
<p id="mntl-sc-block_2-0-38" class="comp mntl-sc-block mntl-sc-block-html">
Everyone knows you can count on your fingers. Did you realize you can use them for multiplication? A simple way to do the "9" multiplication table is to place both hands in front of you with fingers and thumbs extended. To multiply 9 by a number, fold down that number finger, counting from the left.
</p>
<p id="mntl-sc-block_2-0-39" class="comp mntl-sc-block mntl-sc-block-html">
Examples: To multiply 9 by 5, fold down the fifth finger from the left. Count fingers on either side of the "fold" to get the answer. In this case, the answer is 45.
</p>
<p id="mntl-sc-block_2-0-40" class="comp mntl-sc-block mntl-sc-block-html">
To multiply 9 times 6, fold down the sixth finger, giving an answer of 54.
</p>
</div>
</div>
<div id="article-outro_1-0" class="comp article-outro mntl-block"></div>
</div>
]]></description>
            <pubDate>Mon, 20 Nov 2017 18:29:31 GMT</pubDate>
            <guid isPermaLink="false">https://www.thoughtco.com/math-tricks-that-will-blow-your-mind-4154742</guid>
            <link>https://www.thoughtco.com/math-tricks-that-will-blow-your-mind-4154742</link>
            <author><![CDATA[Anne Marie Helmenstine, Ph.D.]]></author>
                <category>Science</category>
                <category>Tech</category>
                <category>Math</category>
        </item>
        <item>
            <title><![CDATA[8 Facts About Infinity That Will Blow Your Mind]]></title>
            <description><![CDATA[<figure>
    <img src="https://www.thoughtco.com/thmb/oWz9nmFVzF4iPqxp-FGGV9cOtEk=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/moebius-strip-522025950-5a085d4013f1290037101e9d.jpg" referrerpolicy="no-referrer">
  </figure>
<div id="list-sc_1-0" class="comp article-content list list-sc mntl-block">
<div id="article-intro_1-0" class="comp article-content__description article-intro mntl-block">
<p id="mntl-sc-block_1-0" class="comp mntl-sc-block mntl-sc-block-html">
Infinity is an abstract concept used to describe something that is endless or boundless. It is important in mathematics, cosmology, physics, computing, and the arts.
</p>
</div>
<div id="list-sc__content_1-0" class="comp list-sc__content mntl-sc-page mntl-block" data-sc-sticky-offset="85" data-sc-ad-label-height="24" data-sc-ad-track-spacing="100" data-sc-min-track-height="250" data-sc-max-track-height="600" data-sc-breakpoint="50em" data-sc-load-immediate="3" data-sc-content-positions="[1, 1050, 1250, 1550, 1950, 2350, 2750, 3150]" data-bind-scroll-on-start="true">
<div id="list-sc-item_1-0" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">01</div>
<div class="total">of 08</div>
</div>
</div>
<span class="heading-toc" id="toc-the-infinity-symbol"></span> <h2 id="mntl-sc-block_2-0" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Infinity Symbol </span> </h2>
<figure id="mntl-sc-block_2-0-1" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
</svg>
<span class="figure-article-caption-text">The infinity symbol is also known as the lemniscate.</span>
<span class="figure-article-caption-owner">Chris Collins / Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-2" class="comp mntl-sc-block mntl-sc-block-html">
Infinity has its own special symbol: ∞. The symbol, sometimes called the lemniscate, was introduced by clergyman and mathematician John Wallis in 1655. The word "lemniscate" comes from the Latin word <em>lemniscus</em>, which means "ribbon," while the word "infinity" comes from the Latin word <em>infinitas</em>, which means "boundless."
</p>
<p id="mntl-sc-block_2-0-3" class="comp mntl-sc-block mntl-sc-block-html">
Wallis may have based the symbol on the Roman numeral for 1000, which the Romans used to indicate "countless" in addition to the number. It's also possible the symbol is based on omega (Ω or ω), the last letter in the Greek alphabet.
</p>
<p id="mntl-sc-block_2-0-4" class="comp mntl-sc-block mntl-sc-block-html">
The concept of infinity was understood long before Wallis gave it the symbol we use today. Around the 4th or 3rd century B.C.E., the Jain mathematical text <em>Surya Prajnapti</em> assigned numbers as either enumerable, innumerable, or infinite. The <a href="https://www.thoughtco.com/timeline-of-greek-and-roman-philosophers-118808" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Greek philosopher</a> Anaximander used the work <em>apeiron</em> to refer to the infinite. Zeno of Elea (born circa 490 B.C.E.) was known for <a href="https://www.thoughtco.com/what-is-a-paradox-1691563" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">paradoxes involving infinity</a>.&nbsp;
</p>
</div>
<div id="list-sc-item_1-0-1" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-1" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">02</div>
<div class="total">of 08</div>
</div>
</div>
<span class="heading-toc" id="toc-zenos-paradox"></span> <h2 id="mntl-sc-block_2-0-5" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Zeno's Paradox </span> </h2>
<figure id="mntl-sc-block_2-0-6" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/BOeejWpYm-lmL61xmmfXe4pQuIw=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/tortoise-and-hare--finish-line-143576837-5a08a081494ec90037e9c6bb.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-1" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
</svg>
<span class="figure-article-caption-text">If the rabbit was forever halving the distance to the tortoise, the tortoise would win the race.</span>
<span class="figure-article-caption-owner">Don Farrall / Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-7" class="comp mntl-sc-block mntl-sc-block-html">
Of all Zeno's paradoxes, the most famous is his paradox of the Tortoise and Achilles. In the paradox, a tortoise challenges the <a href="https://www.thoughtco.com/achilles-greek-hero-of-the-trojan-war-116708" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Greek hero Achilles</a> to a race, providing the tortoise is given a small head start. The tortoise argues he will win the race because as Achilles catches up to him, the tortoise will have gone a bit further, adding to the distance.
</p>
<p id="mntl-sc-block_2-0-8" class="comp mntl-sc-block mntl-sc-block-html">
In simpler terms, consider crossing a room by going half the distance with each stride. First, you cover half the distance, with half remaining. The next step is half of one-half, or a quarter. Three quarters of the distance is covered, yet a quarter remains. Next is 1/8th, then 1/16th, and so on. Although each step brings you closer, you never actually reach the other side of the room. Or rather, you&nbsp;would after taking an infinite number of steps.
</p>
</div>
<div id="list-sc-item_1-0-2" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-2" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">03</div>
<div class="total">of 08</div>
</div>
</div>
<span class="heading-toc" id="toc-pi-as-an-example-of-infinity"></span> <h2 id="mntl-sc-block_2-0-9" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Pi as an Example of Infinity </span> </h2>
<figure id="mntl-sc-block_2-0-10" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/4-rZRqlIALa1g3bEynp65ehpYaI=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/pi-formula-on-blackboard-112303538-5a089c5247c266003765ecbe.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-2" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
</svg>
<span class="figure-article-caption-text">Pi is a number consisting of an infinite number of digits.</span>
<span class="figure-article-caption-owner">Jeffrey Coolidge / Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-11" class="comp mntl-sc-block mntl-sc-block-html">
Another good example of infinity is the <a href="https://www.thoughtco.com/the-number-pi-3-141592654-3126451" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">number π or pi</a>. Mathematicians use a symbol for pi because it's impossible to write the number down. Pi consists of an infinite number of digits. It's often rounded to 3.14 or even 3.14159, yet no matter how many digits you write, it's impossible to get to the end.
</p>
</div>
<div id="list-sc-item_1-0-3" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-3" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">04</div>
<div class="total">of 08</div>
</div>
</div>
<span class="heading-toc" id="toc-the-monkey-theorem"></span> <h2 id="mntl-sc-block_2-0-12" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Monkey Theorem </span> </h2>
<figure id="mntl-sc-block_2-0-13" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/OiESkBYL-gXGXvhaa-p68mj8YrM=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/furry-animal-hands-use-laptop-computer-with-blank-screen-169981126-5a08b2fa845b34003b83cd50.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-3" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
</svg>
<span class="figure-article-caption-text">Given an infinite amount of time, a monkey could write the great American novel.</span>
<span class="figure-article-caption-owner">PeskyMonkey / Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-14" class="comp mntl-sc-block mntl-sc-block-html">
One way to think about infinity is in terms of the monkey theorem. According to the theorem, if you give a monkey a typewriter and an infinite amount of time, eventually it will write Shakespeare's <em>Hamlet</em>. While some people take the theorem to suggest anything is possible, mathematicians see it as evidence of just how improbable certain events are.
</p>
</div>
<div id="list-sc-item_1-0-4" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-4" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">05</div>
<div class="total">of 08</div>
</div>
</div>
<span class="heading-toc" id="toc-fractals-and-infinity"></span> <h2 id="mntl-sc-block_2-0-15" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Fractals and Infinity </span> </h2>
<figure id="mntl-sc-block_2-0-16" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/xCf2m8IHCjJxu4eeb-nxJyXjakw=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/soap-bubbles-spirals-585837119-5a088fba845b34003b7a4f65.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-4" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
</svg>
<span class="figure-article-caption-text">A fractal may be magnified over and over, to infinity, always revealing more detail.</span>
<span class="figure-article-caption-owner">PhotoviewPlus / Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-17" class="comp mntl-sc-block mntl-sc-block-html">
A fractal is an abstract mathematical object, used in art and to simulate natural phenomena. Written as a mathematical equation, most fractals are nowhere differentiable. When viewing an image of a fractal, this means you could zoom in and see new detail. In other words, a fractal is infinitely magnifiable.
</p>
<p id="mntl-sc-block_2-0-18" class="comp mntl-sc-block mntl-sc-block-html">
The Koch snowflake is an interesting example of a fractal. The snowflake starts as an equilateral triangle. For each iteration of the fractal:
</p>
<ol id="mntl-sc-block_2-0-19" class="comp mntl-sc-block mntl-sc-block-html">
<li>Each line segment is divided into three equal segments.</li><li>An equilateral triangle is drawn using the middle segment as its base, pointing outward.</li><li>The line segment serving as the base of the triangle is removed.</li>
</ol>
<p id="mntl-sc-block_2-0-20" class="comp mntl-sc-block mntl-sc-block-html">
The process may be repeated an infinite number of times. The resulting snowflake has a finite area, yet it is bounded by an infinitely long line.
</p>
</div>
<div id="list-sc-item_1-0-5" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-5" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">06</div>
<div class="total">of 08</div>
</div>
</div>
<span class="heading-toc" id="toc-different-sizes-of-infinity"></span> <h2 id="mntl-sc-block_2-0-21" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Different Sizes of Infinity </span> </h2>
<figure id="mntl-sc-block_2-0-22" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/xtQmNnVgTGTV3tzKrr_MIp8EsYg=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/hands-holding-complex-cats-cradle-molecule-network-723497851-5a08a43813f12900372321fd.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-5" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
</svg>
<span class="figure-article-caption-text">Infinity comes in different sizes.</span>
<span class="figure-article-caption-owner">Tang Yau Hoong / Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-23" class="comp mntl-sc-block mntl-sc-block-html">
Infinity is boundless, yet it comes in different sizes. The positive numbers (those greater than 0) and the negative numbers (those smaller than 0) may be considered to be <a href="https://www.thoughtco.com/examples-of-uncountable-sets-3126438" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">infinite sets</a> of equal sizes. Yet, what happens if you combine both sets? You get a set twice as large. As another example, consider all of the even numbers (an infinite set). This represents an infinity half the size of all of the whole numbers.
</p>
<p id="mntl-sc-block_2-0-24" class="comp mntl-sc-bl

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A schematic is defined as a picture that shows something in a simple way, using symbols. A <strong>schematic diagram</strong> is a picture that represents the components of a process, device, or other object using abstract, often standardized symbols and lines. Schematic diagrams only depict the significant components of a system, though some details in the diagram may also be exaggerated or introduced to facilitate the understanding of the system.
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Schematic diagrams do not include details that are not necessary for comprehending the information that the diagram was intended to convey. For example, in a schematic diagram depicting an electrical circuit, you can see how the wires and components are connected together, but not photographs of the circuit itself.
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Key Takeaways: Schematic Diagram</h3>
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<ul><li>A <strong>schematic diagram</strong> is a picture that represents the components of a process, device, or other object using abstract, often standardized symbols and lines.</li><li>Although schematic diagrams are commonly associated with electrical circuits, many examples can be found in other industries.</li></ul></div>
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<span class="heading-toc" id="toc-what-is-a-schematic-diagram"></span> <h2 id="mntl-sc-block_1-0-5" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> What Is a Schematic Diagram? </span> </h2>
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Schematic diagrams can also differ in their level of abstraction. Although they are typically composed of only abstract symbols and lines, some diagrams can also be <strong>semi-schematic</strong> and contain more realistic elements. Some diagrams can also contain words, such as when a process contains multiple elements that have not been standardized.
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More simply, a schematic diagram is a simplified drawing that uses symbols and lines to convey important information. For example, if you are taking the subway you may see a “map” showing you all the stations along a subway line, but that map will not show all the roads and buildings you may pass along the way. In this case, the entire subway system can be represented as differently colored lines depicting the different subway routes, with dots indicating the stops along the lines.
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Though schematic diagrams are most commonly associated with electronics, you have probably encountered many such diagrams, like the subway example above, even if you have never had to wire a circuit. Here are some examples of the many schematic diagrams you may encounter in your work or in your studies.
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<span class="heading-toc" id="toc-examples-of-schematic-diagrams"></span> <h2 id="mntl-sc-block_1-0-13" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Examples of Schematic Diagrams </span> </h2>
<h3 id="mntl-sc-block_1-0-14" class="comp mntl-sc-block reference-sc-block-subheading mntl-sc-block-subheading"> <span class="mntl-sc-block-subheading__text"> Schematic Diagrams in Electronics </span> </h3>
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<span class="figure-article-caption-text">A circuit diagram behind a circuit board. kr7ysztof / Getty Images.</span>
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Schematic diagrams are typically associated with electrical circuits. Also called <strong>wiring diagrams</strong> or <strong>circuit diagrams</strong>, these diagrams show how the different components of a circuit are connected. In these diagrams, lines represent connecting wires, while other elements like resistors, lamps, and switches are represented by standardized symbols called <strong>electrical schematic symbols</strong>.
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In electronics, having a schematic diagram on hand may help a user design an entire circuit before building it, or troubleshoot an electronic that has stopped working.
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Schematic diagrams may also be used to explain the general way that an electronic functions without detailing the hardware or software used in the actual electronic. For example, to explain how a computer projects the words you type on a screen, you might use a schematic diagram that shows how information passes from the keys you press to a word processing program, and finally to the computer screen.
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<span class="figure-article-caption-text">Schematic diagrams of various parts. Eakachai Leesin / EyeEm / Getty Images.</span>
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Schematic diagrams can be used to depict machines as well. For example, in a textbook a car engine may be depicted as a set of shapes that show how the different parts are positioned relative to one another. A <strong>schematic drawing</strong> may also be created by an engineer while they are designing the machine so that they can properly understand how the parts work together, and make any necessary changes before building the actual system.
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Many chemical products are often obtained by performing several different reactions at different steps of the process. A schematic diagram in chemistry can help someone understand all the reactions that were performed to yield a final product, without showing the actual products themselves. This may be portrayed, for example, as a series of boxes connected together with arrows, with words depicting the various elements and conditions that were used throughout the process.
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Like for machines, a schematic diagram may also be used to depict the apparatus that was used for performing the reactions, especially if it is not typically used for reactions, or had been modified from an instrument that was already known.
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<h3 id="mntl-sc-block_1-0-32" class="comp mntl-sc-block reference-sc-block-subheading mntl-sc-block-subheading"> <span class="mntl-sc-block-subheading__text"> Schematic Diagrams in Business </span> </h3>
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<span class="figure-article-caption-text">A flowchart that could be used to depict a business process. Sean Gladwell / Getty Images.</span>
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Schematic diagrams are helpful in conveying the main parts of a complicated business model and showing how they all relate together. For example, a marketing plan might consist of many different elements, such as strategy, objectives, and an action plan. A schematic diagram would then be used to help organize all those elements, including elements within each category, in ways that would convey the main ideas in a clear and concise fashion.
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]]></description>
            <pubDate>Tue, 29 Jan 2019 17:18:26 GMT</pubDate>
            <guid isPermaLink="false">https://www.thoughtco.com/what-is-a-schematic-diagram-4584811</guid>
            <link>https://www.thoughtco.com/what-is-a-schematic-diagram-4584811</link>
            <author><![CDATA[Alane Lim]]></author>
                <category>Science</category>
                <category>Tech</category>
                <category>Math</category>
        </item>
        <item>
            <title><![CDATA[Why You May Have Heard Humans Only Use 10% of the Brain and Why That's Wrong]]></title>
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You may have heard that humans only use 10 percent of their brain power, and that if you could unlock the rest of your brainpower, you could do so much more. You could become a super genius, or acquire psychic powers like mind reading and telekinesis. However, there is a powerful body of evidence debunking the 10 percent myth. Scientists have consistently shown that humans use their entire brain throughout each day.
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Despite the evidence, the 10 percent myth has inspired many references in the cultural imagination. Films like "Limitless" and "Lucy" depict protagonists who develop godlike powers thanks to drugs that unleash the previously inaccessible 90 percent of the brain. A 2013 study showed that about 65 percent of Americans believe the trope, and a 1998 study showed that a full third of psychology majors, who focus on the workings of the brain, fell for it.
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<span class="heading-toc" id="toc-neuropsychology"></span> <h2 id="mntl-sc-block_1-0-4" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Neuropsychology </span> </h2>
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Neuropsychology studies how the anatomy of the brain affects someone’s behavior, emotion, and cognition. Over the years, brain scientists have shown that different parts of the brain are responsible for <a href="https://www.thoughtco.com/anatomy-of-the-brain-373479" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">specific functions</a>, whether it’s <a href="https://www.thoughtco.com/occipital-lobes-anatomy-373224" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">recognizing colors</a> or <a href="https://www.thoughtco.com/frontal-lobes-anatomy-373213" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="3">problem solving</a>. Contrary to the 10 percent myth, scientists have proven that every part of the brain is integral for our daily functioning, thanks to brain imaging techniques like positron emission tomography and functional magnetic resonance imaging.
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Research has yet to find a brain area that is completely inactive. Even studies that measure activity at the level of single neurons have not revealed any <a href="http://www.bmj.com/content/335/7633/1288.full" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">inactive areas of the brain</a>.&nbsp;Many brain imaging studies that measure brain activity when a person is doing a specific task show how different parts of the brain work together. For example, while you are reading this text on your smartphone, some parts of your brain, including those responsible for vision, reading comprehension, and holding your phone, will be more active.
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However, some brain images <a href="https://www.theatlantic.com/technology/archive/2014/07/you-already-use-way-way-more-than-10-percent-of-your-brain/374520/" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">unintentionally support the 10 percent myth</a>,&nbsp;because they often show small bright splotches on an otherwise gray brain. This may imply that only the bright spots have brain activity, but that isn’t the case. Rather, colored splotches represent brain areas that are<em> <a href="http://www.newsobserver.com/news/technology/article23721076.html" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="2">more</a></em><a href="http://www.newsobserver.com/news/technology/article23721076.html" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="3"> active </a>when someone’s doing a task compared to when they’re not. The gray spots are still active, just to a lesser degree.
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A more direct counter to the 10 percent myth lies in individuals who have suffered brain damage–through a stroke, head trauma, or carbon monoxide poisoning–and what they can no longer do as a result of that damage, or can still do just as well. If the 10 percent myth were true, damage to perhaps 90 percent of the brain wouldn’t affect daily functioning.
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Yet studies show that damaging even a very small part of the brain may have devastating consequences. For example, damage to <a href="https://www.thoughtco.com/brocas-area-anatomy-373215" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Broca’s area</a> hinders proper formation of words and fluent speech, though general language comprehension remains intact. In one highly publicized case, a Florida woman permanently lost her “capacity for thoughts, perceptions, memories, and emotions that are the very essence of being human” when a lack of oxygen destroyed half of her <a href="https://www.thoughtco.com/anatomy-of-the-brain-cerebrum-373218" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">cerebrum</a>, which makes up about 85 percent of the brain.
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<span class="heading-toc" id="toc-evolutionary-arguments"></span> <h2 id="mntl-sc-block_1-0-15" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Evolutionary Arguments </span> </h2>
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Another line of evidence against the 10 percent myth comes from evolution. The adult brain only constitutes 2 percent of body mass, yet it consumes over 20 percent of the body’s energy. In comparison, the adult brains of many vertebrate species–including some fish, reptiles, birds, and mammals–consume <a href="https://www.physiology.org/doi/abs/10.1152/ajpregu.1981.241.3.R203%29" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">2 to 8 percent of their body’s energy</a>. The brain has been shaped by millions of years of <a href="https://www.thoughtco.com/survival-of-the-fittest-1224578" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">natural selection</a>, which passes down favorable traits to increase likelihood of survival. It is unlikely that the body would dedicate so much of its energy to keep an entire brain functioning if it only uses 10 percent of the brain.
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<span class="heading-toc" id="toc-the-origin-of-the-myth"></span> <h2 id="mntl-sc-block_1-0-18" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Origin of the Myth </span> </h2>
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The main allure of the 10 percent myth is the idea that you could do so much more <em>if only</em> you could unlock the rest of your brain. Even with ample evidence suggesting the contrary, why do many people still believe that humans only use 10 percent of their brains? It’s unclear how the myth spread in the first place, but it has been popularized by self-help books, and may even also be grounded in older, flawed, neuroscience studies.
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The myth could be aligned with messages espoused by self-improvement books, which show you ways to do better and live up to your "potential."&nbsp;For example, the preface to the notorious "How to Win Friends and Influence People" says that the average person “develops only 10 percent of his latent mental ability.” This statement, which is traced back to psychologist William James, refers to a person’s potential to achieve more rather than how much brain matter they used. Others have even said that Einstein explained his brilliance using the 10 percent myth, though these claims remain unfounded.
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Another possible source of the myth lies in “silent” brain areas from older neuroscience research. In the 1930s, for instance, neurosurgeon Wilder Penfield hooked electrodes to the exposed brains of his epilepsy patients while operating on them. He noticed that particular brain areas triggered the experience various sensations, but while others seemed to <a href="http://www.sci-con.org/2004/09/exploding-the-10-percent-myth/" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1" rel="nofollow">cause no reaction</a>. Still, as technology evolved, researchers found that these “silent” brain areas, which included the <a href="https://misuse.ncbi.nlm.nih.gov/error/abuse.shtml" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="2">prefrontal lobes</a>, had major functions after all.
