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Warnings-and-common-mistakes.tex
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Warnings-and-common-mistakes.tex
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\Section{Warnings and common mistakes}
\begin{WarningBox}{Negation is like multiplication: $-1\times \dots$}
A negative sign before a number is like multiplication,
so apply it \emph{after} an exponent on that number.
\begin{equation*}
-3^2 = -1 \times (3^2) = -9
\end{equation*}
but
\begin{equation*}
(-3)^2 = (-1 \times 3)^2 = 9
\end{equation*}
\end{WarningBox}
\begin{WarningBox}{Substitutions must respect structure}
Sometimes you need extra parentheses:
\begin{equation*}
\Where{u^2}{u = a + b} = \left(a + b\right)^2 \qquad \text{ \emph{and not} } a + b^2
\end{equation*}
In the quadratic formula, for example
\begin{equation*}
\Where{b^2}{b = -3} = (-3)^2 = 9 \qquad \text{ \emph{and not} } -3^2 = -9
\end{equation*}
\end{WarningBox}
\begin{WarningBox}{Fractions can be split on $+$ or $-$ only in the numerator}
Splitting the numerator is okay:
\begin{equation*}
\frac{2 + x}{7} = \frac{2}{7} + \frac{x}{7}
\end{equation*}
But splitting the denominator \emph{does not work:}
\begin{equation*}
\frac{7}{2 + x} \qquad \text{ \emph{is not} } \frac{7}{2} + \frac{7}{x}
\end{equation*}
\end{WarningBox}
\begin{WarningBox}{Fractions can be spread out on multiplication}
When you have multiplication of factors divided by multiplication of factors,
you can spread out the fraction:
\begin{equation*}
\frac{x^2 y}{x^2 (3 + z)}
= \frac{x^2}{x^2} \cdot \frac{y}{1} \cdot \frac{1}{3 + z}
\end{equation*}
But when you have addition or subtraction as the outermost operation in the numerator or denominator, spreading out \emph{does not work:}
\begin{equation*}
\frac{x^2 + y}{x^2 (3 + z)}
\qquad \text{ \emph{is not} }
\frac{x^2}{x^2} + \frac{y}{1} + \frac{1}{3 + z}
\end{equation*}
\end{WarningBox}
\begin{WarningBox}{Cancel only factors}
When you have multiplication of factors divided by multiplication of factors,
canceling a common factor is an option, as long as it's nonzero:
\begin{equation*}
\frac{x^2 y}{x^2 (3 + z)}
= \frac{\cancel{x^2} y}{\cancel{x^2} (3 + z)}
= \frac{y}{3 + z}
\end{equation*}
But when you have addition or subtraction as the outermost operation in the numerator or denominator, canceling \emph{does not work:}
\begin{equation*}
\frac{x^2 + y}{x^2 (3 + z)}
\qquad \text{ \emph{is not} } \frac{y}{3 + z}
\end{equation*}
\begin{equation*}
\frac{x^2 y}{x^2 + (3 + z)}
\qquad \text{ \emph{is not} } \frac{y}{3 + z}
\end{equation*}
\begin{equation*}
\frac{x^2 + y}{x^2 + 3 + z}
\qquad \text{ \emph{is not} } \frac{y}{3 + z}
\end{equation*}
\end{WarningBox}
\begin{WarningBox}{Power applied to $+$ or $-$ requires FOIL}
When a power is applied to \emph{multiplication}, you can distribute the power:
\begin{equation*}
(5 x)^2 = 5^2 x^2 = 25 x^2
\end{equation*}
But when a power is applied to \emph{addition or subtraction}, you \emph{can't} distribute the power.
You do have the option of expanding with FOIL:
\begin{equation*}
(5 + x)^2 = (5 + x)(5 + x) = 25 + 10x + x^2
\qquad \text{ \emph{which is not} } 25 + x^2
\end{equation*}
\end{WarningBox}
\begin{WarningBox}{Root applied to $+$ or $-$ is mostly hopeless}
When a root is applied to \emph{multiplication}, you can distribute the root:
\begin{equation*}
\sqrt{9 x} = \sqrt{9} \cdot \sqrt{x} = 3 \sqrt{x}
\end{equation*}
But when a root is applied to \emph{addition or subtraction}, you \emph{can't} distribute:
\begin{equation*}
\sqrt{9 - x} \qquad \text{ \emph{is not} } 3 - \sqrt{x}
\end{equation*}
and there's usually not much else you can do with that.
\end{WarningBox}
\begin{WarningBox}{When solving, apply operations to all of both sides}
You can multiply all of both sides of an equation by a number:
\begin{equation*}
\begin{split}
\frac{3 + x}{2}
&= 1 + x
\\
2 \cdot \left(\frac{3 + x}{2}\right)
&= \left(1 + x\right) \cdot 2
\qquad \text{ \emph{which is not} } 1 + x \cdot 2
\\
\cancel{2} \cdot \left(\frac{3 + x}{\cancel{2}}\right)
&= 2 + 2x
\qquad \text{ \emph{which is not} } 1 + 2x
\\
3 + x &= 2 + 2x
\end{split}
\end{equation*}
\end{WarningBox}
\begin{WarningBox}{Multiplying a fraction is not the same as changing terms}
Multiply by $1$ to change terms:
\begin{equation*}
\begin{split}
\frac{x}{2} &= 1 \cdot \frac{x}{2}
\\
\frac{x}{2} &= \frac{3}{3} \cdot \frac{x}{2}
\\
\frac{x}{2} &= \frac{3 \cdot x}{3 \cdot 2}
\\
\frac{x}{2} &= \frac{3x}{6}
\end{split}
\end{equation*}
But multiplying a fraction by a number works like this:
\begin{equation*}
\begin{split}
3 \cdot \frac{x}{2} &= \frac{3}{1} \cdot \frac{x}{2}
\qquad \text{ \emph{which is not} } \frac{3 \cdot x}{3 \cdot 2}
\\
3 \cdot \frac{x}{2} &= \frac{3 \cdot x}{1 \cdot 2}
\\
3 \cdot \frac{x}{2} &= \frac{3x}{2}
\end{split}
\end{equation*}
\end{WarningBox}
\begin{WarningBox}{Careful with calculators and denominators}
When using left-to-right notation, calculators group division and multiplication together, so
\begin{equation*}
\mathtt{60 / 3 * 4}
\text{ means }
\left(\nicefrac{60}{3}\right) \cdot 4
= 80
\end{equation*}
If your denominator includes multiplication, you need extra parentheses
\begin{equation*}
\text{ To compute }
\frac{60}{3 \cdot 4}
\text{ enter }
\mathtt{60 / (3 * 4)}
\end{equation*}
\end{WarningBox}
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