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<span class="heading-toc" id="toc-resources-and-further-reading"></span> <h2 id="mntl-sc-block_1-0-25" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Resources and Further Reading </span> </h2>
<ul id="mntl-sc-block_1-0-26" class="comp mntl-sc-block mntl-sc-block-html">
<li><cite>Beyerstein, B.L. “Whence Cometh the Myth That We Only Use 10% of Our Brains?” <em>Mind Myths: Exploring Popular Assumptions about the Mind and Brain</em>, edited by Sergio Della Sala, Wiley, 1999, pp. 3-24.</cite></li>
<li><cite>Broadfoot, Marla Vacek. “<a href="https://www.newsobserver.com/news/technology/article23721076.html" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">How Do Brain Scans Work?</a>” <em>Raleigh News &amp; Observer</em>, 27 Jan. 2013.</cite></li>
<li><cite>“<a href="https://www.sci-con.org/2004/09/exploding-the-10-percent-myth/" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="2" rel="nofollow">Exploding the 10 Percent Myth</a>.” <em>Science &amp; Consciousness Review</em>.</cite></li>
<li><cite>Higbee, Kenneth L., and Samuel L. Clay. “<a href="https://doi.org/10.1080/00223989809599280" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="3">College Students' Beliefs in the Ten-Percent Myth</a>.” <em>The Journal of Psychology</em>, vol. 132, no. 5, 1998, pp. 469-476.</cite></li>
<li><cite>Jarrett, Christian. <em>Great Myths of the Brain</em>. Wiley Blackwell, 2014.</cite></li>
<li><cite>McDougle, Sam. “<a href="https://www.theatlantic.com/technology/archive/2014/07/you-already-use-way-way-more-than-10-percent-of-your-brain/374520/" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="4">You Already Use Way, Way More Than 10 Percent of Your Brain</a>.” <em>The Atlantic</em>, 7 Aug. 2014.</cite></li>
<li><cite>Mink, J. W., et al. “<a href="https://doi.org/10.1152/ajpregu.1981.241.3.R203" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="5">Ratio of Central Nervous System to Body Metabolism in Vertebrates: Its Constancy and Functional Basis</a>.” <em>American Journal of Physiology-Regulatory, Integrative and Comparative Physiology</em>, vol. 241, no. 3, 1 Sept. 1981, pp. R203-R212.</cite></li>
<li><cite>“<a href="https://www.michaeljfox.org/publication/new-survey-finds-americans-care-about-brain-health-misperceptions-abound" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="6">New Survey Finds Americans Care about Brain Health, but Misperceptions Abound</a>.” <em>The Michael J. Fox Foundation for Parkinson's Research</em>, 25 Sept. 2013.</cite></li><span class="mntl-sc-block-adslot mntl-sc-block-adslot-inline"></span>
<li><cite>Tandon, Prakashnarain. “<a href="https://doi.org/10.4103/0028-3886.125242" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="7">Not so ‘Silent’: The Human Prefrontal Cortex</a>.” <em>Neurology India</em>, vol. 61, no. 6, 2013, pp. 578-580.</cite></li>
<li><cite>Vreeman, Rachel C, and Aaron E Carroll. “<a href="https://doi.org/10.1136/bmj.39420.420370.25" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="8">Medical Myths</a>.” <em>BMJ</em>, vol. 335, no. 7633, 20 Dec. 2007, pp. 1288-1289.</cite></li>
<li><cite>Wanjek, Christopher. <em>Bad Medicine: Misconceptions and Misuses Revealed, from Distance Healing to Vitamin O</em>. Wiley, 2003.</cite></li>
</ul>
</div>
</div>
]]></description>
            <pubDate>Thu, 22 Feb 2018 15:31:24 GMT</pubDate>
            <guid isPermaLink="false">https://www.thoughtco.com/percentage-of-human-brain-used-4159438</guid>
            <link>https://www.thoughtco.com/percentage-of-human-brain-used-4159438</link>
            <author><![CDATA[Alane Lim]]></author>
                <category>Science</category>
                <category>Tech</category>
                <category>Math</category>
        </item>
        <item>
            <title><![CDATA[Déjà Vu: The Science Behind the Eerie Feeling of Familiarity]]></title>
            <description><![CDATA[<figure>
    <img src="https://www.thoughtco.com/thmb/llu30lLBRc1W0UWbh86qlp37QN0=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/blurred-motion-on-city-street--hong-kong-852432264-5a8edf2aa18d9e0037a84d01.jpg" referrerpolicy="no-referrer">
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If you’ve ever had the feeling that a situation feels very familiar even though you know it shouldn’t feel familiar at all, like if you’re traveling in a city for the very first time, then you’ve probably experienced <strong>déjà vu.</strong> Déjà vu, which means “already seen” in French, combines <em>objective</em> unfamiliarity – that you know, based on ample evidence, that something shouldn’t be familiar – with <em>subjective</em> familiarity – that feeling that it’s familiar anyway.
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Déjà vu is common. According to a paper published in 2004, more than 50 surveys on déjà vu suggested that about two-thirds of individuals have experienced it at least once in their lifetime, with many reporting multiple experiences. This reported number also appears to be growing as people become more aware of what déjà vu is.
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Most often, déjà vu is described in terms of what you see, but it’s not specific to vision and even people who were born blind can experience it.
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<span class="heading-toc" id="toc-measuring-dj-vu"></span> <h2 id="mntl-sc-block_1-0-6" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Measuring Déjà Vu </span> </h2>
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Déjà vu is difficult to study in the laboratory because it is a fleeting experience, and also because there is no clearly identifiable trigger for it. Nevertheless, researchers have used several tools to study the phenomenon, based on the hypotheses they’ve put forward. Researchers may survey participants; study possibly related processes, especially those involved in memory; or design other experiments to probe déjà vu.
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Because déjà vu is hard to measure, researchers have postulated many explanations for how it works. Below are several of the more prominent hypotheses.
</p>
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<p id="mntl-sc-block_1-0-11" class="comp mntl-sc-block mntl-sc-block-html">
<strong>Memory Explanations</strong>
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Memory explanations of déjà vu are based on the idea that you have previously experienced a situation, or something very much like it, but you don’t <em>consciously</em> remember that you have. Instead, you remember it <em>unconsciously</em>, which is why it feels familiar even though you don’t know why.
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<em>Single element familiarity</em>
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The single element familiarity hypothesis suggests you experience déjà vu if one element of the scene is familiar to you but you don’t consciously recognize it because it’s in a different setting, like if you see your barber out on the street.
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Your brain still finds your barber familiar even if you don’t recognize them, and generalizes that feeling of familiarity to the entire scene. Other researchers have extended this hypothesis to multiple elements as well.
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<em>Gestalt familiarity</em>
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The gestalt familiarity hypothesis focuses on how items are organized in a scene and how déjà vu occurs when you experience something with a similar layout. For example, you may not have seen your friend’s painting in their living room before, but maybe you’ve seen a room that’s laid out like your friend’s living room – a painting hanging over the sofa, across from a bookcase. Since you can’t recall the other room, you experience déjà vu.
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One advantage to the gestalt similarity hypothesis is that it can be more directly tested. In <a href="https://misuse.ncbi.nlm.nih.gov/error/abuse.shtml" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">one study</a>, participants looked at rooms in virtual reality, then were asked how familiar a new room was and whether they felt they were experiencing déjà vu.
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The researchers found that study participants who couldn’t recall the old rooms tended to think a new room was familiar, and that they were experiencing déjà vu, if the new room resembled old ones. Furthermore, the more similar the new room was to an old room, the higher these ratings were.
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<p id="mntl-sc-block_1-0-29" class="comp mntl-sc-block mntl-sc-block-html">
<strong>Neurological Explanations</strong>
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<em>Spontaneous brain activity</em>
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Some explanations posit that&nbsp;déjà vu is experienced when there is spontaneous brain activity unrelated to what you’re currently experiencing. When that happens in the part of your brain dealing with memory, you can have a false feeling of familiarity.
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Some evidence comes from individuals with <a href="https://www.thoughtco.com/temporal-lobes-anatomy-373228" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">temporal lobe </a>epilepsy, when abnormal electrical activity occurs in the part of the brain dealing with memory. When the brains of these patients are electrically stimulated as part of a pre-surgery evaluation, they may experience déjà vu.
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One <a href="https://misuse.ncbi.nlm.nih.gov/error/abuse.shtml" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">researcher</a>&nbsp;suggests that you experience déjà vu when the <a href="https://www.thoughtco.com/hippocampus-anatomy-373221" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">parahippocampal system</a>, which helps identify something as familiar, randomly misfires and makes you think something is familiar when it shouldn’t.&nbsp;
</p>
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<p id="mntl-sc-block_1-0-39" class="comp mntl-sc-block mntl-sc-block-html">
<a href="https://www.clinph-journal.com/article/S1388-2457%2811%2900561-X/fulltext" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1" rel="nofollow">Others have said</a> that déjà vu can’t be isolated to a single familiarity system, but rather involves multiple structures involved in memory and the connections between them.
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<em>Neural transmission speed</em>
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Other hypotheses are based on how fast information travels through your brain. Different areas of your brain transmit information to “higher order” areas that combine the information together to help you make sense of the world. If this complex process is disrupted in any way – perhaps one part sends something more slowly or more quickly than it usually does – then your brain interprets your surroundings incorrectly.
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<span class="heading-toc" id="toc-which-explanation-is-correct"></span> <h2 id="mntl-sc-block_1-0-45" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Which Explanation is Correct? </span> </h2>
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An explanation for déjà vu remains elusive, though the hypotheses above appear to have one common thread: a temporary error in cognitive processing. For now, scientists can continue to design experiments that more directly probe the nature of déjà vu, to be more certain of the correct explanation.
</p>
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<span class="heading-toc" id="toc-sources"></span> <h2 id="mntl-sc-block_1-0-48" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Sources </span> </h2>
<ul id="mntl-sc-block_1-0-49" class="comp mntl-sc-block mntl-sc-block-html">
<li><cite>Tip-of-the-tongue states and related phenomena. Ed. Bennett L. Schwartz and Alan S. Brown. Cambridge University Press. New York, NY 2014. http://www.cambridge.org/gb/academic/subjects/psychology/biological-psychology/tip-tongue-states-and-related-phenomena?format=HB</cite></li>
<li><cite>C. Moulin. The cognitive neuropsychology of déjà vu. Part of the Essays in Cognitive Psychology series. Psychology Press. New York, NY 2018.<a href="https://www.routledge.com/The-Cognitive-Neuropsychology-of-Deja-Vu/Moulin/p/book/9781138696266" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1"> https://www.routledge.com/The-Cognitive-Neuropsychology-of-Deja-Vu/Moulin/p/book/9781138696266</a></cite></li>
<li><cite>Bartolomei, F., Barbeau, E., Gavaret, M., Guye, M., McGonigal, A., Régis, J., and P. Chauvel. “<a href="http://n.neurology.org/content/63/5/858.short" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="2">Cortical stimulation study of the role of rhinal cortex in déjà vu and reminiscence of memories.</a>” <em>Neurology</em>, vol. 63, no. 5, Sept. 2004, pp. 858-864, </cite><cite>doi</cite><cite>:10.1212/01.wnl.0000137037.56916.3f.</cite></li>
<li><cite>J. Spatt. “<a href="https://misuse.ncbi.nlm.nih.gov/error/abuse.shtml" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="3">Déjà vu: possible parahippocampal mechanisms.</a>” <em>The Journal of Neuropsychiatry &amp; Clinical Neurosciences</em>, vol. 14, no. 1, 2002, pp. 6-10, </cite><cite>doi</cite><cite>:10.1176/jnp.14.1.6.</cite></li>
<li><cite>Cleary, A. M., Brown, A. S., Sawyer, B.D., Nomi, J.S., Ajoku, A.C., and A. J. Ryals. “<a href="https://misuse.ncbi.nlm.nih.gov/error/abuse.shtml" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="4">Familiarity from the configuration of objects in 3-dimensional space and its relation to déjà vu: a virtual reality investigation.</a>” <em>Consciousness and Cognition</em>, vol. 21, no. 2, 2012, pp. 969-975, </cite><cite>doi</cite><cite>:10.1016/j.concog.2011.12.010.</cite></li><span class="mntl-sc-block-adslot mntl-sc-block-adslot-inline"></span>
<li><cite>A. S. Brown. The déjà vu experience. Part of the Essays in Cognitive Psychology series. Psychology Press. New York, NY 2004.<a href="https://www.routledge.com/The-Deja-Vu-Experience/Brown/p/book/9780203485446" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="5"> https://www.routledge.com/The-Deja-Vu-Experience/Brown/p/book/9780203485446</a></cite></li>
<li><cite>A. S. Brown. “<a href="https://misuse.ncbi.nlm.nih.gov/error/abuse.shtml" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="6">A review of the déjà vu experience.</a>” <em>Psychology Bulletin</em>, vol. 129, no. 3, 2003, pp. 394-413. </cite><cite>doi</cite><cite>:10.1037/0033-2909.129.3.394.</cite></li>
<li><cite>Bartolomei, F., Barbeau, E. J., Nguyen, T., McGonigal, A., Régis, J., Chauvel, P., and F. Wendling. “<a href="http://www.clinph-journal.com/article/S1388-2457%2811%2900561-X/abstract" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="7" rel="nofollow">Rhinal-hippocampal interactions during déjà vu.</a>” <em>Clinical Neurophysiology</em>, vol. 123, no. 3, March 2012, pp. 489-495. </cite><cite>doi</cite><cite>:10.1016/j.clinph.2011.08.012</cite></li>
</ul>
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</div>
]]></description>
            <pubDate>Thu, 22 Feb 2018 15:31:01 GMT</pubDate>
            <guid isPermaLink="false">https://www.thoughtco.com/causes-of-deja-vu-4159448</guid>
            <link>https://www.thoughtco.com/causes-of-deja-vu-4159448</link>
            <author><![CDATA[Alane Lim]]></author>
                <category>Science</category>
                <category>Tech</category>
                <category>Math</category>
        </item>
        <item>
            <title><![CDATA[7 Extinction Level Events That Could End Life as We Know It]]></title>
            <description><![CDATA[<figure>
    <img src="https://www.thoughtco.com/thmb/v9MJhtNFw_KTAyDdJ8FtqewdvzI=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/GettyImages-724233197-5a834dfc8023b90037be80d3.jpg" referrerpolicy="no-referrer">
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If you've watched the movies "2012" or "Armageddon" or read "On the Beach," you know about some of the threats that could end life as we know it. The <a href="https://www.thoughtco.com/things-you-should-know-about-the-sun-3073449" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Sun</a> could do something nasty. A <a href="https://www.thoughtco.com/astronomy-basics-4133556" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">meteor</a> could strike. We could nuke ourselves out of existence. These are only a few well-known extinction level events. There are so many more ways to die!
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But first, what exactly is an extinction event? An <strong>extinction level event</strong> or ELE is a catastrophe resulting in the extinction of the majority of species on the planet. It's not the normal extinction of species that occurs every day. It isn't necessarily the sterilization of all living organisms. We can identify major extinction events by examining the deposition and chemical composition of rocks, the <a href="https://www.thoughtco.com/famous-fossil-discoveries-1092049" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">fossil record</a>, and evidence of major events on moons and other planets.
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There are dozens of phenomena capable of causing widespread extinctions, but they can be grouped into a few categories:
</p>
</div>
<div id="list-sc__content_1-0" class="comp list-sc__content mntl-sc-page mntl-block" data-sc-sticky-offset="85" data-sc-ad-label-height="24" data-sc-ad-track-spacing="100" data-sc-min-track-height="250" data-sc-max-track-height="600" data-sc-breakpoint="50em" data-sc-load-immediate="3" data-sc-content-positions="[1, 1050, 1250, 1550, 1950, 2350, 2750, 3150]" data-bind-scroll-on-start="true">
<div id="list-sc-item_1-0" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">01</div>
<div class="total">of 09</div>
</div>
</div>
<span class="heading-toc" id="toc-the-sun-will-kill-us"></span> <h2 id="mntl-sc-block_2-0" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Sun Will Kill Us </span> </h2>
<figure id="mntl-sc-block_2-0-1" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
</svg>
<span class="figure-article-caption-text">If a strong solar flare hit the Earth, the results could be devastating.</span>
<span class="figure-article-caption-owner">VICTOR HABBICK VISIONS, Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-2" class="comp mntl-sc-block mntl-sc-block-html">
Life as we know it wouldn't exist without the Sun, but let's be honest. The Sun has it out for planet Earth. Even if none of the other catastrophes on this list ever happen, the Sun will end us. Stars like the Sun burn brighter over time as they fuse hydrogen into helium. In another billion years, it will be about 10 percent brighter. While this might not seem significant, it will cause more water to evaporate. <a href="https://www.thoughtco.com/worst-greenhouse-gases-606789" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Water is&nbsp;a greenhouse gas</a>, so it traps heat in the <a href="https://www.thoughtco.com/most-abundant-gases-in-earths-atmosphere-607594" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">atmosphere</a>, leading to more evaporation. Sunlight will break water into hydrogen and oxygen, so it can <a href="https://www.thoughtco.com/if-earths-atmosphere-vanished-607906" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="3">bleed away into space</a>. Should any life survive, it will meet a fiery fate when the Sun enters its <a href="https://www.thoughtco.com/what-are-stars-really-3073631" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="4">red giant</a> phase, expanding out to the orbit of Mars. It's not likely any life will survive <em>inside</em> the Sun.
</p>
<p id="mntl-sc-block_2-0-3" class="comp mntl-sc-block mntl-sc-block-html">
But, the Sun can kill us any old day it wants <a href="https://www.thoughtco.com/solar-flares-4137226" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">via a coronal mass ejection</a> (CME). As you can guess from the name, this is when our favorite star expels charged particles outward from its corona. Since a CME can sent matter any direction, it doesn't usually shoot directly toward Earth. Sometimes only a tiny fraction of particles reach us, granting us an aurora or a solar storm. However, it's possible for a CME to barbecue the planet.
</p>
<p id="mntl-sc-block_2-0-4" class="comp mntl-sc-block mntl-sc-block-html">
The Sun has pals (and they hate Earth too). A nearby (within 6000 light years) <a href="https://www.thoughtco.com/supernovae-deaths-of-massive-stars-3073301" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">supernova</a>, nova, or gamma ray burst could irradiate organisms and destroy the ozone layer, leaving life&nbsp;at the mercy of the Sun's <a href="https://www.thoughtco.com/definition-of-ultraviolet-radiation-604675" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">ultraviolet radiation</a>. Scientists think a <a href="https://www.thoughtco.com/gamma-ray-burst-destroy-life-earth-3072521" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="3">gamma burst</a> or supernova might have led to the End-Ordovician extinction.
</p>
</div>
<div id="list-sc-item_1-0-1" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-1" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">02</div>
<div class="total">of 09</div>
</div>
</div>
<span class="heading-toc" id="toc-geomagnetic-reversals-may-kill-us"></span> <h2 id="mntl-sc-block_2-0-5" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Geomagnetic Reversals May Kill Us </span> </h2>
<figure id="mntl-sc-block_2-0-6" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/7CfPJl1EDfXgckaZvnvIqPRDuDA=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/GettyImages-693943194-5a84493fd8fdd500375c2f46.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-1" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
</svg>
<span class="figure-article-caption-text">Scientists believe magnetic pole reversals were involved in some past mass extinctions.</span>
<span class="figure-article-caption-owner">siiixth, Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-7" class="comp mntl-sc-block mntl-sc-block-html">
The Earth is a giant magnet that has a love-hate relationship with life. The magnetic field protects us from the worst the Sun throws at us. Every so often, the positions of the <a href="https://www.thoughtco.com/what-is-magnetic-reversal-1435340" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">north and south magnetic poles flip</a>. How often the reversals occur and how long it takes the magnetic field to get settled&nbsp;is highly variable. Scientists aren't completely sure what will happen when the poles flip. Maybe nothing. Or maybe the weakened magnetic field will expose the Earth to the <a href="https://www.thoughtco.com/the-effects-of-solar-storms-3073703" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">solar wind</a>, letting the Sun steal a lot of our oxygen. You know, that gas humans breathe. Scientists say magnetic field reversals aren't always extinction level events. Just sometimes.
</p>
</div>
<div id="list-sc-item_1-0-2" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-2" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">03</div>
<div class="total">of 09</div>
</div>
</div>
<span class="heading-toc" id="toc-the-big-bad-meteor"></span> <h2 id="mntl-sc-block_2-0-8" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Big Bad Meteor </span> </h2>
<figure id="mntl-sc-block_2-0-9" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/fRhKUp4HvIhoOQA3RRt9eSsNA3w=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/GettyImages-532101243-5a844ef13418c6003624983e.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-2" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
</svg>
<span class="figure-article-caption-text">A big meteor impact could be an extinction level event.</span>
<span class="figure-article-caption-owner">Marc Ward/Stocktrek Images, Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-10" class="comp mntl-sc-block mntl-sc-block-html">
You may be surprised to learn the impact of an asteroid or meteor has only been connected with certainty to one mass extinction, the Cretaceous-Paleogene extinction event. Other impacts have been contributing factors to extinctions, but not the primary cause.
</p>
<p id="mntl-sc-block_2-0-11" class="comp mntl-sc-block mntl-sc-block-html">
The good news is that <a href="https://nypost.com/2016/12/14/nasa-scientist-warns-earth-is-due-for-extinction-level-event/" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">NASA claims</a> about 95 percent of comets and asteroids bigger than 1 kilometer in diameter have been identified. The other good news is that scientists estimate an object needs to be about 100 kilometers (60 miles) across to wipe out all life. The bad news is there are another 5 percent out there and not much we can do about a significant threat with our present technology (no, Bruce Willis cannot detonate a nuke and save us).
</p>
<p id="mntl-sc-block_2-0-12" class="comp mntl-sc-block mntl-sc-block-html">
Obviously, living things at ground zero for a meteor strike will die. Many more will die from the shock wave, earthquakes, tsunamis, and firestorms. Those that survive the initial impact would have a hard time finding food, as the debris thrown into the atmosphere would change the climate, leading to mass extinctions. You're probably better off at ground zero for this one.
</p>
</div>
<div id="list-sc-item_1-0-3" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-3" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">04</div>
<div class="total">of 09</div>
</div>
</div>
<span class="heading-toc" id="toc-the-sea"></span> <h2 id="mntl-sc-block_2-0-13" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Sea </span> </h2>
<figure id="mntl-sc-block_2-0-14" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/v_x1oZvDSXc0FtAdEggw2a_eenM=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/GettyImages-128139136-5a8467ef3418c6003645ddc5.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-3" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
</svg>
<span class="figure-article-caption-text">A tsunami is dangerous, but the sea has more lethal tricks.</span>
<span class="figure-article-caption-owner">Bill Romerhaus, Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-15" class="comp mntl-sc-block mntl-sc-block-html">
A day at the beach might seem idyllic, until you realize the blue part of the marble we call Earth is deadlier than all of the sharks in its depths. The ocean has various ways of causing ELEs.
</p>
<p id="mntl-sc-block_2-0-16" class="comp mntl-sc-block mntl-sc-block-html">
Methane clathrates (molecules made of water and methane) sometimes break from the continental shelves, producing a methane eruption called a clathrate gun. The "gun" shoots immense amounts of the greenhouse gas methane into the atmosphere. Such events are linked to the <a href="https://www.thoughtco.com/permian-period-300-250-million-years-1091430" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">end-Permian extinction</a> and Paleocene-Eocene Thermal Maximum.
</p>
<p id="mntl-sc-block_2-0-17" class="comp mntl-sc-block mntl-sc-block-html">
Prolonged sea level rise or fall also leads to extinctions. Falling sea levels are more insidious, as exposing the continental shelf kills off innumerable marine species. This, in turn, upsets the terrestrial ecosystem, leading to an ELE.
</p>
<p id="mntl-sc-block_2-0-18" class="comp mntl-sc-block mntl-sc-block-html">
Chemical imbalances in the sea also cause extinction events. When the middle or upper layers of the ocean become anoxic, a chain reaction of death occurs. The Ordovician-Silurian, late Devonian, Permian-Triassic, and Triassic-Jurassic extinctions all included anoxic events.
</p>
<p id="mntl-sc-block_2-0-19" class="comp mntl-sc-block mntl-sc-block-html">
Sometimes the levels of essential trace elements (e.g., <a href="https://www.thoughtco.com/selenium-facts-606594" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">selenium</a>) fall, leading to mass extinctions. Sometimes the sulfate-reducing bacteria in thermal vents get out of control, releasing an excess of hydrogen sulfide that weakens the ozone layer, exposing life to lethal UV. The ocean also undergoes a periodic overturn in which the high-salinity surface water sinks to the depths. Anoxic deep water rises, killing surface organisms. The late-Devonian and <a href="https://www.thoughtco.com/the-permian-triassic-extinction-event-1092136" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">Permian-Triassic extinctions</a> are associated oceanic overturn.
</p>
<p id="mntl-sc-block_2-0-20" class="comp mntl-sc-block mntl-sc-block-html">
The beach doesn't look so nice now, does it?
</p>
</div>
<div id="list-sc-item_1-0-4" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-4" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">05</div>
<div class="total">of 09</div>
</div>
</div>
<span class="heading-toc" id="toc-and-the-winner-is-volcanoes"></span> <h2 id="mntl-sc-block_2-0-21" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> And the "Winner" Is... Volcanoes </span> </h2>
<figure id="mntl-sc-block_2-0-22" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/B6IEDBoM4dSm5JDu9JNS8j8L2xQ=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/GettyImages-675488626-5a84688e312834003700b60f.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-4" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
</svg>
<span class="figure-article-caption-text">Historically, most extinction level events have been caused by volcanoes.</span>
<span class="figure-article-caption-owner">Mike Lyvers, Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-23" class="comp mntl-sc-block mntl-sc-block-html">
While falling sea level has been associated with 12 extinction events, only seven involved a significant loss of species. On the other hand, <a href="https://www.thoughtco.com/what-happens-when-a-volcano-erupts-4151722" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">volcanoes</a> have led to 11 ELEs, <em>all</em> of them significant. The End-Permian, End-Triassic, and <a href="https://www.thoughtco.com/the-k-t-extinction-event-1092141" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">End-Cretaceous extinctions</a> are&nbsp;associated with volcanic eruptions called flood basalt events. Volcanoes kill by releasing dust, sulfur oxides, and <a href="https://www.thoughtco.com/carbon-dioxide-molecular-formula-608475" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="3">carbon dioxide</a> that collapse food chains by inhibiting photosynthesis, poison the land and sea with acid rain, and produce global warming. The 

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http://localhost:1200/thoughtco/math-4133545 - Success ✔️

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        <description><![CDATA[Struggling with scatterplots? Can't quite wrap your head around circumference? Here are resources and tutorials for all the major functions, formulas, equations, and theories you'll encounter in math class. Teachers can find useful math resources for the classroom. - Made with love by RSSHub(https://github.com/DIYgod/RSSHub)]]></description>
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            <title><![CDATA[Why Mathematics Is a Language]]></title>
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Mathematics is called the language of science. Italian astronomer and physicist <a href="https://www.thoughtco.com/galileo-galilei-biography-1991864" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Galileo Galilei</a> is attributed with the quote, "<em>Mathematics is the language in which God has written the universe</em>." Most likely this quote is a summary of his statement in&nbsp;<em>Opere Il Saggiatore:</em>
</p>
<blockquote id="mntl-sc-block_1-0-1" class="comp mntl-sc-block mntl-sc-block-html">
[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.
</blockquote>
<p id="mntl-sc-block_1-0-2" class="comp mntl-sc-block mntl-sc-block-html">
Yet, is mathematics truly a language, like English or Chinese? To answer the question, it helps to know what language is and how the vocabulary and grammar of mathematics are used to construct sentences.
</p>
<div id="mntl-sc-block_1-0-3" class="comp theme-keytakeaways mntl-sc-block reference-sc-block-callout mntl-sc-block-callout mntl-block" data-tracking-id="mntl-sc-block-callout" data-tracking-container="true">
<h3 id="mntl-sc-block-callout-heading_1-0" class="comp mntl-sc-block-callout-heading mntl-text-block">
Key Takeaways: Why Math is a Language</h3>
<div id="mntl-sc-block-callout-body_1-0" class="comp mntl-sc-block-callout-body mntl-text-block">
<ul><li>In order to be considered a language, a system of communication must have vocabulary, grammar, syntax, and people who use and understand it.</li><li>Mathematics meets this definition of a language. Linguists who don't consider math a language cite its use as a written rather than spoken form of communication.</li><li>Math is a universal language. The symbols and organization to form equations are the same in every country of the world.</li></ul></div>
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<div id="list-sc__content_1-0" class="comp list-sc__content mntl-sc-page mntl-block" data-sc-sticky-offset="85" data-sc-ad-label-height="24" data-sc-ad-track-spacing="100" data-sc-min-track-height="250" data-sc-max-track-height="600" data-sc-breakpoint="50em" data-sc-load-immediate="3" data-sc-content-positions="[1, 1050, 1250, 1550, 1950, 2350, 2750, 3150]" data-bind-scroll-on-start="true">
<div id="list-sc-item_1-0" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0" class="comp list-marker list-marker--none"></div>
<span class="heading-toc" id="toc-what-is-a-language"></span> <h2 id="mntl-sc-block_2-0" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> What Is a Language? </span> </h2>
<p id="mntl-sc-block_2-0-1" class="comp mntl-sc-block mntl-sc-block-html">
There are multiple definitions of "<a href="https://www.thoughtco.com/what-is-a-language-1691218" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">language</a>." A language may be a system of words or codes used within a discipline. Language may refer to a system of communication using symbols or sounds. <a href="https://www.thoughtco.com/what-is-a-linguist-1691239" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">Linguist</a> Noam Chomsky defined language as a set of sentences constructed using a finite set of elements. Some linguists believe language should be able to represent events and abstract concepts.
</p>
<p id="mntl-sc-block_2-0-2" class="comp mntl-sc-block mntl-sc-block-html">
Whichever definition is used, a language contains the following components:
</p>
<ul id="mntl-sc-block_2-0-3" class="comp mntl-sc-block mntl-sc-block-html">
<li>There must be a <strong>vocabulary</strong> of words or symbols.</li><li><strong>Meaning</strong> must be attached to the words or symbols.</li><li>A language employs <strong>grammar</strong>, which is a set of rules that outline how vocabulary is used.</li><li>A <strong>syntax</strong> organizes symbols into linear structures or propositions.</li><li>A <strong>narrative</strong> or discourse consists of strings of syntactic propositions.</li><li>There must be (or have been) a group of people who use and understand the symbols.</li>
</ul>
<p id="mntl-sc-block_2-0-4" class="comp mntl-sc-block mntl-sc-block-html">
Mathematics meets all of these requirements. The symbols, their meanings, syntax, and grammar are the same throughout the world. Mathematicians, scientists, and others use math to communicate concepts. Mathematics describes itself (a field called meta-mathematics), real-world phenomena, and abstract concepts.
</p>
</div>
<div id="list-sc-item_1-0-1" class="comp mntl-sc-list-item list-sc-item mntl-block">
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<span class="heading-toc" id="toc-vocabulary-grammar-and-syntax-in-mathematics"></span> <h2 id="mntl-sc-block_2-0-5" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Vocabulary, Grammar, and Syntax in Mathematics </span> </h2>
<figure id="mntl-sc-block_2-0-6" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/N2hMLp5mKoTWbfT8wpxt1VZ25eM=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/student-writing-on-blackboard-695556138-5a6e40fdeb97de0037e8a0c6.jpg" referrerpolicy="no-referrer">
  </figure>
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<span class="figure-article-caption-owner">Emilija Manevska / Getty Images</span>
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<p id="mntl-sc-block_2-0-7" class="comp mntl-sc-block mntl-sc-block-html">
The vocabulary of math draws from many different alphabets and includes symbols unique to math. A mathematical equation may be stated in words to form a sentence that has a noun and a verb, just like a sentence in a spoken language. For example:
</p>
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3 + 5 = 8
</p>
<p id="mntl-sc-block_2-0-9" class="comp mntl-sc-block mntl-sc-block-html">
could be stated as "Three added to five equals eight."
</p>
<p id="mntl-sc-block_2-0-10" class="comp mntl-sc-block mntl-sc-block-html">
Breaking this down, <a href="https://www.thoughtco.com/what-is-concrete-noun-1689904" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">nouns</a> in math include:
</p>
<ul id="mntl-sc-block_2-0-11" class="comp mntl-sc-block mntl-sc-block-html">
<li>Arabic numerals (0, 5, 123.7)</li><li>Fractions (1⁄4, 5⁄9, 2 1⁄3)</li><li>Variables (a, b, c, x, y, z)</li><li>Expressions (3x, x<sup>2</sup>, 4 + x)</li><li>Diagrams or visual elements (circle, angle, triangle, tensor, matrix)</li><li>Infinity (∞)</li><li>Pi (π)</li><li>Imaginary numbers (i, -i)</li><li>The speed of light (c)</li>
</ul>
<p id="mntl-sc-block_2-0-12" class="comp mntl-sc-block mntl-sc-block-html">
Verbs include symbols including:
</p>
<ul id="mntl-sc-block_2-0-13" class="comp mntl-sc-block mntl-sc-block-html">
<li>Equalities or inequalities (=, &lt;, &gt;)</li><li>Actions such as addition, subtraction, multiplication, and division (+, -, x or *, ÷ or /)</li><li>Other operations (sin, cos, tan, sec)</li>
</ul>
<p id="mntl-sc-block_2-0-14" class="comp mntl-sc-block mntl-sc-block-html">
If you try to perform a sentence diagram on a mathematical sentence, you'll find infinitives, conjunctions, adjectives, etc. As in other languages, the role played by a symbol depends on its context.
</p>
<span class="heading-toc" id="toc-international-rules"></span> <h2 id="mntl-sc-block_2-0-15" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> International Rules </span> </h2>
<p id="mntl-sc-block_2-0-16" class="comp mntl-sc-block mntl-sc-block-html">
Mathematics grammar and syntax, like vocabulary, are international. No matter what country you're from or what language you speak, the structure of the mathematical language is the same.
</p>
<ul id="mntl-sc-block_2-0-17" class="comp mntl-sc-block mntl-sc-block-html">
<li>Formulas are read from left to right.</li>
<li>The Latin alphabet is used for parameters and variables. To some extent, the Greek alphabet is also used. <a href="https://www.thoughtco.com/understanding-classification-of-numbers-2312407" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Integers</a> are usually drawn from <em>i</em>, <em>j</em>, <em>k</em>, <em>l</em>, <em>m</em>, <em>n</em>. <a href="https://www.thoughtco.com/what-is-a-real-number-3126307" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">Real numbers</a> are represented by&nbsp;<em>a</em>,&nbsp;<em>b</em>,&nbsp;<em>c</em>, α<em>, β</em>, γ. Complex numbers are indicated by <em>w</em> and <em>z</em>. Unknowns are <em>x</em>, <em>y</em>, <em>z</em>. Names of functions are usually <em>f</em>, <em>g</em>, <em>h</em>.</li>
<li>The Greek alphabet is used to represent specific concepts. For example, λ is used to indicate wavelength and ρ means density.</li>
<li>Parentheses and brackets indicate the <a href="https://www.thoughtco.com/definition-of-bedmas-2312372" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="3">order in which the symbols interact</a>.</li>
<li>The way functions, integrals, and derivatives are phrased is uniform.</li>
</ul>
</div>
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<span class="heading-toc" id="toc-language-as-a-teaching-tool"></span> <h2 id="mntl-sc-block_2-0-18" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Language as a Teaching Tool </span> </h2>
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    <img src="https://www.thoughtco.com/thmb/3FBkjxj7Vj_djIbgjhs5Asro2ho=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/question-marks-on-black-background-184837701-5a6df53904d1cf00378895cb.jpg" referrerpolicy="no-referrer">
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<span class="figure-article-caption-owner">StockFinland / Getty Images</span>
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<p id="mntl-sc-block_2-0-20" class="comp mntl-sc-block mntl-sc-block-html">
Understanding how mathematical sentences work is helpful when teaching or learning math. Students often find numbers and symbols intimidating, so putting an equation into a familiar language makes the subject more approachable. Basically, it's like translating a foreign language into a known one.
</p>
<p id="mntl-sc-block_2-0-21" class="comp mntl-sc-block mntl-sc-block-html">
While students typically dislike word problems, extracting the nouns, verbs, and modifiers from a spoken/written language and translating them into a mathematical equation is a valuable skill to have. Word problems improve comprehension and increase problem-solving skills.
</p>
<p id="mntl-sc-block_2-0-22" class="comp mntl-sc-block mntl-sc-block-html">
Because mathematics is the same all over the world, math can act as a universal language. A phrase or formula has the same meaning, regardless of another language that accompanies it. In this way, math helps people learn and communicate, even if other communication barriers exist.
</p>
</div>
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<span class="heading-toc" id="toc-the-argument-against-math-as-a-language"></span> <h2 id="mntl-sc-block_2-0-23" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Argument Against Math as a Language </span> </h2>
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    <img src="https://www.thoughtco.com/thmb/ZMMuwx9pPqpL8vJSpD30sCDqkjc=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/maxwellsequations-5a6dea1deb97de0037dbb20b.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-2" class="comp mntl-figure-caption figure-article-caption">
<span class="figure-article-caption-owner">Anne Helmenstine</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-25" class="comp mntl-sc-block mntl-sc-block-html">
Not everyone agrees that mathematics is a language. Some definitions of "language" describe it as a spoken form of communication. Mathematics is a written form of communication. While it may be easy to read a simple addition statement aloud (e.g., 1 + 1 = 2), it's much harder to read other equations aloud (e.g., Maxwell's equations). Also, the spoken statements would be rendered in the speaker's native language, not a universal tongue.
</p>
<p id="mntl-sc-block_2-0-26" class="comp mntl-sc-block mntl-sc-block-html">
However, sign language would also be disqualified based on this criterion. Most linguists accept sign language as a true language.&nbsp;There are a handful of dead languages that no one alive knows how to pronounce or even read anymore.
</p>
<p id="mntl-sc-block_2-0-27" class="comp mntl-sc-block mntl-sc-block-html">
A strong case for mathematics as a language is that modern elementary-high school curricula uses techniques from language education for teaching mathematics. Educational psychologist Paul Riccomini and colleagues wrote that students learning mathematics require "a robust vocabulary knowledge base; flexibility; fluency and proficiency with numbers, symbols, words, and diagrams; and comprehension skills."
</p>
<span class="heading-toc" id="toc-sources"></span> <h2 id="mntl-sc-block_2-0-28" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Sources </span> </h2>
<ul id="mntl-sc-block_2-0-29" class="comp mntl-sc-block mntl-sc-block-html">
<li><cite>Ford, Alan, and F. David Peat. "<a href="https://doi.org/10.1007/BF01889434" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">The Role of Language in Science</a>." <em>Foundations of Physics</em> 18.12 (1988): 1233–42.&nbsp;</cite></li>
<li><cite>Galilei, Galileo. "'The Assayer' ('Il Saggiatore' in Italian) (Rome, 1623)." <em>The Controversy on the Comets of 1618</em>. Eds. Drake, Stillman and C. D. O'Malley. Philadelphia: University of Pennsylvania Press, 1960.&nbsp;</cite></li>
<li><cite>Klima, Edward S., and Ursula Bellugi. "The Signs of Language. "Cambridge, MA: Harvard University Press, 1979.&nbsp;</cite></li>
<li><cite>Riccomini, Paul J., et al. "<a href="https://doi.org/10.1080/10573569.2015.1030995" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="2">The Language of Mathematics: The Importance of Teaching and Learning Mathematical Vocabulary</a>." <em>Reading &amp; Writing Quarterly</em> 31.3 (2015): 235-52. Print.</cite></li>
</ul>
</div>
</div>
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]]></description>
            <pubDate>Mon, 29 Jan 2018 19:48:57 GMT</pubDate>
            <guid isPermaLink="false">https://www.thoughtco.com/why-mathematics-is-a-language-4158142</guid>
            <link>https://www.thoughtco.com/why-mathematics-is-a-language-4158142</link>
            <author><![CDATA[Anne Marie Helmenstine, Ph.D.]]></author>
                <category>Science</category>
                <category>Tech</category>
                <category>Math</category>
        </item>
        <item>
            <title><![CDATA[Here's How Equivalent Systems of Equations Work]]></title>
            <description><![CDATA[<figure>
    <img src="https://www.thoughtco.com/thmb/xr1EIfRZtWAHfml2WyLPPa0MWrI=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/GettyImages-595348759-5372ddd37cf44fdb957bdb81028ef7ed.jpg" referrerpolicy="no-referrer">
  </figure>
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<p id="mntl-sc-block_1-0" class="comp mntl-sc-block mntl-sc-block-html">
Equivalent equations are systems of equations that have the same solutions. Identifying and solving equivalent equations is a valuable skill, not only in <a href="https://www.thoughtco.com/what-is-algebra-why-take-algebra-2311937" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">algebra class</a> but also in everyday life. Take a look at examples of equivalent equations, how to solve them for one or more variables, and how you might use this skill outside a classroom.
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<h3 id="mntl-sc-block-callout-heading_1-0" class="comp mntl-sc-block-callout-heading mntl-text-block">
Key Takeaways</h3>
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<ul><li>Equivalent equations are algebraic equations that have identical solutions or roots.</li><li>Adding or subtracting the same number or expression to both sides of an equation produces an equivalent equation.</li><li>Multiplying or dividing both sides of an equation by the same non-zero number produces an equivalent equation.</li></ul></div>
</div>
<span class="heading-toc" id="toc-linear-equations-with-one-variable"></span> <h2 id="mntl-sc-block_1-0-3" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Linear Equations With One Variable </span> </h2>
<p id="mntl-sc-block_1-0-4" class="comp mntl-sc-block mntl-sc-block-html">
The simplest examples of equivalent equations don't have any variables. For example, these three equations are equivalent to each other:
</p>
<div id="mntl-sc-block_1-0-5" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-6" class="comp mntl-sc-block mntl-sc-block-html">
<li>3 + 2 = 5</li><li>4 + 1 = 5</li><li>5 + 0 = 5</li>
</ul>
<div id="mntl-sc-block_1-0-7" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-8" class="comp mntl-sc-block mntl-sc-block-html">
Recognizing these equations are equivalent is great, but not particularly useful. Usually, an equivalent equation problem asks you to solve for a variable to see if it is the same (the same <strong>root</strong>) as the one in another equation.
</p>
<div id="mntl-sc-block_1-0-9" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-10" class="comp mntl-sc-block mntl-sc-block-html">
For example, the following equations are equivalent:
</p>
<div id="mntl-sc-block_1-0-11" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-12" class="comp mntl-sc-block mntl-sc-block-html">
<li>x = 5</li><li>-2x = -10</li>
</ul>
<div id="mntl-sc-block_1-0-13" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-14" class="comp mntl-sc-block mntl-sc-block-html">
In both cases, x = 5. How do we know this? How do you solve this for the "-2x = -10" equation? The first step is to know the rules of equivalent equations:
</p>
<div id="mntl-sc-block_1-0-15" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-16" class="comp mntl-sc-block mntl-sc-block-html">
<li><a href="https://www.thoughtco.com/3-digit-addition-worksheets-with-regrouping-2311922" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Adding</a> or subtracting the same number or expression to both sides of an equation produces an equivalent equation.</li>
<li>Multiplying or dividing both sides of an equation by the same non-zero number produces an equivalent equation.</li>
<li>Raising both sides of the equation to the <a href="https://www.thoughtco.com/exponents-and-bases-2312002" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">same odd power</a> or taking the same odd root will produce an equivalent equation.</li>
<li>If both sides of an equation are non-<a href="https://www.thoughtco.com/calculations-with-negative-numbers-2312514" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="3">negative</a>, raising both sides of an equation to the same even power or taking the same even root will give an equivalent equation.</li>
</ul>
<div id="mntl-sc-block_1-0-17" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<span class="heading-toc" id="toc-example"></span> <h2 id="mntl-sc-block_1-0-18" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Example </span> </h2>
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Putting these rules into practice, determine whether these two equations are equivalent:
</p>
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<ul id="mntl-sc-block_1-0-21" class="comp mntl-sc-block mntl-sc-block-html">
<li>x + 2 = 7</li><li>2x + 1 = 11</li>
</ul>
<div id="mntl-sc-block_1-0-22" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-23" class="comp mntl-sc-block mntl-sc-block-html">
To solve this, you need to find "x" for each <a href="https://www.thoughtco.com/exercise-worksheets-using-foil-2312026" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">equation</a>. If "x" is the same for both equations, then they are equivalent. If "x" is different (i.e., the equations have different roots), then the equations are not equivalent. For the first equation:
</p>
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<ul id="mntl-sc-block_1-0-25" class="comp mntl-sc-block mntl-sc-block-html">
<li>x + 2 = 7</li><li>x + 2 - 2 = 7 - 2 (subtracting both sides by same number)</li><li>x = 5</li>
</ul>
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<p id="mntl-sc-block_1-0-27" class="comp mntl-sc-block mntl-sc-block-html">
For the second equation:
</p>
<div id="mntl-sc-block_1-0-28" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-29" class="comp mntl-sc-block mntl-sc-block-html">
<li>2x + 1 = 11</li><li>2x + 1 - 1 = 11 - 1 (subtracting both sides by the same number)</li><li>2x = 10</li><li>2x/2&nbsp;= 10/2 (dividing both sides of the equation by the same number)</li><li>x = 5</li>
</ul>
<div id="mntl-sc-block_1-0-30" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-31" class="comp mntl-sc-block mntl-sc-block-html">
So, yes, the two equations are equivalent because x = 5 in each case.
</p>
<div id="mntl-sc-block_1-0-32" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<span class="heading-toc" id="toc-practical-equivalent-equations"></span> <h2 id="mntl-sc-block_1-0-33" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Practical Equivalent Equations </span> </h2>
<p id="mntl-sc-block_1-0-34" class="comp mntl-sc-block mntl-sc-block-html">
You can use equivalent equations in daily life. It's particularly helpful when shopping. For example, you like a particular shirt. One company offers the shirt for $6 and has $12 shipping, while another company offers the shirt for $7.50 and has $9 shipping. Which shirt has the best price? How many shirts (maybe you want to get them for friends) would you have to buy for the price to be the same for both companies?
</p>
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<p id="mntl-sc-block_1-0-36" class="comp mntl-sc-block mntl-sc-block-html">
To solve this problem, let "x" be the number of shirts. To start with, set x =1 for the purchase of one shirt. For company #1:
</p>
<div id="mntl-sc-block_1-0-37" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-38" class="comp mntl-sc-block mntl-sc-block-html">
<li>Price = 6x + 12 = (6)(1) + 12 = 6 + 12 = $18</li>
</ul>
<div id="mntl-sc-block_1-0-39" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-40" class="comp mntl-sc-block mntl-sc-block-html">
For company #2:
</p>
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<ul id="mntl-sc-block_1-0-42" class="comp mntl-sc-block mntl-sc-block-html">
<li>Price = 7.5x + 9 = (1)(7.5) + 9 = 7.5 + 9 = $16.50</li>
</ul>
<div id="mntl-sc-block_1-0-43" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-44" class="comp mntl-sc-block mntl-sc-block-html">
So, if you're buying one shirt, the second company offers a better deal.
</p>
<div id="mntl-sc-block_1-0-45" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-46" class="comp mntl-sc-block mntl-sc-block-html">
To find the point where prices are equal, let "x" remain the number of shirts, but set the two equations equal to each other. Solve for "x" to find how many shirts you'd have to buy:
</p>
<div id="mntl-sc-block_1-0-47" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-48" class="comp mntl-sc-block mntl-sc-block-html">
<li>6x + 12 = 7.5x + 9</li>
<li>6x - 7.5x = 9 - 12 (<a href="https://www.thoughtco.com/2-digit-subtraction-worksheets-with-regrouping-2311924" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">subtracting</a> the same numbers or expressions from each side)</li>
<li>-1.5x = -3</li>
<li>1.5x = 3 (dividing both sides by the same number, -1)</li>
<li>x = 3/1.5 (dividing both sides by 1.5)</li>
<li>x = 2</li>
</ul>
<div id="mntl-sc-block_1-0-49" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-50" class="comp mntl-sc-block mntl-sc-block-html">
If you buy two shirts, the price is the same, no matter where you get it. You can use the same math to determine which company gives you a better deal with larger orders and also to calculate how much you'll save using one company over the other. See, algebra is useful!
</p>
<div id="mntl-sc-block_1-0-51" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<span class="heading-toc" id="toc-equivalent-equations-with-two-variables"></span> <h2 id="mntl-sc-block_1-0-52" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Equivalent Equations With Two Variables </span> </h2>
<p id="mntl-sc-block_1-0-53" class="comp mntl-sc-block mntl-sc-block-html">
If you have two equations and two unknowns (x and y), you can determine whether two sets of linear equations are equivalent.
</p>
<div id="mntl-sc-block_1-0-54" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-55" class="comp mntl-sc-block mntl-sc-block-html">
For example, if you're given the equations:
</p>
<div id="mntl-sc-block_1-0-56" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-57" class="comp mntl-sc-block mntl-sc-block-html">
<li>-3x + 12y = 15</li><li>7x - 10y = -2</li>
</ul>
<div id="mntl-sc-block_1-0-58" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-59" class="comp mntl-sc-block mntl-sc-block-html">
You can determine whether the following system is equivalent:
</p>
<div id="mntl-sc-block_1-0-60" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-61" class="comp mntl-sc-block mntl-sc-block-html">
<li>-x + 4y = 5</li><li>7x -10y = -2</li>
</ul>
<div id="mntl-sc-block_1-0-62" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-63" class="comp mntl-sc-block mntl-sc-block-html">
To <a href="https://www.thoughtco.com/problem-solving-in-mathematics-2311775" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">solve this problem</a>, find "x" and "y" for each system of equations. If the values are the same, then the systems of equations are equivalent.
</p>
<div id="mntl-sc-block_1-0-64" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-65" class="comp mntl-sc-block mntl-sc-block-html">
Start with the first set. To solve two <a href="https://www.thoughtco.com/solve-algebra-problems-step-by-step-2311970" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">equations</a> with two <a href="https://www.thoughtco.com/what-is-a-scientific-variable-3975929" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">variables</a>, isolate one variable and plug its solution into the other equation. To isolate the "y" variable:
</p>
<div id="mntl-sc-block_1-0-66" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-67" class="comp mntl-sc-block mntl-sc-block-html">
<li>-3x + 12y = 15</li><li>-3x = 15 - 12y</li><li>x = -(15 - 12y)/3 = -5 + 4y (plug in for "x" in the second equation)</li><li>7x - 10y = -2</li><li>7(-5 + 4y) - 10y = -2</li><li>-35 + 28y - 10y = -2</li><li>18y = 33</li><li>y = 33/18 = 11/6</li>
</ul>
<div id="mntl-sc-block_1-0-68" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-69" class="comp mntl-sc-block mntl-sc-block-html">
Now, plug "y" back into either equation to solve for "x":
</p>
<div id="mntl-sc-block_1-0-70" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-71" class="comp mntl-sc-block mntl-sc-block-html">
<li>7x - 10y = -2</li><li>7x = -2 + 10(11/6)</li>
</ul>
<div id="mntl-sc-block_1-0-72" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-73" class="comp mntl-sc-block mntl-sc-block-html">
Working through this, you'll eventually&nbsp;get x = 7/3.
</p>
<div id="mntl-sc-block_1-0-74" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-75" class="comp mntl-sc-block mntl-sc-block-html">
To answer the question,&nbsp;you <em>could</em> apply the same principles to the second set of equations to solve for "x" and "y" to find that yes, they are indeed equivalent. It's easy to get bogged down in the algebra, so it's a good idea to check your work using an <a href="http://www.webmath.com/solver2.html" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">online equation solver</a>.
</p>
<div id="mntl-sc-block_1-0-76" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-77" class="comp mntl-sc-block mntl-sc-block-html">
However, the clever student will notice the two sets of equations are equivalent <em>without doing any difficult calculations at all.</em> The only difference between the first equation in each set is that the first one is three times the second one (equivalent). The second equation is exactly the same.
</p>
</div>
</div>
]]></description>
            <pubDate>Wed, 17 Jan 2018 18:49:52 GMT</pubDate>
            <guid isPermaLink="false">https://www.thoughtco.com/understanding-equivalent-equations-4157661</guid>
            <link>https://www.thoughtco.com/understanding-equivalent-equations-4157661</link>
            <author><![CDATA[Anne Marie Helmenstine, Ph.D.]]></author>
                <category>Science</category>
                <category>Tech</category>
                <category>Math</category>
        </item>
        <item>
            <title><![CDATA[10 Math Tricks That Will Blow Your Mind]]></title>
            <description><![CDATA[<figure>
    <img src="https://www.thoughtco.com/thmb/FGRV9FpTnS_WqfKU6xTvkI9vI28=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/Math_class_in_Da_Ji_Junior_High_School_2006-12-1-89e9cb05eb2a4d53a503a97e3f3b1ae1.jpg" referrerpolicy="no-referrer">
  </figure>
<div id="list-sc_1-0" class="comp article-content list list-sc mntl-block">
<div id="article-intro_1-0" class="comp article-content__description article-intro mntl-block">
<p id="mntl-sc-block_1-0" class="comp mntl-sc-block mntl-sc-block-html">
Are you ready to give your mathematics skills a boost? These simple math tricks can help you perform calculations more quickly and easily. They also come in handy if you want to impress your teacher, parents, or friends.
</p>
</div>
<div id="list-sc__content_1-0" class="comp list-sc__content mntl-sc-page mntl-block" data-sc-sticky-offset="85" data-sc-ad-label-height="24" data-sc-ad-track-spacing="100" data-sc-min-track-height="250" data-sc-max-track-height="600" data-sc-breakpoint="50em" data-sc-load-immediate="3" data-sc-content-positions="[1, 1050, 1250, 1550, 1950, 2350, 2750, 3150]" data-bind-scroll-on-start="true">
<div id="list-sc-item_1-0" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">01</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-multiplying-by-6"></span> <h2 id="mntl-sc-block_2-0" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Multiplying by 6 </span> </h2>
<p id="mntl-sc-block_2-0-1" class="comp mntl-sc-block mntl-sc-block-html">
If you multiply 6 by an even number, the answer will end with the same digit. The number in the ten's place will be half of the number in the one's place.
</p>
<p id="mntl-sc-block_2-0-2" class="comp mntl-sc-block mntl-sc-block-html">
Example: 6 x 4 = 24.
</p>
</div>
<div id="list-sc-item_1-0-1" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-1" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">02</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-the-answer-is-2"></span> <h2 id="mntl-sc-block_2-0-3" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Answer Is 2 </span> </h2>
<ol id="mntl-sc-block_2-0-4" class="comp mntl-sc-block mntl-sc-block-html">
<li>Think of a number.</li><li>Multiply it by 3.</li><li>Add 6.</li><li>Divide this number by 3.</li><li>Subtract the number from Step 1 from the answer in Step 4.</li>
</ol>
<p id="mntl-sc-block_2-0-5" class="comp mntl-sc-block mntl-sc-block-html">
The answer is 2.
</p>
</div>
<div id="list-sc-item_1-0-2" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-2" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">03</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-same-three-digit-number"></span> <h2 id="mntl-sc-block_2-0-6" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Same Three-Digit Number </span> </h2>
<ol id="mntl-sc-block_2-0-7" class="comp mntl-sc-block mntl-sc-block-html">
<li>Think of any three-digit number in which each of the digits is the same. Examples include 333, 666, 777, and 999.</li>
<li>Add up the digits.</li>
<li><a href="https://www.thoughtco.com/basic-division-facts-no-remainder-2312184" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Divide</a> the three-digit number by the answer in Step 2.</li>
</ol>
<p id="mntl-sc-block_2-0-8" class="comp mntl-sc-block mntl-sc-block-html">
The answer is 37.
</p>
</div>
<div id="list-sc-item_1-0-3" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-3" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">04</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-six-digits-become-three"></span> <h2 id="mntl-sc-block_2-0-9" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Six Digits Become Three </span> </h2>
<ol id="mntl-sc-block_2-0-10" class="comp mntl-sc-block mntl-sc-block-html">
<li>Take any three-digit number and write it twice to make a six-digit number. Examples include 371371 or 552552.</li><li>Divide the number by 7.</li><li>Divide it by 11.</li><li>Divide it by 13.</li>
</ol>
<p id="mntl-sc-block_2-0-11" class="comp mntl-sc-block mntl-sc-block-html">
The order in which you do the division is unimportant!
</p>
<p id="mntl-sc-block_2-0-12" class="comp mntl-sc-block mntl-sc-block-html">
The answer is the three-digit number.
</p>
<p id="mntl-sc-block_2-0-13" class="comp mntl-sc-block mntl-sc-block-html">
Examples: 371371 gives you 371 or 552552 gives you 552.
</p>
<ol id="mntl-sc-block_2-0-14" class="comp mntl-sc-block mntl-sc-block-html">
<li>A related trick is to take any three-digit number.</li>
<li><a href="https://www.thoughtco.com/magical-multiplication-tricks-2086556" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Multiply</a> it by 7, 11, and 13.</li>
</ol>
<p id="mntl-sc-block_2-0-15" class="comp mntl-sc-block mntl-sc-block-html">
The result will be a six-digit number that repeats the three-digit number.
</p>
<p id="mntl-sc-block_2-0-16" class="comp mntl-sc-block mntl-sc-block-html">
Example: 456 becomes 456456.
</p>
</div>
<div id="list-sc-item_1-0-4" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-4" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">05</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-the-11-rule"></span> <h2 id="mntl-sc-block_2-0-17" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The 11 Rule </span> </h2>
<p id="mntl-sc-block_2-0-18" class="comp mntl-sc-block mntl-sc-block-html">
This is a quick way to multiply two-digit numbers by 11 in your head.
</p>
<ol id="mntl-sc-block_2-0-19" class="comp mntl-sc-block mntl-sc-block-html">
<li>Separate the two digits in your mind.</li><li>Add the two digits together.</li><li>Place the number from Step 2 between the two digits. If the number from Step 2 is greater than 9, put the one's digit in the space and carry the ten's digit.</li>
</ol>
<p id="mntl-sc-block_2-0-20" class="comp mntl-sc-block mntl-sc-block-html">
Examples: 72 x 11 = 792.
</p>
<p id="mntl-sc-block_2-0-21" class="comp mntl-sc-block mntl-sc-block-html">
57 x 11 = 5 _ 7, but 5 + 7 = 12, so put 2 in the space and add the 1 to the 5 to get 627
</p>
</div>
<div id="list-sc-item_1-0-5" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-5" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">06</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-memorizing-pi"></span> <h2 id="mntl-sc-block_2-0-22" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Memorizing Pi </span> </h2>
<p id="mntl-sc-block_2-0-23" class="comp mntl-sc-block mntl-sc-block-html">
To remember the first seven <a href="https://www.thoughtco.com/the-number-pi-3-141592654-3126451" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">digits of pi</a>, count the number of letters in each word of the sentence:
</p>
<p id="mntl-sc-block_2-0-24" class="comp mntl-sc-block mntl-sc-block-html">
"How I wish I could calculate pi."
</p>
<p id="mntl-sc-block_2-0-25" class="comp mntl-sc-block mntl-sc-block-html">
This becomes 3.141592.
</p>
</div>
<div id="list-sc-item_1-0-6" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-6" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">07</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-contains-the-digits-1-2-4-5-7-8"></span> <h2 id="mntl-sc-block_2-0-26" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Contains the Digits 1, 2, 4, 5, 7, 8 </span> </h2>
<ol id="mntl-sc-block_2-0-27" class="comp mntl-sc-block mntl-sc-block-html">
<li>Select a number from 1 to 6.</li><li>Multiply the number by 9.</li><li>Multiply it by 111.</li><li>Multiply it by 1001.</li><li>Divide the answer by 7.</li>
</ol>
<p id="mntl-sc-block_2-0-28" class="comp mntl-sc-block mntl-sc-block-html">
The number will contain the digits 1, 2, 4, 5, 7, and 8.&nbsp;
</p>
<p id="mntl-sc-block_2-0-29" class="comp mntl-sc-block mntl-sc-block-html">
Example: The number 6 yields the answer 714285.
</p>
</div>
<div id="list-sc-item_1-0-7" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-7" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">08</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-multiply-large-numbers-in-your-head"></span> <h2 id="mntl-sc-block_2-0-30" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Multiply Large Numbers in Your Head </span> </h2>
<p id="mntl-sc-block_2-0-31" class="comp mntl-sc-block mntl-sc-block-html">
To easily multiply two double-digit numbers, use their distance from 100 to simplify the math:
</p>
<ol id="mntl-sc-block_2-0-32" class="comp mntl-sc-block mntl-sc-block-html">
<li><a href="https://www.thoughtco.com/2-digit-subtraction-worksheets-with-regrouping-2311924" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Subtract</a> each number from 100.</li>
<li>Add these values together.</li>
<li>100 minus this number is the first part of the answer.</li>
<li>Multiply the digits from Step 1 to get the second part of the answer.</li>
</ol>
</div>
<div id="list-sc-item_1-0-8" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-8" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">09</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-super-simple-divisibility-rules"></span> <h2 id="mntl-sc-block_2-0-33" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Super Simple Divisibility Rules </span> </h2>
<p id="mntl-sc-block_2-0-34" class="comp mntl-sc-block mntl-sc-block-html">
You've got 210 pieces of pizza and want to know whether or not you can split them evenly within your group. Rather than whip out the calculator, use these simple shortcuts to do the math in your head:
</p>
<ul id="mntl-sc-block_2-0-35" class="comp mntl-sc-block mntl-sc-block-html">
<li>Divisible by 2 if the last digit is a multiple of 2 (210).</li>
<li>Divisible by 3 if&nbsp;the sum of the digits is divisible by 3&nbsp;(522 because the digits add up to 9, which is divisible by 3).</li>
<li>Divisible by 4 if&nbsp;the last two digits are divisible by 4 (2540 because 40 is divisible by 4).</li>
<li>Divisible by 5 if&nbsp;the last digit is 0 or 5 (9905).</li>
<li>Divisible by 6 if it passes the rules for both 2 and 3 (408).</li>
<li>Divisible by 9 if the <a href="https://www.thoughtco.com/3-digit-addition-worksheets-no-regrouping-2311907" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">sum</a> of the digits is divisible by 9 (6390 since 6 + 3 + 9 + 0 = 18, which is divisible&nbsp;by 9).</li>
<li>Divisible by 10 if the number ends in a 0 (8910).</li>
<li>Divisible by 12 if the rules for divisibility by 3 and 4 apply.</li>
</ul>
<p id="mntl-sc-block_2-0-36" class="comp mntl-sc-block mntl-sc-block-html">
Example: The 210 slices of pizza may be evenly distributed into groups of 2, 3, 5, 6, 10.
</p>
</div>
<div id="list-sc-item_1-0-9" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-9" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">10</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-finger-multiplication-tables"></span> <h2 id="mntl-sc-block_2-0-37" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Finger Multiplication Tables </span> </h2>
<p id="mntl-sc-block_2-0-38" class="comp mntl-sc-block mntl-sc-block-html">
Everyone knows you can count on your fingers. Did you realize you can use them for multiplication? A simple way to do the "9" multiplication table is to place both hands in front of you with fingers and thumbs extended. To multiply 9 by a number, fold down that number finger, counting from the left.
</p>
<p id="mntl-sc-block_2-0-39" class="comp mntl-sc-block mntl-sc-block-html">
Examples: To multiply 9 by 5, fold down the fifth finger from the left. Count fingers on either side of the "fold" to get the answer. In this case, the answer is 45.
</p>
<p id="mntl-sc-block_2-0-40" class="comp mntl-sc-block mntl-sc-block-html">
To multiply 9 times 6, fold down the sixth finger, giving an answer of 54.
</p>
</div>
</div>
<div id="article-outro_1-0" class="comp article-outro mntl-block"></div>
</div>
]]></description>
            <pubDate>Mon, 20 Nov 2017 18:29:31 GMT</pubDate>
            <guid isPermaLink="false">https://www.thoughtco.com/math-tricks-that-will-blow-your-mind-4154742</guid>
            <link>https://www.thoughtco.com/math-tricks-that-will-blow-your-mind-4154742</link>
            <author><![CDATA[Anne Marie Helmenstine, Ph.D.]]></author>
                <category>Science</category>
                <category>Tech</category>
                <category>Math</category>
        </item>
        <item>
            <title><![CDATA[8 Facts About Infinity That Will Blow Your Mind]]></title>
            <description><![CDATA[<figure>
    <img src="https://www.thoughtco.com/thmb/oWz9nmFVzF4iPqxp-FGGV9cOtEk=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/moebius-strip-522025950-5a085d4013f1290037101e9d.jpg" referrerpolicy="no-referrer">
  </figure>
<div id="list-sc_1-0" class="comp article-content list list-sc mntl-block">
<div id="article-intro_1-0" class="comp article-content__description article-intro mntl-block">
<p id="mntl-sc-block_1-0" class="comp mntl-sc-block mntl-sc-block-html">
Infinity is an abstract concept used to describe something that is endless or boundless. It is important in mathematics, cosmology, physics, computing, and the arts.
</p>
</div>
<div id="list-sc__content_1-0" class="comp list-sc__content mntl-sc-page mntl-block" data-sc-sticky-offset="85" data-sc-ad-label-height="24" data-sc-ad-track-spacing="100" data-sc-min-track-height="250" data-sc-max-track-height="600" data-sc-breakpoint="50em" data-sc-load-immediate="3" data-sc-content-positions="[1, 1050, 1250, 1550, 1950, 2350, 2750, 3150]" data-bind-scroll-on-start="true">
<div id="list-sc-item_1-0" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">01</div>
<div class="total">of 08</div>
</div>
</div>
<span class="heading-toc" id="toc-the-infinity-symbol"></span> <h2 id="mntl-sc-block_2-0" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Infinity Symbol </span> </h2>
<figure id="mntl-sc-block_2-0-1" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
</svg>
<span class="figure-article-caption-text">The infinity symbol is also known as the lemniscate.</span>
<span class="figure-article-caption-owner">Chris Collins / Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-2" class="comp mntl-sc-block mntl-sc-block-html">
Infinity has its own special symbol: ∞. The symbol, sometimes called the lemniscate, was introduced by clergyman and mathematician John Wallis in 1655. The word "lemniscate" comes from the Latin word <em>lemniscus</em>, which means "ribbon," while the word "infinity" comes from the Latin word <em>infinitas</em>, which means "boundless."
</p>
<p id="mntl-sc-block_2-0-3" class="comp mntl-sc-block mntl-sc-block-html">
Wallis may have based the symbol on the Roman numeral for 1000, which the Romans used to indicate "countless" in addition to the number. It's also possible the symbol is based on omega (Ω or ω), the last letter in the Greek alphabet.
</p>
<p id="mntl-sc-block_2-0-4" class="comp mntl-sc-block mntl-sc-block-html">
The concept of infinity was understood long before Wallis gave it the symbol we use today. Around the 4th or 3rd century B.C.E., the Jain mathematical text <em>Surya Prajnapti</em> assigned numbers as either enumerable, innumerable, or infinite. The <a href="https://www.thoughtco.com/timeline-of-greek-and-roman-philosophers-118808" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Greek philosopher</a> Anaximander used the work <em>apeiron</em> to refer to the infinite. Zeno of Elea (born circa 490 B.C.E.) was known for <a href="https://www.thoughtco.com/what-is-a-paradox-1691563" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">paradoxes involving infinity</a>.&nbsp;
</p>
</div>
<div id="list-sc-item_1-0-1" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-1" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">02</div>
<div class="total">of 08</div>
</div>
</div>
<span class="heading-toc" id="toc-zenos-paradox"></span> <h2 id="mntl-sc-block_2-0-5" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Zeno's Paradox </span> </h2>
<figure id="mntl-sc-block_2-0-6" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/BOeejWpYm-lmL61xmmfXe4pQuIw=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/tortoise-and-hare--finish-line-143576837-5a08a081494ec90037e9c6bb.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-1" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
</svg>
<span class="figure-article-caption-text">If the rabbit was forever halving the distance to the tortoise, the tortoise would win the race.</span>
<span class="figure-article-caption-owner">Don Farrall / Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-7" class="comp mntl-sc-block mntl-sc-block-html">
Of all Zeno's paradoxes, the most famous is his paradox of the Tortoise and Achilles. In the paradox, a tortoise challenges the <a href="https://www.thoughtco.com/achilles-greek-hero-of-the-trojan-war-116708" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Greek hero Achilles</a> to a race, providing the tortoise is given a small head start. The tortoise argues he will win the race because as Achilles catches up to him, the tortoise will have gone a bit further, adding to the distance.
</p>
<p id="mntl-sc-block_2-0-8" class="comp mntl-sc-block mntl-sc-block-html">
In simpler terms, consider crossing a room by going half the distance with each stride. First, you cover half the distance, with half remaining. The next step is half of one-half, or a quarter. Three quarters of the distance is covered, yet a quarter remains. Next is 1/8th, then 1/16th, and so on. Although each step brings you closer, you never actually reach the other side of the room. Or rather, you&nbsp;would after taking an infinite number of steps.
</p>
</div>
<div id="list-sc-item_1-0-2" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-2" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">03</div>
<div class="total">of 08</div>
</div>
</div>
<span class="heading-toc" id="toc-pi-as-an-example-of-infinity"></span> <h2 id="mntl-sc-block_2-0-9" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Pi as an Example of Infinity </span> </h2>
<figure id="mntl-sc-block_2-0-10" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/4-rZRqlIALa1g3bEynp65ehpYaI=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/pi-formula-on-blackboard-112303538-5a089c5247c266003765ecbe.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-2" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
</svg>
<span class="figure-article-caption-text">Pi is a number consisting of an infinite number of digits.</span>
<span class="figure-article-caption-owner">Jeffrey Coolidge / Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-11" class="comp mntl-sc-block mntl-sc-block-html">
Another good example of infinity is the <a href="https://www.thoughtco.com/the-number-pi-3-141592654-3126451" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">number π or pi</a>. Mathematicians use a symbol for pi because it's impossible to write the number down. Pi consists of an infinite number of digits. It's often rounded to 3.14 or even 3.14159, yet no matter how many digits you write, it's impossible to get to the end.
</p>
</div>
<div id="list-sc-item_1-0-3" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-3" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">04</div>
<div class="total">of 08</div>
</div>
</div>
<span class="heading-toc" id="toc-the-monkey-theorem"></span> <h2 id="mntl-sc-block_2-0-12" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Monkey Theorem </span> </h2>
<figure id="mntl-sc-block_2-0-13" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/OiESkBYL-gXGXvhaa-p68mj8YrM=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/furry-animal-hands-use-laptop-computer-with-blank-screen-169981126-5a08b2fa845b34003b83cd50.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-3" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
</svg>
<span class="figure-article-caption-text">Given an infinite amount of time, a monkey could write the great American novel.</span>
<span class="figure-article-caption-owner">PeskyMonkey / Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-14" class="comp mntl-sc-block mntl-sc-block-html">
One way to think about infinity is in terms of the monkey theorem. According to the theorem, if you give a monkey a typewriter and an infinite amount of time, eventually it will write Shakespeare's <em>Hamlet</em>. While some people take the theorem to suggest anything is possible, mathematicians see it as evidence of just how improbable certain events are.
</p>
</div>
<div id="list-sc-item_1-0-4" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-4" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">05</div>
<div class="total">of 08</div>
</div>
</div>
<span class="heading-toc" id="toc-fractals-and-infinity"></span> <h2 id="mntl-sc-block_2-0-15" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Fractals and Infinity </span> </h2>
<figure id="mntl-sc-block_2-0-16" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/xCf2m8IHCjJxu4eeb-nxJyXjakw=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/soap-bubbles-spirals-585837119-5a088fba845b34003b7a4f65.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-4" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
</svg>
<span class="figure-article-caption-text">A fractal may be magnified over and over, to infinity, always revealing more detail.</span>
<span class="figure-article-caption-owner">PhotoviewPlus / Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-17" class="comp mntl-sc-block mntl-sc-block-html">
A fractal is an abstract mathematical object, used in art and to simulate natural phenomena. Written as a mathematical equation, most fractals are nowhere differentiable. When viewing an image of a fractal, this means you could zoom in and see new detail. In other words, a fractal is infinitely magnifiable.
</p>
<p id="mntl-sc-block_2-0-18" class="comp mntl-sc-block mntl-sc-block-html">
The Koch snowflake is an interesting example of a fractal. The snowflake starts as an equilateral triangle. For each iteration of the fractal:
</p>
<ol id="mntl-sc-block_2-0-19" class="comp mntl-sc-block mntl-sc-block-html">
<li>Each line segment is divided into three equal segments.</li><li>An equilateral triangle is drawn using the middle segment as its base, pointing outward.</li><li>The line segment serving as the base of the triangle is removed.</li>
</ol>
<p id="mntl-sc-block_2-0-20" class="comp mntl-sc-block mntl-sc-block-html">
The process may be repeated an infinite number of times. The resulting snowflake has a finite area, yet it is bounded by an infinitely long line.
</p>
</div>
<div id="list-sc-item_1-0-5" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-5" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">06</div>
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<span class="heading-toc" id="toc-different-sizes-of-infinity"></span> <h2 id="mntl-sc-block_2-0-21" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Different Sizes of Infinity </span> </h2>
<figure id="mntl-sc-block_2-0-22" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/xtQmNnVgTGTV3tzKrr_MIp8EsYg=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/hands-holding-complex-cats-cradle-molecule-network-723497851-5a08a43813f12900372321fd.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-5" class="comp mntl-figure-caption figure-article-caption">
<svg class="icon icon-camera figure-article-caption-icon">
<use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-camera"></use>
</svg>
<span class="figure-article-caption-text">Infinity comes in different sizes.</span>
<span class="figure-article-caption-owner">Tang Yau Hoong / Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-23" class="comp mntl-sc-block mntl-sc-block-html">
Infinity is boundless, yet it comes in different sizes. The positive numbers (those greater than 0) and the negative numbers (those smaller than 0) may be considered to be <a href="https://www.thoughtco.com/examples-of-uncountable-sets-3126438" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">infinite sets</a> of equal sizes. Yet, what happens if you combine both sets? You get a set twice as large. As another example, consider all of the even numbers (an infinite set). This represents an infinity half the size of all of the whole numbers.
</p>
<p id="mntl-sc-block_2-0-24" class="comp mntl-sc-bl

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            <title><![CDATA[What Is a Schematic Diagram?]]></title>
            <description><![CDATA[<figure>
    <img src="https://www.thoughtco.com/thmb/zDnzif-Ssfn68ePLIyVEWdG-P_Y=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/MontyRakusen-5c4df34cc9e77c0001d760bd.jpg" referrerpolicy="no-referrer">
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<p id="mntl-sc-block_1-0" class="comp mntl-sc-block mntl-sc-block-html">
A schematic is defined as a picture that shows something in a simple way, using symbols. A <strong>schematic diagram</strong> is a picture that represents the components of a process, device, or other object using abstract, often standardized symbols and lines. Schematic diagrams only depict the significant components of a system, though some details in the diagram may also be exaggerated or introduced to facilitate the understanding of the system.
</p>
<div id="mntl-sc-block_1-0-1" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-2" class="comp mntl-sc-block mntl-sc-block-html">
Schematic diagrams do not include details that are not necessary for comprehending the information that the diagram was intended to convey. For example, in a schematic diagram depicting an electrical circuit, you can see how the wires and components are connected together, but not photographs of the circuit itself.
</p>
<div id="mntl-sc-block_1-0-3" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
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<h3 id="mntl-sc-block-callout-heading_1-0" class="comp mntl-sc-block-callout-heading mntl-text-block">
Key Takeaways: Schematic Diagram</h3>
<div id="mntl-sc-block-callout-body_1-0" class="comp mntl-sc-block-callout-body mntl-text-block">
<ul><li>A <strong>schematic diagram</strong> is a picture that represents the components of a process, device, or other object using abstract, often standardized symbols and lines.</li><li>Although schematic diagrams are commonly associated with electrical circuits, many examples can be found in other industries.</li></ul></div>
</div>
<span class="heading-toc" id="toc-what-is-a-schematic-diagram"></span> <h2 id="mntl-sc-block_1-0-5" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> What Is a Schematic Diagram? </span> </h2>
<p id="mntl-sc-block_1-0-6" class="comp mntl-sc-block mntl-sc-block-html">
Schematic diagrams can also differ in their level of abstraction. Although they are typically composed of only abstract symbols and lines, some diagrams can also be <strong>semi-schematic</strong> and contain more realistic elements. Some diagrams can also contain words, such as when a process contains multiple elements that have not been standardized.
</p>
<div id="mntl-sc-block_1-0-7" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-8" class="comp mntl-sc-block mntl-sc-block-html">
More simply, a schematic diagram is a simplified drawing that uses symbols and lines to convey important information. For example, if you are taking the subway you may see a “map” showing you all the stations along a subway line, but that map will not show all the roads and buildings you may pass along the way. In this case, the entire subway system can be represented as differently colored lines depicting the different subway routes, with dots indicating the stops along the lines.
</p>
<div id="mntl-sc-block_1-0-9" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<figure id="mntl-sc-block_1-0-10" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-square figure-high-res">
  <figure>
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<span class="figure-article-caption-text">An example of a public transportation map, using differnetly colored lines to represent the diffrent lines and dots to represent the stations along each line. Yuri_Arcurs / Getty Images.</span>
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Though schematic diagrams are most commonly associated with electronics, you have probably encountered many such diagrams, like the subway example above, even if you have never had to wire a circuit. Here are some examples of the many schematic diagrams you may encounter in your work or in your studies.
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<span class="heading-toc" id="toc-examples-of-schematic-diagrams"></span> <h2 id="mntl-sc-block_1-0-13" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Examples of Schematic Diagrams </span> </h2>
<h3 id="mntl-sc-block_1-0-14" class="comp mntl-sc-block reference-sc-block-subheading mntl-sc-block-subheading"> <span class="mntl-sc-block-subheading__text"> Schematic Diagrams in Electronics </span> </h3>
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<span class="figure-article-caption-text">A circuit diagram behind a circuit board. kr7ysztof / Getty Images.</span>
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Schematic diagrams are typically associated with electrical circuits. Also called <strong>wiring diagrams</strong> or <strong>circuit diagrams</strong>, these diagrams show how the different components of a circuit are connected. In these diagrams, lines represent connecting wires, while other elements like resistors, lamps, and switches are represented by standardized symbols called <strong>electrical schematic symbols</strong>.
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In electronics, having a schematic diagram on hand may help a user design an entire circuit before building it, or troubleshoot an electronic that has stopped working.
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Schematic diagrams may also be used to explain the general way that an electronic functions without detailing the hardware or software used in the actual electronic. For example, to explain how a computer projects the words you type on a screen, you might use a schematic diagram that shows how information passes from the keys you press to a word processing program, and finally to the computer screen.
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<h3 id="mntl-sc-block_1-0-22" class="comp mntl-sc-block reference-sc-block-subheading mntl-sc-block-subheading"> <span class="mntl-sc-block-subheading__text"> Schematic Diagrams in Manufacturing </span> </h3>
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<span class="figure-article-caption-text">Schematic diagrams of various parts. Eakachai Leesin / EyeEm / Getty Images.</span>
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Schematic diagrams can be used to depict machines as well. For example, in a textbook a car engine may be depicted as a set of shapes that show how the different parts are positioned relative to one another. A <strong>schematic drawing</strong> may also be created by an engineer while they are designing the machine so that they can properly understand how the parts work together, and make any necessary changes before building the actual system.
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<h3 id="mntl-sc-block_1-0-26" class="comp mntl-sc-block reference-sc-block-subheading mntl-sc-block-subheading"> <span class="mntl-sc-block-subheading__text"> Schematic Diagrams in Chemistry </span> </h3>
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    <img src="https://www.thoughtco.com/thmb/SgrFNorGm300D4On0HLEbk86usI=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/chemistry-blackboard-186738050-5c50896546e0fb00018decd2.jpg" referrerpolicy="no-referrer">
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<span class="figure-article-caption-owner">traffic_analyzer / Getty Images</span>
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Many chemical products are often obtained by performing several different reactions at different steps of the process. A schematic diagram in chemistry can help someone understand all the reactions that were performed to yield a final product, without showing the actual products themselves. This may be portrayed, for example, as a series of boxes connected together with arrows, with words depicting the various elements and conditions that were used throughout the process.
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Like for machines, a schematic diagram may also be used to depict the apparatus that was used for performing the reactions, especially if it is not typically used for reactions, or had been modified from an instrument that was already known.
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<div id="mntl-sc-block_1-0-31" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<h3 id="mntl-sc-block_1-0-32" class="comp mntl-sc-block reference-sc-block-subheading mntl-sc-block-subheading"> <span class="mntl-sc-block-subheading__text"> Schematic Diagrams in Business </span> </h3>
<figure id="mntl-sc-block_1-0-33" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
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    <img src="https://www.thoughtco.com/thmb/BilEEwA3hhB8083zp-U8WOOrM8g=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/SeanGladwellbusinessmodel-5c4df1f7c9e77c00014afaee.jpg" referrerpolicy="no-referrer">
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<figcaption id="mntl-figure-caption_1-0-4" class="comp mntl-figure-caption figure-article-caption">
<span class="figure-article-caption-text">A flowchart that could be used to depict a business process. Sean Gladwell / Getty Images.</span>
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<p id="mntl-sc-block_1-0-34" class="comp mntl-sc-block mntl-sc-block-html">
Schematic diagrams are helpful in conveying the main parts of a complicated business model and showing how they all relate together. For example, a marketing plan might consist of many different elements, such as strategy, objectives, and an action plan. A schematic diagram would then be used to help organize all those elements, including elements within each category, in ways that would convey the main ideas in a clear and concise fashion.
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]]></description>
            <pubDate>Tue, 29 Jan 2019 17:18:26 GMT</pubDate>
            <guid isPermaLink="false">https://www.thoughtco.com/what-is-a-schematic-diagram-4584811</guid>
            <link>https://www.thoughtco.com/what-is-a-schematic-diagram-4584811</link>
            <author><![CDATA[Alane Lim]]></author>
                <category>Science</category>
                <category>Tech</category>
                <category>Math</category>
        </item>
        <item>
            <title><![CDATA[Why You May Have Heard Humans Only Use 10% of the Brain and Why That's Wrong]]></title>
            <description><![CDATA[<figure>
    <img src="https://www.thoughtco.com/thmb/NkUaOtX5HTe8Pg-VKTJp-xzXzFA=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/GettyImages-1034250008-83ed606ca5c845cda1d05f21a8972609.jpg" referrerpolicy="no-referrer">
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You may have heard that humans only use 10 percent of their brain power, and that if you could unlock the rest of your brainpower, you could do so much more. You could become a super genius, or acquire psychic powers like mind reading and telekinesis. However, there is a powerful body of evidence debunking the 10 percent myth. Scientists have consistently shown that humans use their entire brain throughout each day.
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Despite the evidence, the 10 percent myth has inspired many references in the cultural imagination. Films like "Limitless" and "Lucy" depict protagonists who develop godlike powers thanks to drugs that unleash the previously inaccessible 90 percent of the brain. A 2013 study showed that about 65 percent of Americans believe the trope, and a 1998 study showed that a full third of psychology majors, who focus on the workings of the brain, fell for it.
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<span class="heading-toc" id="toc-neuropsychology"></span> <h2 id="mntl-sc-block_1-0-4" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Neuropsychology </span> </h2>
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Neuropsychology studies how the anatomy of the brain affects someone’s behavior, emotion, and cognition. Over the years, brain scientists have shown that different parts of the brain are responsible for <a href="https://www.thoughtco.com/anatomy-of-the-brain-373479" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">specific functions</a>, whether it’s <a href="https://www.thoughtco.com/occipital-lobes-anatomy-373224" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">recognizing colors</a> or <a href="https://www.thoughtco.com/frontal-lobes-anatomy-373213" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="3">problem solving</a>. Contrary to the 10 percent myth, scientists have proven that every part of the brain is integral for our daily functioning, thanks to brain imaging techniques like positron emission tomography and functional magnetic resonance imaging.
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Research has yet to find a brain area that is completely inactive. Even studies that measure activity at the level of single neurons have not revealed any <a href="http://www.bmj.com/content/335/7633/1288.full" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">inactive areas of the brain</a>.&nbsp;Many brain imaging studies that measure brain activity when a person is doing a specific task show how different parts of the brain work together. For example, while you are reading this text on your smartphone, some parts of your brain, including those responsible for vision, reading comprehension, and holding your phone, will be more active.
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However, some brain images <a href="https://www.theatlantic.com/technology/archive/2014/07/you-already-use-way-way-more-than-10-percent-of-your-brain/374520/" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">unintentionally support the 10 percent myth</a>,&nbsp;because they often show small bright splotches on an otherwise gray brain. This may imply that only the bright spots have brain activity, but that isn’t the case. Rather, colored splotches represent brain areas that are<em> <a href="http://www.newsobserver.com/news/technology/article23721076.html" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="2">more</a></em><a href="http://www.newsobserver.com/news/technology/article23721076.html" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="3"> active </a>when someone’s doing a task compared to when they’re not. The gray spots are still active, just to a lesser degree.
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A more direct counter to the 10 percent myth lies in individuals who have suffered brain damage–through a stroke, head trauma, or carbon monoxide poisoning–and what they can no longer do as a result of that damage, or can still do just as well. If the 10 percent myth were true, damage to perhaps 90 percent of the brain wouldn’t affect daily functioning.
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Yet studies show that damaging even a very small part of the brain may have devastating consequences. For example, damage to <a href="https://www.thoughtco.com/brocas-area-anatomy-373215" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Broca’s area</a> hinders proper formation of words and fluent speech, though general language comprehension remains intact. In one highly publicized case, a Florida woman permanently lost her “capacity for thoughts, perceptions, memories, and emotions that are the very essence of being human” when a lack of oxygen destroyed half of her <a href="https://www.thoughtco.com/anatomy-of-the-brain-cerebrum-373218" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">cerebrum</a>, which makes up about 85 percent of the brain.
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<span class="heading-toc" id="toc-evolutionary-arguments"></span> <h2 id="mntl-sc-block_1-0-15" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Evolutionary Arguments </span> </h2>
<p id="mntl-sc-block_1-0-16" class="comp mntl-sc-block mntl-sc-block-html">
Another line of evidence against the 10 percent myth comes from evolution. The adult brain only constitutes 2 percent of body mass, yet it consumes over 20 percent of the body’s energy. In comparison, the adult brains of many vertebrate species–including some fish, reptiles, birds, and mammals–consume <a href="https://www.physiology.org/doi/abs/10.1152/ajpregu.1981.241.3.R203%29" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">2 to 8 percent of their body’s energy</a>. The brain has been shaped by millions of years of <a href="https://www.thoughtco.com/survival-of-the-fittest-1224578" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">natural selection</a>, which passes down favorable traits to increase likelihood of survival. It is unlikely that the body would dedicate so much of its energy to keep an entire brain functioning if it only uses 10 percent of the brain.
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<span class="heading-toc" id="toc-the-origin-of-the-myth"></span> <h2 id="mntl-sc-block_1-0-18" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Origin of the Myth </span> </h2>
<p id="mntl-sc-block_1-0-19" class="comp mntl-sc-block mntl-sc-block-html">
The main allure of the 10 percent myth is the idea that you could do so much more <em>if only</em> you could unlock the rest of your brain. Even with ample evidence suggesting the contrary, why do many people still believe that humans only use 10 percent of their brains? It’s unclear how the myth spread in the first place, but it has been popularized by self-help books, and may even also be grounded in older, flawed, neuroscience studies.
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The myth could be aligned with messages espoused by self-improvement books, which show you ways to do better and live up to your "potential."&nbsp;For example, the preface to the notorious "How to Win Friends and Influence People" says that the average person “develops only 10 percent of his latent mental ability.” This statement, which is traced back to psychologist William James, refers to a person’s potential to achieve more rather than how much brain matter they used. Others have even said that Einstein explained his brilliance using the 10 percent myth, though these claims remain unfounded.
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Another possible source of the myth lies in “silent” brain areas from older neuroscience research. In the 1930s, for instance, neurosurgeon Wilder Penfield hooked electrodes to the exposed brains of his epilepsy patients while operating on them. He noticed that particular brain areas triggered the experience various sensations, but while others seemed to <a href="http://www.sci-con.org/2004/09/exploding-the-10-percent-myth/" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1" rel="nofollow">cause no reaction</a>. Still, as technology evolved, researchers found that these “silent” brain areas, which included the <a href="https://misuse.ncbi.nlm.nih.gov/error/abuse.shtml" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="2">prefrontal lobes</a>, had major functions after all.
</p>
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<span class="heading-toc" id="toc-resources-and-further-reading"></span> <h2 id="mntl-sc-block_1-0-25" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Resources and Further Reading </span> </h2>
<ul id="mntl-sc-block_1-0-26" class="comp mntl-sc-block mntl-sc-block-html">
<li><cite>Beyerstein, B.L. “Whence Cometh the Myth That We Only Use 10% of Our Brains?” <em>Mind Myths: Exploring Popular Assumptions about the Mind and Brain</em>, edited by Sergio Della Sala, Wiley, 1999, pp. 3-24.</cite></li>
<li><cite>Broadfoot, Marla Vacek. “<a href="https://www.newsobserver.com/news/technology/article23721076.html" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">How Do Brain Scans Work?</a>” <em>Raleigh News &amp; Observer</em>, 27 Jan. 2013.</cite></li>
<li><cite>“<a href="https://www.sci-con.org/2004/09/exploding-the-10-percent-myth/" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="2" rel="nofollow">Exploding the 10 Percent Myth</a>.” <em>Science &amp; Consciousness Review</em>.</cite></li>
<li><cite>Higbee, Kenneth L., and Samuel L. Clay. “<a href="https://doi.org/10.1080/00223989809599280" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="3">College Students' Beliefs in the Ten-Percent Myth</a>.” <em>The Journal of Psychology</em>, vol. 132, no. 5, 1998, pp. 469-476.</cite></li>
<li><cite>Jarrett, Christian. <em>Great Myths of the Brain</em>. Wiley Blackwell, 2014.</cite></li>
<li><cite>McDougle, Sam. “<a href="https://www.theatlantic.com/technology/archive/2014/07/you-already-use-way-way-more-than-10-percent-of-your-brain/374520/" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="4">You Already Use Way, Way More Than 10 Percent of Your Brain</a>.” <em>The Atlantic</em>, 7 Aug. 2014.</cite></li>
<li><cite>Mink, J. W., et al. “<a href="https://doi.org/10.1152/ajpregu.1981.241.3.R203" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="5">Ratio of Central Nervous System to Body Metabolism in Vertebrates: Its Constancy and Functional Basis</a>.” <em>American Journal of Physiology-Regulatory, Integrative and Comparative Physiology</em>, vol. 241, no. 3, 1 Sept. 1981, pp. R203-R212.</cite></li>
<li><cite>“<a href="https://www.michaeljfox.org/publication/new-survey-finds-americans-care-about-brain-health-misperceptions-abound" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="6">New Survey Finds Americans Care about Brain Health, but Misperceptions Abound</a>.” <em>The Michael J. Fox Foundation for Parkinson's Research</em>, 25 Sept. 2013.</cite></li><span class="mntl-sc-block-adslot mntl-sc-block-adslot-inline"></span>
<li><cite>Tandon, Prakashnarain. “<a href="https://doi.org/10.4103/0028-3886.125242" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="7">Not so ‘Silent’: The Human Prefrontal Cortex</a>.” <em>Neurology India</em>, vol. 61, no. 6, 2013, pp. 578-580.</cite></li>
<li><cite>Vreeman, Rachel C, and Aaron E Carroll. “<a href="https://doi.org/10.1136/bmj.39420.420370.25" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="8">Medical Myths</a>.” <em>BMJ</em>, vol. 335, no. 7633, 20 Dec. 2007, pp. 1288-1289.</cite></li>
<li><cite>Wanjek, Christopher. <em>Bad Medicine: Misconceptions and Misuses Revealed, from Distance Healing to Vitamin O</em>. Wiley, 2003.</cite></li>
</ul>
</div>
</div>
]]></description>
            <pubDate>Thu, 22 Feb 2018 15:31:24 GMT</pubDate>
            <guid isPermaLink="false">https://www.thoughtco.com/percentage-of-human-brain-used-4159438</guid>
            <link>https://www.thoughtco.com/percentage-of-human-brain-used-4159438</link>
            <author><![CDATA[Alane Lim]]></author>
                <category>Science</category>
                <category>Tech</category>
                <category>Math</category>
        </item>
        <item>
            <title><![CDATA[Déjà Vu: The Science Behind the Eerie Feeling of Familiarity]]></title>
            <description><![CDATA[<figure>
    <img src="https://www.thoughtco.com/thmb/llu30lLBRc1W0UWbh86qlp37QN0=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/blurred-motion-on-city-street--hong-kong-852432264-5a8edf2aa18d9e0037a84d01.jpg" referrerpolicy="no-referrer">
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If you’ve ever had the feeling that a situation feels very familiar even though you know it shouldn’t feel familiar at all, like if you’re traveling in a city for the very first time, then you’ve probably experienced <strong>déjà vu.</strong> Déjà vu, which means “already seen” in French, combines <em>objective</em> unfamiliarity – that you know, based on ample evidence, that something shouldn’t be familiar – with <em>subjective</em> familiarity – that feeling that it’s familiar anyway.
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Déjà vu is common. According to a paper published in 2004, more than 50 surveys on déjà vu suggested that about two-thirds of individuals have experienced it at least once in their lifetime, with many reporting multiple experiences. This reported number also appears to be growing as people become more aware of what déjà vu is.
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Most often, déjà vu is described in terms of what you see, but it’s not specific to vision and even people who were born blind can experience it.
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<span class="heading-toc" id="toc-measuring-dj-vu"></span> <h2 id="mntl-sc-block_1-0-6" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Measuring Déjà Vu </span> </h2>
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Déjà vu is difficult to study in the laboratory because it is a fleeting experience, and also because there is no clearly identifiable trigger for it. Nevertheless, researchers have used several tools to study the phenomenon, based on the hypotheses they’ve put forward. Researchers may survey participants; study possibly related processes, especially those involved in memory; or design other experiments to probe déjà vu.
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Because déjà vu is hard to measure, researchers have postulated many explanations for how it works. Below are several of the more prominent hypotheses.
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<p id="mntl-sc-block_1-0-11" class="comp mntl-sc-block mntl-sc-block-html">
<strong>Memory Explanations</strong>
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Memory explanations of déjà vu are based on the idea that you have previously experienced a situation, or something very much like it, but you don’t <em>consciously</em> remember that you have. Instead, you remember it <em>unconsciously</em>, which is why it feels familiar even though you don’t know why.
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<em>Single element familiarity</em>
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The single element familiarity hypothesis suggests you experience déjà vu if one element of the scene is familiar to you but you don’t consciously recognize it because it’s in a different setting, like if you see your barber out on the street.
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Your brain still finds your barber familiar even if you don’t recognize them, and generalizes that feeling of familiarity to the entire scene. Other researchers have extended this hypothesis to multiple elements as well.
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<em>Gestalt familiarity</em>
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The gestalt familiarity hypothesis focuses on how items are organized in a scene and how déjà vu occurs when you experience something with a similar layout. For example, you may not have seen your friend’s painting in their living room before, but maybe you’ve seen a room that’s laid out like your friend’s living room – a painting hanging over the sofa, across from a bookcase. Since you can’t recall the other room, you experience déjà vu.
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One advantage to the gestalt similarity hypothesis is that it can be more directly tested. In <a href="https://misuse.ncbi.nlm.nih.gov/error/abuse.shtml" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">one study</a>, participants looked at rooms in virtual reality, then were asked how familiar a new room was and whether they felt they were experiencing déjà vu.
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The researchers found that study participants who couldn’t recall the old rooms tended to think a new room was familiar, and that they were experiencing déjà vu, if the new room resembled old ones. Furthermore, the more similar the new room was to an old room, the higher these ratings were.
</p>
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<p id="mntl-sc-block_1-0-29" class="comp mntl-sc-block mntl-sc-block-html">
<strong>Neurological Explanations</strong>
</p>
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<p id="mntl-sc-block_1-0-31" class="comp mntl-sc-block mntl-sc-block-html">
<em>Spontaneous brain activity</em>
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Some explanations posit that&nbsp;déjà vu is experienced when there is spontaneous brain activity unrelated to what you’re currently experiencing. When that happens in the part of your brain dealing with memory, you can have a false feeling of familiarity.
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Some evidence comes from individuals with <a href="https://www.thoughtco.com/temporal-lobes-anatomy-373228" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">temporal lobe </a>epilepsy, when abnormal electrical activity occurs in the part of the brain dealing with memory. When the brains of these patients are electrically stimulated as part of a pre-surgery evaluation, they may experience déjà vu.
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One <a href="https://misuse.ncbi.nlm.nih.gov/error/abuse.shtml" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">researcher</a>&nbsp;suggests that you experience déjà vu when the <a href="https://www.thoughtco.com/hippocampus-anatomy-373221" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">parahippocampal system</a>, which helps identify something as familiar, randomly misfires and makes you think something is familiar when it shouldn’t.&nbsp;
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<div id="mntl-sc-block_1-0-38" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-39" class="comp mntl-sc-block mntl-sc-block-html">
<a href="https://www.clinph-journal.com/article/S1388-2457%2811%2900561-X/fulltext" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1" rel="nofollow">Others have said</a> that déjà vu can’t be isolated to a single familiarity system, but rather involves multiple structures involved in memory and the connections between them.
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<em>Neural transmission speed</em>
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Other hypotheses are based on how fast information travels through your brain. Different areas of your brain transmit information to “higher order” areas that combine the information together to help you make sense of the world. If this complex process is disrupted in any way – perhaps one part sends something more slowly or more quickly than it usually does – then your brain interprets your surroundings incorrectly.
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<span class="heading-toc" id="toc-which-explanation-is-correct"></span> <h2 id="mntl-sc-block_1-0-45" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Which Explanation is Correct? </span> </h2>
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An explanation for déjà vu remains elusive, though the hypotheses above appear to have one common thread: a temporary error in cognitive processing. For now, scientists can continue to design experiments that more directly probe the nature of déjà vu, to be more certain of the correct explanation.
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<div id="mntl-sc-block_1-0-47" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<span class="heading-toc" id="toc-sources"></span> <h2 id="mntl-sc-block_1-0-48" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Sources </span> </h2>
<ul id="mntl-sc-block_1-0-49" class="comp mntl-sc-block mntl-sc-block-html">
<li><cite>Tip-of-the-tongue states and related phenomena. Ed. Bennett L. Schwartz and Alan S. Brown. Cambridge University Press. New York, NY 2014. http://www.cambridge.org/gb/academic/subjects/psychology/biological-psychology/tip-tongue-states-and-related-phenomena?format=HB</cite></li>
<li><cite>C. Moulin. The cognitive neuropsychology of déjà vu. Part of the Essays in Cognitive Psychology series. Psychology Press. New York, NY 2018.<a href="https://www.routledge.com/The-Cognitive-Neuropsychology-of-Deja-Vu/Moulin/p/book/9781138696266" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1"> https://www.routledge.com/The-Cognitive-Neuropsychology-of-Deja-Vu/Moulin/p/book/9781138696266</a></cite></li>
<li><cite>Bartolomei, F., Barbeau, E., Gavaret, M., Guye, M., McGonigal, A., Régis, J., and P. Chauvel. “<a href="http://n.neurology.org/content/63/5/858.short" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="2">Cortical stimulation study of the role of rhinal cortex in déjà vu and reminiscence of memories.</a>” <em>Neurology</em>, vol. 63, no. 5, Sept. 2004, pp. 858-864, </cite><cite>doi</cite><cite>:10.1212/01.wnl.0000137037.56916.3f.</cite></li>
<li><cite>J. Spatt. “<a href="https://misuse.ncbi.nlm.nih.gov/error/abuse.shtml" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="3">Déjà vu: possible parahippocampal mechanisms.</a>” <em>The Journal of Neuropsychiatry &amp; Clinical Neurosciences</em>, vol. 14, no. 1, 2002, pp. 6-10, </cite><cite>doi</cite><cite>:10.1176/jnp.14.1.6.</cite></li>
<li><cite>Cleary, A. M., Brown, A. S., Sawyer, B.D., Nomi, J.S., Ajoku, A.C., and A. J. Ryals. “<a href="https://misuse.ncbi.nlm.nih.gov/error/abuse.shtml" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="4">Familiarity from the configuration of objects in 3-dimensional space and its relation to déjà vu: a virtual reality investigation.</a>” <em>Consciousness and Cognition</em>, vol. 21, no. 2, 2012, pp. 969-975, </cite><cite>doi</cite><cite>:10.1016/j.concog.2011.12.010.</cite></li><span class="mntl-sc-block-adslot mntl-sc-block-adslot-inline"></span>
<li><cite>A. S. Brown. The déjà vu experience. Part of the Essays in Cognitive Psychology series. Psychology Press. New York, NY 2004.<a href="https://www.routledge.com/The-Deja-Vu-Experience/Brown/p/book/9780203485446" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="5"> https://www.routledge.com/The-Deja-Vu-Experience/Brown/p/book/9780203485446</a></cite></li>
<li><cite>A. S. Brown. “<a href="https://misuse.ncbi.nlm.nih.gov/error/abuse.shtml" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="6">A review of the déjà vu experience.</a>” <em>Psychology Bulletin</em>, vol. 129, no. 3, 2003, pp. 394-413. </cite><cite>doi</cite><cite>:10.1037/0033-2909.129.3.394.</cite></li>
<li><cite>Bartolomei, F., Barbeau, E. J., Nguyen, T., McGonigal, A., Régis, J., Chauvel, P., and F. Wendling. “<a href="http://www.clinph-journal.com/article/S1388-2457%2811%2900561-X/abstract" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="7" rel="nofollow">Rhinal-hippocampal interactions during déjà vu.</a>” <em>Clinical Neurophysiology</em>, vol. 123, no. 3, March 2012, pp. 489-495. </cite><cite>doi</cite><cite>:10.1016/j.clinph.2011.08.012</cite></li>
</ul>
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]]></description>
            <pubDate>Thu, 22 Feb 2018 15:31:01 GMT</pubDate>
            <guid isPermaLink="false">https://www.thoughtco.com/causes-of-deja-vu-4159448</guid>
            <link>https://www.thoughtco.com/causes-of-deja-vu-4159448</link>
            <author><![CDATA[Alane Lim]]></author>
                <category>Science</category>
                <category>Tech</category>
                <category>Math</category>
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            <title><![CDATA[7 Extinction Level Events That Could End Life as We Know It]]></title>
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If you've watched the movies "2012" or "Armageddon" or read "On the Beach," you know about some of the threats that could end life as we know it. The <a href="https://www.thoughtco.com/things-you-should-know-about-the-sun-3073449" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Sun</a> could do something nasty. A <a href="https://www.thoughtco.com/astronomy-basics-4133556" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">meteor</a> could strike. We could nuke ourselves out of existence. These are only a few well-known extinction level events. There are so many more ways to die!
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But first, what exactly is an extinction event? An <strong>extinction level event</strong> or ELE is a catastrophe resulting in the extinction of the majority of species on the planet. It's not the normal extinction of species that occurs every day. It isn't necessarily the sterilization of all living organisms. We can identify major extinction events by examining the deposition and chemical composition of rocks, the <a href="https://www.thoughtco.com/famous-fossil-discoveries-1092049" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">fossil record</a>, and evidence of major events on moons and other planets.
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There are dozens of phenomena capable of causing widespread extinctions, but they can be grouped into a few categories:
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<span class="heading-toc" id="toc-the-sun-will-kill-us"></span> <h2 id="mntl-sc-block_2-0" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Sun Will Kill Us </span> </h2>
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<span class="figure-article-caption-text">If a strong solar flare hit the Earth, the results could be devastating.</span>
<span class="figure-article-caption-owner">VICTOR HABBICK VISIONS, Getty Images</span>
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Life as we know it wouldn't exist without the Sun, but let's be honest. The Sun has it out for planet Earth. Even if none of the other catastrophes on this list ever happen, the Sun will end us. Stars like the Sun burn brighter over time as they fuse hydrogen into helium. In another billion years, it will be about 10 percent brighter. While this might not seem significant, it will cause more water to evaporate. <a href="https://www.thoughtco.com/worst-greenhouse-gases-606789" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Water is&nbsp;a greenhouse gas</a>, so it traps heat in the <a href="https://www.thoughtco.com/most-abundant-gases-in-earths-atmosphere-607594" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">atmosphere</a>, leading to more evaporation. Sunlight will break water into hydrogen and oxygen, so it can <a href="https://www.thoughtco.com/if-earths-atmosphere-vanished-607906" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="3">bleed away into space</a>. Should any life survive, it will meet a fiery fate when the Sun enters its <a href="https://www.thoughtco.com/what-are-stars-really-3073631" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="4">red giant</a> phase, expanding out to the orbit of Mars. It's not likely any life will survive <em>inside</em> the Sun.
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But, the Sun can kill us any old day it wants <a href="https://www.thoughtco.com/solar-flares-4137226" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">via a coronal mass ejection</a> (CME). As you can guess from the name, this is when our favorite star expels charged particles outward from its corona. Since a CME can sent matter any direction, it doesn't usually shoot directly toward Earth. Sometimes only a tiny fraction of particles reach us, granting us an aurora or a solar storm. However, it's possible for a CME to barbecue the planet.
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The Sun has pals (and they hate Earth too). A nearby (within 6000 light years) <a href="https://www.thoughtco.com/supernovae-deaths-of-massive-stars-3073301" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">supernova</a>, nova, or gamma ray burst could irradiate organisms and destroy the ozone layer, leaving life&nbsp;at the mercy of the Sun's <a href="https://www.thoughtco.com/definition-of-ultraviolet-radiation-604675" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">ultraviolet radiation</a>. Scientists think a <a href="https://www.thoughtco.com/gamma-ray-burst-destroy-life-earth-3072521" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="3">gamma burst</a> or supernova might have led to the End-Ordovician extinction.
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</div>
<span class="heading-toc" id="toc-geomagnetic-reversals-may-kill-us"></span> <h2 id="mntl-sc-block_2-0-5" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Geomagnetic Reversals May Kill Us </span> </h2>
<figure id="mntl-sc-block_2-0-6" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/7CfPJl1EDfXgckaZvnvIqPRDuDA=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/GettyImages-693943194-5a84493fd8fdd500375c2f46.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-1" class="comp mntl-figure-caption figure-article-caption">
<span class="figure-article-caption-text">Scientists believe magnetic pole reversals were involved in some past mass extinctions.</span>
<span class="figure-article-caption-owner">siiixth, Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-7" class="comp mntl-sc-block mntl-sc-block-html">
The Earth is a giant magnet that has a love-hate relationship with life. The magnetic field protects us from the worst the Sun throws at us. Every so often, the positions of the <a href="https://www.thoughtco.com/what-is-magnetic-reversal-1435340" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">north and south magnetic poles flip</a>. How often the reversals occur and how long it takes the magnetic field to get settled&nbsp;is highly variable. Scientists aren't completely sure what will happen when the poles flip. Maybe nothing. Or maybe the weakened magnetic field will expose the Earth to the <a href="https://www.thoughtco.com/the-effects-of-solar-storms-3073703" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">solar wind</a>, letting the Sun steal a lot of our oxygen. You know, that gas humans breathe. Scientists say magnetic field reversals aren't always extinction level events. Just sometimes.
</p>
</div>
<div id="list-sc-item_1-0-2" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-2" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">03</div>
<div class="total">of 09</div>
</div>
</div>
<span class="heading-toc" id="toc-the-big-bad-meteor"></span> <h2 id="mntl-sc-block_2-0-8" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Big Bad Meteor </span> </h2>
<figure id="mntl-sc-block_2-0-9" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/fRhKUp4HvIhoOQA3RRt9eSsNA3w=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/GettyImages-532101243-5a844ef13418c6003624983e.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-2" class="comp mntl-figure-caption figure-article-caption">
<span class="figure-article-caption-text">A big meteor impact could be an extinction level event.</span>
<span class="figure-article-caption-owner">Marc Ward/Stocktrek Images, Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-10" class="comp mntl-sc-block mntl-sc-block-html">
You may be surprised to learn the impact of an asteroid or meteor has only been connected with certainty to one mass extinction, the Cretaceous-Paleogene extinction event. Other impacts have been contributing factors to extinctions, but not the primary cause.
</p>
<p id="mntl-sc-block_2-0-11" class="comp mntl-sc-block mntl-sc-block-html">
The good news is that <a href="https://nypost.com/2016/12/14/nasa-scientist-warns-earth-is-due-for-extinction-level-event/" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">NASA claims</a> about 95 percent of comets and asteroids bigger than 1 kilometer in diameter have been identified. The other good news is that scientists estimate an object needs to be about 100 kilometers (60 miles) across to wipe out all life. The bad news is there are another 5 percent out there and not much we can do about a significant threat with our present technology (no, Bruce Willis cannot detonate a nuke and save us).
</p>
<p id="mntl-sc-block_2-0-12" class="comp mntl-sc-block mntl-sc-block-html">
Obviously, living things at ground zero for a meteor strike will die. Many more will die from the shock wave, earthquakes, tsunamis, and firestorms. Those that survive the initial impact would have a hard time finding food, as the debris thrown into the atmosphere would change the climate, leading to mass extinctions. You're probably better off at ground zero for this one.
</p>
</div>
<div id="list-sc-item_1-0-3" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-3" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">04</div>
<div class="total">of 09</div>
</div>
</div>
<span class="heading-toc" id="toc-the-sea"></span> <h2 id="mntl-sc-block_2-0-13" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Sea </span> </h2>
<figure id="mntl-sc-block_2-0-14" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/v_x1oZvDSXc0FtAdEggw2a_eenM=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/GettyImages-128139136-5a8467ef3418c6003645ddc5.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-3" class="comp mntl-figure-caption figure-article-caption">
<span class="figure-article-caption-text">A tsunami is dangerous, but the sea has more lethal tricks.</span>
<span class="figure-article-caption-owner">Bill Romerhaus, Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-15" class="comp mntl-sc-block mntl-sc-block-html">
A day at the beach might seem idyllic, until you realize the blue part of the marble we call Earth is deadlier than all of the sharks in its depths. The ocean has various ways of causing ELEs.
</p>
<p id="mntl-sc-block_2-0-16" class="comp mntl-sc-block mntl-sc-block-html">
Methane clathrates (molecules made of water and methane) sometimes break from the continental shelves, producing a methane eruption called a clathrate gun. The "gun" shoots immense amounts of the greenhouse gas methane into the atmosphere. Such events are linked to the <a href="https://www.thoughtco.com/permian-period-300-250-million-years-1091430" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">end-Permian extinction</a> and Paleocene-Eocene Thermal Maximum.
</p>
<p id="mntl-sc-block_2-0-17" class="comp mntl-sc-block mntl-sc-block-html">
Prolonged sea level rise or fall also leads to extinctions. Falling sea levels are more insidious, as exposing the continental shelf kills off innumerable marine species. This, in turn, upsets the terrestrial ecosystem, leading to an ELE.
</p>
<p id="mntl-sc-block_2-0-18" class="comp mntl-sc-block mntl-sc-block-html">
Chemical imbalances in the sea also cause extinction events. When the middle or upper layers of the ocean become anoxic, a chain reaction of death occurs. The Ordovician-Silurian, late Devonian, Permian-Triassic, and Triassic-Jurassic extinctions all included anoxic events.
</p>
<p id="mntl-sc-block_2-0-19" class="comp mntl-sc-block mntl-sc-block-html">
Sometimes the levels of essential trace elements (e.g., <a href="https://www.thoughtco.com/selenium-facts-606594" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">selenium</a>) fall, leading to mass extinctions. Sometimes the sulfate-reducing bacteria in thermal vents get out of control, releasing an excess of hydrogen sulfide that weakens the ozone layer, exposing life to lethal UV. The ocean also undergoes a periodic overturn in which the high-salinity surface water sinks to the depths. Anoxic deep water rises, killing surface organisms. The late-Devonian and <a href="https://www.thoughtco.com/the-permian-triassic-extinction-event-1092136" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">Permian-Triassic extinctions</a> are associated oceanic overturn.
</p>
<p id="mntl-sc-block_2-0-20" class="comp mntl-sc-block mntl-sc-block-html">
The beach doesn't look so nice now, does it?
</p>
</div>
<div id="list-sc-item_1-0-4" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-4" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">05</div>
<div class="total">of 09</div>
</div>
</div>
<span class="heading-toc" id="toc-and-the-winner-is-volcanoes"></span> <h2 id="mntl-sc-block_2-0-21" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> And the "Winner" Is... Volcanoes </span> </h2>
<figure id="mntl-sc-block_2-0-22" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/B6IEDBoM4dSm5JDu9JNS8j8L2xQ=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/GettyImages-675488626-5a84688e312834003700b60f.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-4" class="comp mntl-figure-caption figure-article-caption">
<span class="figure-article-caption-text">Historically, most extinction level events have been caused by volcanoes.</span>
<span class="figure-article-caption-owner">Mike Lyvers, Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-23" class="comp mntl-sc-block mntl-sc-block-html">
While falling sea level has been associated with 12 extinction events, only seven involved a significant loss of species. On the other hand, <a href="https://www.thoughtco.com/what-happens-when-a-volcano-erupts-4151722" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">volcanoes</a> have led to 11 ELEs, <em>all</em> of them significant. The End-Permian, End-Triassic, and <a href="https://www.thoughtco.com/the-k-t-extinction-event-1092141" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">End-Cretaceous extinctions</a> are&nbsp;associated with volcanic eruptions called flood basalt events. Volcanoes kill by releasing dust, sulfur oxides, and <a href="https://www.thoughtco.com/carbon-dioxide-molecular-formula-608475" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="3">carbon dioxide</a> that collapse food chains by inhibiting photosynthesis, poison the land and sea with acid rain, and produce global warming. The next time you vacation at Yellowstone, take a moment to stop and ponder the implications when the volcano erupts. At least the volcanoes in Hawaii aren't planet killers.
</p>
</div>
<div id="list-sc-item_1-0-5" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-5" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">06</div>
<div class="total">of 09</div>
</div>
</div>
<span class="heading-toc" id="toc-global-warming-and-cooling"></span> <h2 id="mntl-sc-block_2-0-24" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Global Warming and Cooling </span> </h2>
<figure id="mntl-sc-block_2-0-25" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/KPZo0YTjUutXXK-gqDCEzkG_VVo=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/GettyImages-143745638-5a8468fdd8fdd500378595ad.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-5" class="comp mntl-figure-caption figure-article-caption">
<span class="figure-article-caption-text">Runaway global warming could make 

...

[github-actions]

http://localhost:1200/thoughtco/math-4133545 - Success ✔️

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        <description><![CDATA[Struggling with scatterplots? Can't quite wrap your head around circumference? Here are resources and tutorials for all the major functions, formulas, equations, and theories you'll encounter in math class. Teachers can find useful math resources for the classroom. - Made with love by RSSHub(https://github.com/DIYgod/RSSHub)]]></description>
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            <title><![CDATA[Why Mathematics Is a Language]]></title>
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<p id="mntl-sc-block_1-0" class="comp mntl-sc-block mntl-sc-block-html">
Mathematics is called the language of science. Italian astronomer and physicist <a href="https://www.thoughtco.com/galileo-galilei-biography-1991864" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Galileo Galilei</a> is attributed with the quote, "<em>Mathematics is the language in which God has written the universe</em>." Most likely this quote is a summary of his statement in&nbsp;<em>Opere Il Saggiatore:</em>
</p>
<blockquote id="mntl-sc-block_1-0-1" class="comp mntl-sc-block mntl-sc-block-html">
[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.
</blockquote>
<p id="mntl-sc-block_1-0-2" class="comp mntl-sc-block mntl-sc-block-html">
Yet, is mathematics truly a language, like English or Chinese? To answer the question, it helps to know what language is and how the vocabulary and grammar of mathematics are used to construct sentences.
</p>
<div id="mntl-sc-block_1-0-3" class="comp theme-keytakeaways mntl-sc-block reference-sc-block-callout mntl-sc-block-callout mntl-block" data-tracking-id="mntl-sc-block-callout" data-tracking-container="true">
<h3 id="mntl-sc-block-callout-heading_1-0" class="comp mntl-sc-block-callout-heading mntl-text-block">
Key Takeaways: Why Math is a Language</h3>
<div id="mntl-sc-block-callout-body_1-0" class="comp mntl-sc-block-callout-body mntl-text-block">
<ul><li>In order to be considered a language, a system of communication must have vocabulary, grammar, syntax, and people who use and understand it.</li><li>Mathematics meets this definition of a language. Linguists who don't consider math a language cite its use as a written rather than spoken form of communication.</li><li>Math is a universal language. The symbols and organization to form equations are the same in every country of the world.</li></ul></div>
</div>
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<div id="list-sc__content_1-0" class="comp list-sc__content mntl-sc-page mntl-block" data-sc-sticky-offset="85" data-sc-ad-label-height="24" data-sc-ad-track-spacing="100" data-sc-min-track-height="250" data-sc-max-track-height="600" data-sc-breakpoint="50em" data-sc-load-immediate="3" data-sc-content-positions="[1, 1050, 1250, 1550, 1950, 2350, 2750, 3150]" data-bind-scroll-on-start="true">
<div id="list-sc-item_1-0" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0" class="comp list-marker list-marker--none"></div>
<span class="heading-toc" id="toc-what-is-a-language"></span> <h2 id="mntl-sc-block_2-0" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> What Is a Language? </span> </h2>
<p id="mntl-sc-block_2-0-1" class="comp mntl-sc-block mntl-sc-block-html">
There are multiple definitions of "<a href="https://www.thoughtco.com/what-is-a-language-1691218" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">language</a>." A language may be a system of words or codes used within a discipline. Language may refer to a system of communication using symbols or sounds. <a href="https://www.thoughtco.com/what-is-a-linguist-1691239" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">Linguist</a> Noam Chomsky defined language as a set of sentences constructed using a finite set of elements. Some linguists believe language should be able to represent events and abstract concepts.
</p>
<p id="mntl-sc-block_2-0-2" class="comp mntl-sc-block mntl-sc-block-html">
Whichever definition is used, a language contains the following components:
</p>
<ul id="mntl-sc-block_2-0-3" class="comp mntl-sc-block mntl-sc-block-html">
<li>There must be a <strong>vocabulary</strong> of words or symbols.</li><li><strong>Meaning</strong> must be attached to the words or symbols.</li><li>A language employs <strong>grammar</strong>, which is a set of rules that outline how vocabulary is used.</li><li>A <strong>syntax</strong> organizes symbols into linear structures or propositions.</li><li>A <strong>narrative</strong> or discourse consists of strings of syntactic propositions.</li><li>There must be (or have been) a group of people who use and understand the symbols.</li>
</ul>
<p id="mntl-sc-block_2-0-4" class="comp mntl-sc-block mntl-sc-block-html">
Mathematics meets all of these requirements. The symbols, their meanings, syntax, and grammar are the same throughout the world. Mathematicians, scientists, and others use math to communicate concepts. Mathematics describes itself (a field called meta-mathematics), real-world phenomena, and abstract concepts.
</p>
</div>
<div id="list-sc-item_1-0-1" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-1" class="comp list-marker list-marker--none"></div>
<span class="heading-toc" id="toc-vocabulary-grammar-and-syntax-in-mathematics"></span> <h2 id="mntl-sc-block_2-0-5" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Vocabulary, Grammar, and Syntax in Mathematics </span> </h2>
<figure id="mntl-sc-block_2-0-6" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/N2hMLp5mKoTWbfT8wpxt1VZ25eM=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/student-writing-on-blackboard-695556138-5a6e40fdeb97de0037e8a0c6.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0" class="comp mntl-figure-caption figure-article-caption">
<span class="figure-article-caption-owner">Emilija Manevska / Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-7" class="comp mntl-sc-block mntl-sc-block-html">
The vocabulary of math draws from many different alphabets and includes symbols unique to math. A mathematical equation may be stated in words to form a sentence that has a noun and a verb, just like a sentence in a spoken language. For example:
</p>
<p id="mntl-sc-block_2-0-8" class="comp mntl-sc-block mntl-sc-block-html">
3 + 5 = 8
</p>
<p id="mntl-sc-block_2-0-9" class="comp mntl-sc-block mntl-sc-block-html">
could be stated as "Three added to five equals eight."
</p>
<p id="mntl-sc-block_2-0-10" class="comp mntl-sc-block mntl-sc-block-html">
Breaking this down, <a href="https://www.thoughtco.com/what-is-concrete-noun-1689904" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">nouns</a> in math include:
</p>
<ul id="mntl-sc-block_2-0-11" class="comp mntl-sc-block mntl-sc-block-html">
<li>Arabic numerals (0, 5, 123.7)</li><li>Fractions (1⁄4, 5⁄9, 2 1⁄3)</li><li>Variables (a, b, c, x, y, z)</li><li>Expressions (3x, x<sup>2</sup>, 4 + x)</li><li>Diagrams or visual elements (circle, angle, triangle, tensor, matrix)</li><li>Infinity (∞)</li><li>Pi (π)</li><li>Imaginary numbers (i, -i)</li><li>The speed of light (c)</li>
</ul>
<p id="mntl-sc-block_2-0-12" class="comp mntl-sc-block mntl-sc-block-html">
Verbs include symbols including:
</p>
<ul id="mntl-sc-block_2-0-13" class="comp mntl-sc-block mntl-sc-block-html">
<li>Equalities or inequalities (=, &lt;, &gt;)</li><li>Actions such as addition, subtraction, multiplication, and division (+, -, x or *, ÷ or /)</li><li>Other operations (sin, cos, tan, sec)</li>
</ul>
<p id="mntl-sc-block_2-0-14" class="comp mntl-sc-block mntl-sc-block-html">
If you try to perform a sentence diagram on a mathematical sentence, you'll find infinitives, conjunctions, adjectives, etc. As in other languages, the role played by a symbol depends on its context.
</p>
<span class="heading-toc" id="toc-international-rules"></span> <h2 id="mntl-sc-block_2-0-15" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> International Rules </span> </h2>
<p id="mntl-sc-block_2-0-16" class="comp mntl-sc-block mntl-sc-block-html">
Mathematics grammar and syntax, like vocabulary, are international. No matter what country you're from or what language you speak, the structure of the mathematical language is the same.
</p>
<ul id="mntl-sc-block_2-0-17" class="comp mntl-sc-block mntl-sc-block-html">
<li>Formulas are read from left to right.</li>
<li>The Latin alphabet is used for parameters and variables. To some extent, the Greek alphabet is also used. <a href="https://www.thoughtco.com/understanding-classification-of-numbers-2312407" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Integers</a> are usually drawn from <em>i</em>, <em>j</em>, <em>k</em>, <em>l</em>, <em>m</em>, <em>n</em>. <a href="https://www.thoughtco.com/what-is-a-real-number-3126307" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">Real numbers</a> are represented by&nbsp;<em>a</em>,&nbsp;<em>b</em>,&nbsp;<em>c</em>, α<em>, β</em>, γ. Complex numbers are indicated by <em>w</em> and <em>z</em>. Unknowns are <em>x</em>, <em>y</em>, <em>z</em>. Names of functions are usually <em>f</em>, <em>g</em>, <em>h</em>.</li>
<li>The Greek alphabet is used to represent specific concepts. For example, λ is used to indicate wavelength and ρ means density.</li>
<li>Parentheses and brackets indicate the <a href="https://www.thoughtco.com/definition-of-bedmas-2312372" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="3">order in which the symbols interact</a>.</li>
<li>The way functions, integrals, and derivatives are phrased is uniform.</li>
</ul>
</div>
<div id="list-sc-item_1-0-2" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-2" class="comp list-marker list-marker--none"></div>
<span class="heading-toc" id="toc-language-as-a-teaching-tool"></span> <h2 id="mntl-sc-block_2-0-18" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Language as a Teaching Tool </span> </h2>
<figure id="mntl-sc-block_2-0-19" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/3FBkjxj7Vj_djIbgjhs5Asro2ho=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/question-marks-on-black-background-184837701-5a6df53904d1cf00378895cb.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-1" class="comp mntl-figure-caption figure-article-caption">
<span class="figure-article-caption-owner">StockFinland / Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-20" class="comp mntl-sc-block mntl-sc-block-html">
Understanding how mathematical sentences work is helpful when teaching or learning math. Students often find numbers and symbols intimidating, so putting an equation into a familiar language makes the subject more approachable. Basically, it's like translating a foreign language into a known one.
</p>
<p id="mntl-sc-block_2-0-21" class="comp mntl-sc-block mntl-sc-block-html">
While students typically dislike word problems, extracting the nouns, verbs, and modifiers from a spoken/written language and translating them into a mathematical equation is a valuable skill to have. Word problems improve comprehension and increase problem-solving skills.
</p>
<p id="mntl-sc-block_2-0-22" class="comp mntl-sc-block mntl-sc-block-html">
Because mathematics is the same all over the world, math can act as a universal language. A phrase or formula has the same meaning, regardless of another language that accompanies it. In this way, math helps people learn and communicate, even if other communication barriers exist.
</p>
</div>
<div id="list-sc-item_1-0-3" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-3" class="comp list-marker list-marker--none"></div>
<span class="heading-toc" id="toc-the-argument-against-math-as-a-language"></span> <h2 id="mntl-sc-block_2-0-23" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Argument Against Math as a Language </span> </h2>
<figure id="mntl-sc-block_2-0-24" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-square figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/ZMMuwx9pPqpL8vJSpD30sCDqkjc=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/maxwellsequations-5a6dea1deb97de0037dbb20b.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-2" class="comp mntl-figure-caption figure-article-caption">
<span class="figure-article-caption-owner">Anne Helmenstine</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-25" class="comp mntl-sc-block mntl-sc-block-html">
Not everyone agrees that mathematics is a language. Some definitions of "language" describe it as a spoken form of communication. Mathematics is a written form of communication. While it may be easy to read a simple addition statement aloud (e.g., 1 + 1 = 2), it's much harder to read other equations aloud (e.g., Maxwell's equations). Also, the spoken statements would be rendered in the speaker's native language, not a universal tongue.
</p>
<p id="mntl-sc-block_2-0-26" class="comp mntl-sc-block mntl-sc-block-html">
However, sign language would also be disqualified based on this criterion. Most linguists accept sign language as a true language.&nbsp;There are a handful of dead languages that no one alive knows how to pronounce or even read anymore.
</p>
<p id="mntl-sc-block_2-0-27" class="comp mntl-sc-block mntl-sc-block-html">
A strong case for mathematics as a language is that modern elementary-high school curricula uses techniques from language education for teaching mathematics. Educational psychologist Paul Riccomini and colleagues wrote that students learning mathematics require "a robust vocabulary knowledge base; flexibility; fluency and proficiency with numbers, symbols, words, and diagrams; and comprehension skills."
</p>
<span class="heading-toc" id="toc-sources"></span> <h2 id="mntl-sc-block_2-0-28" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Sources </span> </h2>
<ul id="mntl-sc-block_2-0-29" class="comp mntl-sc-block mntl-sc-block-html">
<li><cite>Ford, Alan, and F. David Peat. "<a href="https://doi.org/10.1007/BF01889434" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">The Role of Language in Science</a>." <em>Foundations of Physics</em> 18.12 (1988): 1233–42.&nbsp;</cite></li>
<li><cite>Galilei, Galileo. "'The Assayer' ('Il Saggiatore' in Italian) (Rome, 1623)." <em>The Controversy on the Comets of 1618</em>. Eds. Drake, Stillman and C. D. O'Malley. Philadelphia: University of Pennsylvania Press, 1960.&nbsp;</cite></li>
<li><cite>Klima, Edward S., and Ursula Bellugi. "The Signs of Language. "Cambridge, MA: Harvard University Press, 1979.&nbsp;</cite></li>
<li><cite>Riccomini, Paul J., et al. "<a href="https://doi.org/10.1080/10573569.2015.1030995" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="2">The Language of Mathematics: The Importance of Teaching and Learning Mathematical Vocabulary</a>." <em>Reading &amp; Writing Quarterly</em> 31.3 (2015): 235-52. Print.</cite></li>
</ul>
</div>
</div>
<div id="article-outro_1-0" class="comp article-outro mntl-block"></div>
</div>
]]></description>
            <pubDate>Mon, 29 Jan 2018 19:48:57 GMT</pubDate>
            <guid isPermaLink="false">https://www.thoughtco.com/why-mathematics-is-a-language-4158142</guid>
            <link>https://www.thoughtco.com/why-mathematics-is-a-language-4158142</link>
            <author><![CDATA[Anne Marie Helmenstine, Ph.D.]]></author>
                <category>Science</category>
                <category>Tech</category>
                <category>Math</category>
        </item>
        <item>
            <title><![CDATA[Here's How Equivalent Systems of Equations Work]]></title>
            <description><![CDATA[<figure>
    <img src="https://www.thoughtco.com/thmb/xr1EIfRZtWAHfml2WyLPPa0MWrI=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/GettyImages-595348759-5372ddd37cf44fdb957bdb81028ef7ed.jpg" referrerpolicy="no-referrer">
  </figure>
<div id="article-content_1-0" class="comp article-content mntl-block">
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<p id="mntl-sc-block_1-0" class="comp mntl-sc-block mntl-sc-block-html">
Equivalent equations are systems of equations that have the same solutions. Identifying and solving equivalent equations is a valuable skill, not only in <a href="https://www.thoughtco.com/what-is-algebra-why-take-algebra-2311937" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">algebra class</a> but also in everyday life. Take a look at examples of equivalent equations, how to solve them for one or more variables, and how you might use this skill outside a classroom.
</p>
<div id="mntl-sc-block_1-0-1" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
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<h3 id="mntl-sc-block-callout-heading_1-0" class="comp mntl-sc-block-callout-heading mntl-text-block">
Key Takeaways</h3>
<div id="mntl-sc-block-callout-body_1-0" class="comp mntl-sc-block-callout-body mntl-text-block">
<ul><li>Equivalent equations are algebraic equations that have identical solutions or roots.</li><li>Adding or subtracting the same number or expression to both sides of an equation produces an equivalent equation.</li><li>Multiplying or dividing both sides of an equation by the same non-zero number produces an equivalent equation.</li></ul></div>
</div>
<span class="heading-toc" id="toc-linear-equations-with-one-variable"></span> <h2 id="mntl-sc-block_1-0-3" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Linear Equations With One Variable </span> </h2>
<p id="mntl-sc-block_1-0-4" class="comp mntl-sc-block mntl-sc-block-html">
The simplest examples of equivalent equations don't have any variables. For example, these three equations are equivalent to each other:
</p>
<div id="mntl-sc-block_1-0-5" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-6" class="comp mntl-sc-block mntl-sc-block-html">
<li>3 + 2 = 5</li><li>4 + 1 = 5</li><li>5 + 0 = 5</li>
</ul>
<div id="mntl-sc-block_1-0-7" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-8" class="comp mntl-sc-block mntl-sc-block-html">
Recognizing these equations are equivalent is great, but not particularly useful. Usually, an equivalent equation problem asks you to solve for a variable to see if it is the same (the same <strong>root</strong>) as the one in another equation.
</p>
<div id="mntl-sc-block_1-0-9" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-10" class="comp mntl-sc-block mntl-sc-block-html">
For example, the following equations are equivalent:
</p>
<div id="mntl-sc-block_1-0-11" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-12" class="comp mntl-sc-block mntl-sc-block-html">
<li>x = 5</li><li>-2x = -10</li>
</ul>
<div id="mntl-sc-block_1-0-13" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-14" class="comp mntl-sc-block mntl-sc-block-html">
In both cases, x = 5. How do we know this? How do you solve this for the "-2x = -10" equation? The first step is to know the rules of equivalent equations:
</p>
<div id="mntl-sc-block_1-0-15" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-16" class="comp mntl-sc-block mntl-sc-block-html">
<li><a href="https://www.thoughtco.com/3-digit-addition-worksheets-with-regrouping-2311922" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Adding</a> or subtracting the same number or expression to both sides of an equation produces an equivalent equation.</li>
<li>Multiplying or dividing both sides of an equation by the same non-zero number produces an equivalent equation.</li>
<li>Raising both sides of the equation to the <a href="https://www.thoughtco.com/exponents-and-bases-2312002" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">same odd power</a> or taking the same odd root will produce an equivalent equation.</li>
<li>If both sides of an equation are non-<a href="https://www.thoughtco.com/calculations-with-negative-numbers-2312514" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="3">negative</a>, raising both sides of an equation to the same even power or taking the same even root will give an equivalent equation.</li>
</ul>
<div id="mntl-sc-block_1-0-17" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<span class="heading-toc" id="toc-example"></span> <h2 id="mntl-sc-block_1-0-18" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Example </span> </h2>
<p id="mntl-sc-block_1-0-19" class="comp mntl-sc-block mntl-sc-block-html">
Putting these rules into practice, determine whether these two equations are equivalent:
</p>
<div id="mntl-sc-block_1-0-20" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-21" class="comp mntl-sc-block mntl-sc-block-html">
<li>x + 2 = 7</li><li>2x + 1 = 11</li>
</ul>
<div id="mntl-sc-block_1-0-22" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-23" class="comp mntl-sc-block mntl-sc-block-html">
To solve this, you need to find "x" for each <a href="https://www.thoughtco.com/exercise-worksheets-using-foil-2312026" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">equation</a>. If "x" is the same for both equations, then they are equivalent. If "x" is different (i.e., the equations have different roots), then the equations are not equivalent. For the first equation:
</p>
<div id="mntl-sc-block_1-0-24" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-25" class="comp mntl-sc-block mntl-sc-block-html">
<li>x + 2 = 7</li><li>x + 2 - 2 = 7 - 2 (subtracting both sides by same number)</li><li>x = 5</li>
</ul>
<div id="mntl-sc-block_1-0-26" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-27" class="comp mntl-sc-block mntl-sc-block-html">
For the second equation:
</p>
<div id="mntl-sc-block_1-0-28" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-29" class="comp mntl-sc-block mntl-sc-block-html">
<li>2x + 1 = 11</li><li>2x + 1 - 1 = 11 - 1 (subtracting both sides by the same number)</li><li>2x = 10</li><li>2x/2&nbsp;= 10/2 (dividing both sides of the equation by the same number)</li><li>x = 5</li>
</ul>
<div id="mntl-sc-block_1-0-30" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-31" class="comp mntl-sc-block mntl-sc-block-html">
So, yes, the two equations are equivalent because x = 5 in each case.
</p>
<div id="mntl-sc-block_1-0-32" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<span class="heading-toc" id="toc-practical-equivalent-equations"></span> <h2 id="mntl-sc-block_1-0-33" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Practical Equivalent Equations </span> </h2>
<p id="mntl-sc-block_1-0-34" class="comp mntl-sc-block mntl-sc-block-html">
You can use equivalent equations in daily life. It's particularly helpful when shopping. For example, you like a particular shirt. One company offers the shirt for $6 and has $12 shipping, while another company offers the shirt for $7.50 and has $9 shipping. Which shirt has the best price? How many shirts (maybe you want to get them for friends) would you have to buy for the price to be the same for both companies?
</p>
<div id="mntl-sc-block_1-0-35" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-36" class="comp mntl-sc-block mntl-sc-block-html">
To solve this problem, let "x" be the number of shirts. To start with, set x =1 for the purchase of one shirt. For company #1:
</p>
<div id="mntl-sc-block_1-0-37" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-38" class="comp mntl-sc-block mntl-sc-block-html">
<li>Price = 6x + 12 = (6)(1) + 12 = 6 + 12 = $18</li>
</ul>
<div id="mntl-sc-block_1-0-39" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-40" class="comp mntl-sc-block mntl-sc-block-html">
For company #2:
</p>
<div id="mntl-sc-block_1-0-41" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-42" class="comp mntl-sc-block mntl-sc-block-html">
<li>Price = 7.5x + 9 = (1)(7.5) + 9 = 7.5 + 9 = $16.50</li>
</ul>
<div id="mntl-sc-block_1-0-43" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-44" class="comp mntl-sc-block mntl-sc-block-html">
So, if you're buying one shirt, the second company offers a better deal.
</p>
<div id="mntl-sc-block_1-0-45" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-46" class="comp mntl-sc-block mntl-sc-block-html">
To find the point where prices are equal, let "x" remain the number of shirts, but set the two equations equal to each other. Solve for "x" to find how many shirts you'd have to buy:
</p>
<div id="mntl-sc-block_1-0-47" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-48" class="comp mntl-sc-block mntl-sc-block-html">
<li>6x + 12 = 7.5x + 9</li>
<li>6x - 7.5x = 9 - 12 (<a href="https://www.thoughtco.com/2-digit-subtraction-worksheets-with-regrouping-2311924" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">subtracting</a> the same numbers or expressions from each side)</li>
<li>-1.5x = -3</li>
<li>1.5x = 3 (dividing both sides by the same number, -1)</li>
<li>x = 3/1.5 (dividing both sides by 1.5)</li>
<li>x = 2</li>
</ul>
<div id="mntl-sc-block_1-0-49" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-50" class="comp mntl-sc-block mntl-sc-block-html">
If you buy two shirts, the price is the same, no matter where you get it. You can use the same math to determine which company gives you a better deal with larger orders and also to calculate how much you'll save using one company over the other. See, algebra is useful!
</p>
<div id="mntl-sc-block_1-0-51" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<span class="heading-toc" id="toc-equivalent-equations-with-two-variables"></span> <h2 id="mntl-sc-block_1-0-52" class="comp mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Equivalent Equations With Two Variables </span> </h2>
<p id="mntl-sc-block_1-0-53" class="comp mntl-sc-block mntl-sc-block-html">
If you have two equations and two unknowns (x and y), you can determine whether two sets of linear equations are equivalent.
</p>
<div id="mntl-sc-block_1-0-54" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-55" class="comp mntl-sc-block mntl-sc-block-html">
For example, if you're given the equations:
</p>
<div id="mntl-sc-block_1-0-56" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-57" class="comp mntl-sc-block mntl-sc-block-html">
<li>-3x + 12y = 15</li><li>7x - 10y = -2</li>
</ul>
<div id="mntl-sc-block_1-0-58" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-59" class="comp mntl-sc-block mntl-sc-block-html">
You can determine whether the following system is equivalent:
</p>
<div id="mntl-sc-block_1-0-60" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-61" class="comp mntl-sc-block mntl-sc-block-html">
<li>-x + 4y = 5</li><li>7x -10y = -2</li>
</ul>
<div id="mntl-sc-block_1-0-62" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-63" class="comp mntl-sc-block mntl-sc-block-html">
To <a href="https://www.thoughtco.com/problem-solving-in-mathematics-2311775" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">solve this problem</a>, find "x" and "y" for each system of equations. If the values are the same, then the systems of equations are equivalent.
</p>
<div id="mntl-sc-block_1-0-64" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-65" class="comp mntl-sc-block mntl-sc-block-html">
Start with the first set. To solve two <a href="https://www.thoughtco.com/solve-algebra-problems-step-by-step-2311970" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">equations</a> with two <a href="https://www.thoughtco.com/what-is-a-scientific-variable-3975929" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">variables</a>, isolate one variable and plug its solution into the other equation. To isolate the "y" variable:
</p>
<div id="mntl-sc-block_1-0-66" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-67" class="comp mntl-sc-block mntl-sc-block-html">
<li>-3x + 12y = 15</li><li>-3x = 15 - 12y</li><li>x = -(15 - 12y)/3 = -5 + 4y (plug in for "x" in the second equation)</li><li>7x - 10y = -2</li><li>7(-5 + 4y) - 10y = -2</li><li>-35 + 28y - 10y = -2</li><li>18y = 33</li><li>y = 33/18 = 11/6</li>
</ul>
<div id="mntl-sc-block_1-0-68" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-69" class="comp mntl-sc-block mntl-sc-block-html">
Now, plug "y" back into either equation to solve for "x":
</p>
<div id="mntl-sc-block_1-0-70" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<ul id="mntl-sc-block_1-0-71" class="comp mntl-sc-block mntl-sc-block-html">
<li>7x - 10y = -2</li><li>7x = -2 + 10(11/6)</li>
</ul>
<div id="mntl-sc-block_1-0-72" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-73" class="comp mntl-sc-block mntl-sc-block-html">
Working through this, you'll eventually&nbsp;get x = 7/3.
</p>
<div id="mntl-sc-block_1-0-74" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-75" class="comp mntl-sc-block mntl-sc-block-html">
To answer the question,&nbsp;you <em>could</em> apply the same principles to the second set of equations to solve for "x" and "y" to find that yes, they are indeed equivalent. It's easy to get bogged down in the algebra, so it's a good idea to check your work using an <a href="http://www.webmath.com/solver2.html" data-component="link" data-source="inlineLink" data-type="externalLink" data-ordinal="1">online equation solver</a>.
</p>
<div id="mntl-sc-block_1-0-76" class="comp mntl-sc-block mntl-sc-block-adslot mntl-block"></div>
<p id="mntl-sc-block_1-0-77" class="comp mntl-sc-block mntl-sc-block-html">
However, the clever student will notice the two sets of equations are equivalent <em>without doing any difficult calculations at all.</em> The only difference between the first equation in each set is that the first one is three times the second one (equivalent). The second equation is exactly the same.
</p>
</div>
</div>
]]></description>
            <pubDate>Wed, 17 Jan 2018 18:49:52 GMT</pubDate>
            <guid isPermaLink="false">https://www.thoughtco.com/understanding-equivalent-equations-4157661</guid>
            <link>https://www.thoughtco.com/understanding-equivalent-equations-4157661</link>
            <author><![CDATA[Anne Marie Helmenstine, Ph.D.]]></author>
                <category>Science</category>
                <category>Tech</category>
                <category>Math</category>
        </item>
        <item>
            <title><![CDATA[10 Math Tricks That Will Blow Your Mind]]></title>
            <description><![CDATA[<figure>
    <img src="https://www.thoughtco.com/thmb/FGRV9FpTnS_WqfKU6xTvkI9vI28=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/Math_class_in_Da_Ji_Junior_High_School_2006-12-1-89e9cb05eb2a4d53a503a97e3f3b1ae1.jpg" referrerpolicy="no-referrer">
  </figure>
<div id="list-sc_1-0" class="comp article-content list list-sc mntl-block">
<div id="article-intro_1-0" class="comp article-content__description article-intro mntl-block">
<p id="mntl-sc-block_1-0" class="comp mntl-sc-block mntl-sc-block-html">
Are you ready to give your mathematics skills a boost? These simple math tricks can help you perform calculations more quickly and easily. They also come in handy if you want to impress your teacher, parents, or friends.
</p>
</div>
<div id="list-sc__content_1-0" class="comp list-sc__content mntl-sc-page mntl-block" data-sc-sticky-offset="85" data-sc-ad-label-height="24" data-sc-ad-track-spacing="100" data-sc-min-track-height="250" data-sc-max-track-height="600" data-sc-breakpoint="50em" data-sc-load-immediate="3" data-sc-content-positions="[1, 1050, 1250, 1550, 1950, 2350, 2750, 3150]" data-bind-scroll-on-start="true">
<div id="list-sc-item_1-0" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">01</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-multiplying-by-6"></span> <h2 id="mntl-sc-block_2-0" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Multiplying by 6 </span> </h2>
<p id="mntl-sc-block_2-0-1" class="comp mntl-sc-block mntl-sc-block-html">
If you multiply 6 by an even number, the answer will end with the same digit. The number in the ten's place will be half of the number in the one's place.
</p>
<p id="mntl-sc-block_2-0-2" class="comp mntl-sc-block mntl-sc-block-html">
Example: 6 x 4 = 24.
</p>
</div>
<div id="list-sc-item_1-0-1" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-1" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">02</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-the-answer-is-2"></span> <h2 id="mntl-sc-block_2-0-3" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Answer Is 2 </span> </h2>
<ol id="mntl-sc-block_2-0-4" class="comp mntl-sc-block mntl-sc-block-html">
<li>Think of a number.</li><li>Multiply it by 3.</li><li>Add 6.</li><li>Divide this number by 3.</li><li>Subtract the number from Step 1 from the answer in Step 4.</li>
</ol>
<p id="mntl-sc-block_2-0-5" class="comp mntl-sc-block mntl-sc-block-html">
The answer is 2.
</p>
</div>
<div id="list-sc-item_1-0-2" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-2" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">03</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-same-three-digit-number"></span> <h2 id="mntl-sc-block_2-0-6" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Same Three-Digit Number </span> </h2>
<ol id="mntl-sc-block_2-0-7" class="comp mntl-sc-block mntl-sc-block-html">
<li>Think of any three-digit number in which each of the digits is the same. Examples include 333, 666, 777, and 999.</li>
<li>Add up the digits.</li>
<li><a href="https://www.thoughtco.com/basic-division-facts-no-remainder-2312184" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Divide</a> the three-digit number by the answer in Step 2.</li>
</ol>
<p id="mntl-sc-block_2-0-8" class="comp mntl-sc-block mntl-sc-block-html">
The answer is 37.
</p>
</div>
<div id="list-sc-item_1-0-3" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-3" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">04</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-six-digits-become-three"></span> <h2 id="mntl-sc-block_2-0-9" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Six Digits Become Three </span> </h2>
<ol id="mntl-sc-block_2-0-10" class="comp mntl-sc-block mntl-sc-block-html">
<li>Take any three-digit number and write it twice to make a six-digit number. Examples include 371371 or 552552.</li><li>Divide the number by 7.</li><li>Divide it by 11.</li><li>Divide it by 13.</li>
</ol>
<p id="mntl-sc-block_2-0-11" class="comp mntl-sc-block mntl-sc-block-html">
The order in which you do the division is unimportant!
</p>
<p id="mntl-sc-block_2-0-12" class="comp mntl-sc-block mntl-sc-block-html">
The answer is the three-digit number.
</p>
<p id="mntl-sc-block_2-0-13" class="comp mntl-sc-block mntl-sc-block-html">
Examples: 371371 gives you 371 or 552552 gives you 552.
</p>
<ol id="mntl-sc-block_2-0-14" class="comp mntl-sc-block mntl-sc-block-html">
<li>A related trick is to take any three-digit number.</li>
<li><a href="https://www.thoughtco.com/magical-multiplication-tricks-2086556" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Multiply</a> it by 7, 11, and 13.</li>
</ol>
<p id="mntl-sc-block_2-0-15" class="comp mntl-sc-block mntl-sc-block-html">
The result will be a six-digit number that repeats the three-digit number.
</p>
<p id="mntl-sc-block_2-0-16" class="comp mntl-sc-block mntl-sc-block-html">
Example: 456 becomes 456456.
</p>
</div>
<div id="list-sc-item_1-0-4" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-4" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">05</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-the-11-rule"></span> <h2 id="mntl-sc-block_2-0-17" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The 11 Rule </span> </h2>
<p id="mntl-sc-block_2-0-18" class="comp mntl-sc-block mntl-sc-block-html">
This is a quick way to multiply two-digit numbers by 11 in your head.
</p>
<ol id="mntl-sc-block_2-0-19" class="comp mntl-sc-block mntl-sc-block-html">
<li>Separate the two digits in your mind.</li><li>Add the two digits together.</li><li>Place the number from Step 2 between the two digits. If the number from Step 2 is greater than 9, put the one's digit in the space and carry the ten's digit.</li>
</ol>
<p id="mntl-sc-block_2-0-20" class="comp mntl-sc-block mntl-sc-block-html">
Examples: 72 x 11 = 792.
</p>
<p id="mntl-sc-block_2-0-21" class="comp mntl-sc-block mntl-sc-block-html">
57 x 11 = 5 _ 7, but 5 + 7 = 12, so put 2 in the space and add the 1 to the 5 to get 627
</p>
</div>
<div id="list-sc-item_1-0-5" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-5" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">06</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-memorizing-pi"></span> <h2 id="mntl-sc-block_2-0-22" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Memorizing Pi </span> </h2>
<p id="mntl-sc-block_2-0-23" class="comp mntl-sc-block mntl-sc-block-html">
To remember the first seven <a href="https://www.thoughtco.com/the-number-pi-3-141592654-3126451" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">digits of pi</a>, count the number of letters in each word of the sentence:
</p>
<p id="mntl-sc-block_2-0-24" class="comp mntl-sc-block mntl-sc-block-html">
"How I wish I could calculate pi."
</p>
<p id="mntl-sc-block_2-0-25" class="comp mntl-sc-block mntl-sc-block-html">
This becomes 3.141592.
</p>
</div>
<div id="list-sc-item_1-0-6" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-6" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">07</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-contains-the-digits-1-2-4-5-7-8"></span> <h2 id="mntl-sc-block_2-0-26" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Contains the Digits 1, 2, 4, 5, 7, 8 </span> </h2>
<ol id="mntl-sc-block_2-0-27" class="comp mntl-sc-block mntl-sc-block-html">
<li>Select a number from 1 to 6.</li><li>Multiply the number by 9.</li><li>Multiply it by 111.</li><li>Multiply it by 1001.</li><li>Divide the answer by 7.</li>
</ol>
<p id="mntl-sc-block_2-0-28" class="comp mntl-sc-block mntl-sc-block-html">
The number will contain the digits 1, 2, 4, 5, 7, and 8.&nbsp;
</p>
<p id="mntl-sc-block_2-0-29" class="comp mntl-sc-block mntl-sc-block-html">
Example: The number 6 yields the answer 714285.
</p>
</div>
<div id="list-sc-item_1-0-7" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-7" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">08</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-multiply-large-numbers-in-your-head"></span> <h2 id="mntl-sc-block_2-0-30" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Multiply Large Numbers in Your Head </span> </h2>
<p id="mntl-sc-block_2-0-31" class="comp mntl-sc-block mntl-sc-block-html">
To easily multiply two double-digit numbers, use their distance from 100 to simplify the math:
</p>
<ol id="mntl-sc-block_2-0-32" class="comp mntl-sc-block mntl-sc-block-html">
<li><a href="https://www.thoughtco.com/2-digit-subtraction-worksheets-with-regrouping-2311924" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Subtract</a> each number from 100.</li>
<li>Add these values together.</li>
<li>100 minus this number is the first part of the answer.</li>
<li>Multiply the digits from Step 1 to get the second part of the answer.</li>
</ol>
</div>
<div id="list-sc-item_1-0-8" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-8" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">09</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-super-simple-divisibility-rules"></span> <h2 id="mntl-sc-block_2-0-33" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Super Simple Divisibility Rules </span> </h2>
<p id="mntl-sc-block_2-0-34" class="comp mntl-sc-block mntl-sc-block-html">
You've got 210 pieces of pizza and want to know whether or not you can split them evenly within your group. Rather than whip out the calculator, use these simple shortcuts to do the math in your head:
</p>
<ul id="mntl-sc-block_2-0-35" class="comp mntl-sc-block mntl-sc-block-html">
<li>Divisible by 2 if the last digit is a multiple of 2 (210).</li>
<li>Divisible by 3 if&nbsp;the sum of the digits is divisible by 3&nbsp;(522 because the digits add up to 9, which is divisible by 3).</li>
<li>Divisible by 4 if&nbsp;the last two digits are divisible by 4 (2540 because 40 is divisible by 4).</li>
<li>Divisible by 5 if&nbsp;the last digit is 0 or 5 (9905).</li>
<li>Divisible by 6 if it passes the rules for both 2 and 3 (408).</li>
<li>Divisible by 9 if the <a href="https://www.thoughtco.com/3-digit-addition-worksheets-no-regrouping-2311907" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">sum</a> of the digits is divisible by 9 (6390 since 6 + 3 + 9 + 0 = 18, which is divisible&nbsp;by 9).</li>
<li>Divisible by 10 if the number ends in a 0 (8910).</li>
<li>Divisible by 12 if the rules for divisibility by 3 and 4 apply.</li>
</ul>
<p id="mntl-sc-block_2-0-36" class="comp mntl-sc-block mntl-sc-block-html">
Example: The 210 slices of pizza may be evenly distributed into groups of 2, 3, 5, 6, 10.
</p>
</div>
<div id="list-sc-item_1-0-9" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-9" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">10</div>
<div class="total">of 10</div>
</div>
</div>
<span class="heading-toc" id="toc-finger-multiplication-tables"></span> <h2 id="mntl-sc-block_2-0-37" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Finger Multiplication Tables </span> </h2>
<p id="mntl-sc-block_2-0-38" class="comp mntl-sc-block mntl-sc-block-html">
Everyone knows you can count on your fingers. Did you realize you can use them for multiplication? A simple way to do the "9" multiplication table is to place both hands in front of you with fingers and thumbs extended. To multiply 9 by a number, fold down that number finger, counting from the left.
</p>
<p id="mntl-sc-block_2-0-39" class="comp mntl-sc-block mntl-sc-block-html">
Examples: To multiply 9 by 5, fold down the fifth finger from the left. Count fingers on either side of the "fold" to get the answer. In this case, the answer is 45.
</p>
<p id="mntl-sc-block_2-0-40" class="comp mntl-sc-block mntl-sc-block-html">
To multiply 9 times 6, fold down the sixth finger, giving an answer of 54.
</p>
</div>
</div>
<div id="article-outro_1-0" class="comp article-outro mntl-block"></div>
</div>
]]></description>
            <pubDate>Mon, 20 Nov 2017 18:29:31 GMT</pubDate>
            <guid isPermaLink="false">https://www.thoughtco.com/math-tricks-that-will-blow-your-mind-4154742</guid>
            <link>https://www.thoughtco.com/math-tricks-that-will-blow-your-mind-4154742</link>
            <author><![CDATA[Anne Marie Helmenstine, Ph.D.]]></author>
                <category>Science</category>
                <category>Tech</category>
                <category>Math</category>
        </item>
        <item>
            <title><![CDATA[8 Facts About Infinity That Will Blow Your Mind]]></title>
            <description><![CDATA[<figure>
    <img src="https://www.thoughtco.com/thmb/oWz9nmFVzF4iPqxp-FGGV9cOtEk=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/moebius-strip-522025950-5a085d4013f1290037101e9d.jpg" referrerpolicy="no-referrer">
  </figure>
<div id="list-sc_1-0" class="comp article-content list list-sc mntl-block">
<div id="article-intro_1-0" class="comp article-content__description article-intro mntl-block">
<p id="mntl-sc-block_1-0" class="comp mntl-sc-block mntl-sc-block-html">
Infinity is an abstract concept used to describe something that is endless or boundless. It is important in mathematics, cosmology, physics, computing, and the arts.
</p>
</div>
<div id="list-sc__content_1-0" class="comp list-sc__content mntl-sc-page mntl-block" data-sc-sticky-offset="85" data-sc-ad-label-height="24" data-sc-ad-track-spacing="100" data-sc-min-track-height="250" data-sc-max-track-height="600" data-sc-breakpoint="50em" data-sc-load-immediate="3" data-sc-content-positions="[1, 1050, 1250, 1550, 1950, 2350, 2750, 3150]" data-bind-scroll-on-start="true">
<div id="list-sc-item_1-0" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">01</div>
<div class="total">of 08</div>
</div>
</div>
<span class="heading-toc" id="toc-the-infinity-symbol"></span> <h2 id="mntl-sc-block_2-0" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Infinity Symbol </span> </h2>
<figure id="mntl-sc-block_2-0-1" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0" class="comp mntl-figure-caption figure-article-caption">
<span class="figure-article-caption-text">The infinity symbol is also known as the lemniscate.</span>
<span class="figure-article-caption-owner">Chris Collins / Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-2" class="comp mntl-sc-block mntl-sc-block-html">
Infinity has its own special symbol: ∞. The symbol, sometimes called the lemniscate, was introduced by clergyman and mathematician John Wallis in 1655. The word "lemniscate" comes from the Latin word <em>lemniscus</em>, which means "ribbon," while the word "infinity" comes from the Latin word <em>infinitas</em>, which means "boundless."
</p>
<p id="mntl-sc-block_2-0-3" class="comp mntl-sc-block mntl-sc-block-html">
Wallis may have based the symbol on the Roman numeral for 1000, which the Romans used to indicate "countless" in addition to the number. It's also possible the symbol is based on omega (Ω or ω), the last letter in the Greek alphabet.
</p>
<p id="mntl-sc-block_2-0-4" class="comp mntl-sc-block mntl-sc-block-html">
The concept of infinity was understood long before Wallis gave it the symbol we use today. Around the 4th or 3rd century B.C.E., the Jain mathematical text <em>Surya Prajnapti</em> assigned numbers as either enumerable, innumerable, or infinite. The <a href="https://www.thoughtco.com/timeline-of-greek-and-roman-philosophers-118808" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Greek philosopher</a> Anaximander used the work <em>apeiron</em> to refer to the infinite. Zeno of Elea (born circa 490 B.C.E.) was known for <a href="https://www.thoughtco.com/what-is-a-paradox-1691563" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="2">paradoxes involving infinity</a>.&nbsp;
</p>
</div>
<div id="list-sc-item_1-0-1" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-1" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">02</div>
<div class="total">of 08</div>
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</div>
<span class="heading-toc" id="toc-zenos-paradox"></span> <h2 id="mntl-sc-block_2-0-5" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Zeno's Paradox </span> </h2>
<figure id="mntl-sc-block_2-0-6" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/BOeejWpYm-lmL61xmmfXe4pQuIw=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/tortoise-and-hare--finish-line-143576837-5a08a081494ec90037e9c6bb.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-1" class="comp mntl-figure-caption figure-article-caption">
<span class="figure-article-caption-text">If the rabbit was forever halving the distance to the tortoise, the tortoise would win the race.</span>
<span class="figure-article-caption-owner">Don Farrall / Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-7" class="comp mntl-sc-block mntl-sc-block-html">
Of all Zeno's paradoxes, the most famous is his paradox of the Tortoise and Achilles. In the paradox, a tortoise challenges the <a href="https://www.thoughtco.com/achilles-greek-hero-of-the-trojan-war-116708" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">Greek hero Achilles</a> to a race, providing the tortoise is given a small head start. The tortoise argues he will win the race because as Achilles catches up to him, the tortoise will have gone a bit further, adding to the distance.
</p>
<p id="mntl-sc-block_2-0-8" class="comp mntl-sc-block mntl-sc-block-html">
In simpler terms, consider crossing a room by going half the distance with each stride. First, you cover half the distance, with half remaining. The next step is half of one-half, or a quarter. Three quarters of the distance is covered, yet a quarter remains. Next is 1/8th, then 1/16th, and so on. Although each step brings you closer, you never actually reach the other side of the room. Or rather, you&nbsp;would after taking an infinite number of steps.
</p>
</div>
<div id="list-sc-item_1-0-2" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-2" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">03</div>
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<span class="heading-toc" id="toc-pi-as-an-example-of-infinity"></span> <h2 id="mntl-sc-block_2-0-9" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Pi as an Example of Infinity </span> </h2>
<figure id="mntl-sc-block_2-0-10" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/4-rZRqlIALa1g3bEynp65ehpYaI=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/pi-formula-on-blackboard-112303538-5a089c5247c266003765ecbe.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-2" class="comp mntl-figure-caption figure-article-caption">
<span class="figure-article-caption-text">Pi is a number consisting of an infinite number of digits.</span>
<span class="figure-article-caption-owner">Jeffrey Coolidge / Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-11" class="comp mntl-sc-block mntl-sc-block-html">
Another good example of infinity is the <a href="https://www.thoughtco.com/the-number-pi-3-141592654-3126451" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">number π or pi</a>. Mathematicians use a symbol for pi because it's impossible to write the number down. Pi consists of an infinite number of digits. It's often rounded to 3.14 or even 3.14159, yet no matter how many digits you write, it's impossible to get to the end.
</p>
</div>
<div id="list-sc-item_1-0-3" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-3" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">04</div>
<div class="total">of 08</div>
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<span class="heading-toc" id="toc-the-monkey-theorem"></span> <h2 id="mntl-sc-block_2-0-12" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> The Monkey Theorem </span> </h2>
<figure id="mntl-sc-block_2-0-13" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/OiESkBYL-gXGXvhaa-p68mj8YrM=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/furry-animal-hands-use-laptop-computer-with-blank-screen-169981126-5a08b2fa845b34003b83cd50.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-3" class="comp mntl-figure-caption figure-article-caption">
<span class="figure-article-caption-text">Given an infinite amount of time, a monkey could write the great American novel.</span>
<span class="figure-article-caption-owner">PeskyMonkey / Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-14" class="comp mntl-sc-block mntl-sc-block-html">
One way to think about infinity is in terms of the monkey theorem. According to the theorem, if you give a monkey a typewriter and an infinite amount of time, eventually it will write Shakespeare's <em>Hamlet</em>. While some people take the theorem to suggest anything is possible, mathematicians see it as evidence of just how improbable certain events are.
</p>
</div>
<div id="list-sc-item_1-0-4" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-4" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">05</div>
<div class="total">of 08</div>
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</div>
<span class="heading-toc" id="toc-fractals-and-infinity"></span> <h2 id="mntl-sc-block_2-0-15" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Fractals and Infinity </span> </h2>
<figure id="mntl-sc-block_2-0-16" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/xCf2m8IHCjJxu4eeb-nxJyXjakw=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/soap-bubbles-spirals-585837119-5a088fba845b34003b7a4f65.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-4" class="comp mntl-figure-caption figure-article-caption">
<span class="figure-article-caption-text">A fractal may be magnified over and over, to infinity, always revealing more detail.</span>
<span class="figure-article-caption-owner">PhotoviewPlus / Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-17" class="comp mntl-sc-block mntl-sc-block-html">
A fractal is an abstract mathematical object, used in art and to simulate natural phenomena. Written as a mathematical equation, most fractals are nowhere differentiable. When viewing an image of a fractal, this means you could zoom in and see new detail. In other words, a fractal is infinitely magnifiable.
</p>
<p id="mntl-sc-block_2-0-18" class="comp mntl-sc-block mntl-sc-block-html">
The Koch snowflake is an interesting example of a fractal. The snowflake starts as an equilateral triangle. For each iteration of the fractal:
</p>
<ol id="mntl-sc-block_2-0-19" class="comp mntl-sc-block mntl-sc-block-html">
<li>Each line segment is divided into three equal segments.</li><li>An equilateral triangle is drawn using the middle segment as its base, pointing outward.</li><li>The line segment serving as the base of the triangle is removed.</li>
</ol>
<p id="mntl-sc-block_2-0-20" class="comp mntl-sc-block mntl-sc-block-html">
The process may be repeated an infinite number of times. The resulting snowflake has a finite area, yet it is bounded by an infinitely long line.
</p>
</div>
<div id="list-sc-item_1-0-5" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-5" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">06</div>
<div class="total">of 08</div>
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</div>
<span class="heading-toc" id="toc-different-sizes-of-infinity"></span> <h2 id="mntl-sc-block_2-0-21" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Different Sizes of Infinity </span> </h2>
<figure id="mntl-sc-block_2-0-22" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/thmb/xtQmNnVgTGTV3tzKrr_MIp8EsYg=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/hands-holding-complex-cats-cradle-molecule-network-723497851-5a08a43813f12900372321fd.jpg" referrerpolicy="no-referrer">
  </figure>
<figcaption id="mntl-figure-caption_1-0-5" class="comp mntl-figure-caption figure-article-caption">
<span class="figure-article-caption-text">Infinity comes in different sizes.</span>
<span class="figure-article-caption-owner">Tang Yau Hoong / Getty Images</span>
</figcaption>
</figure>
<p id="mntl-sc-block_2-0-23" class="comp mntl-sc-block mntl-sc-block-html">
Infinity is boundless, yet it comes in different sizes. The positive numbers (those greater than 0) and the negative numbers (those smaller than 0) may be considered to be <a href="https://www.thoughtco.com/examples-of-uncountable-sets-3126438" data-component="link" data-source="inlineLink" data-type="internalLink" data-ordinal="1">infinite sets</a> of equal sizes. Yet, what happens if you combine both sets? You get a set twice as large. As another example, consider all of the even numbers (an infinite set). This represents an infinity half the size of all of the whole numbers.
</p>
<p id="mntl-sc-block_2-0-24" class="comp mntl-sc-block mntl-sc-block-html">
Another example is simply adding 1 to infinity. The number ∞ + 1 &gt; ∞.
</p>
</div>
<div id="list-sc-item_1-0-6" class="comp mntl-sc-list-item list-sc-item mntl-block">
<div id="list-marker_1-0-6" class="comp list-marker list-marker--numbers">
<div class="content-list-number">
<div class="item-number">07</div>
<div class="total">of 08</div>
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</div>
<span class="heading-toc" id="toc-cosmology-and-infinity"></span> <h2 id="mntl-sc-block_2-0-25" class="comp mntl-sc-list-item-title mntl-sc-block reference-sc-block-heading mntl-sc-block-heading"> <span class="mntl-sc-block-heading__text"> Cosmology and Infinity </span> </h2>
<figure id="mntl-sc-block_2-0-26" class="comp mntl-sc-block-image mntl-sc-block reference-sc-block-image mntl-sc-block-universal-image figure-landscape figure-high-res">
  <figure>
    <img src="https://www.thoughtco.com/t