% LESSON EIGHT
          
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	\textbf{Lesson Eight: Over/Under and Boxes}
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\textbf{Overline and Underline}

In some cases, we like to underline important notions, such as \underline{cardinality}.

A common notation for the cardinality of the set $A$ is $\overline{\overline{A}}$.\\

\textbf{Overbrace and Underbrace}

Can't remember those pesky exponent rules? Happens to all of us.
\footnote{Well, not to me, of course.}

Suppose $m,n\in\mathbb{N}$. Then we have the following
$$a^m a^n = (\underbrace{a\cdot a\cdot \ldots \cdot a}_{m\text{ times}})
(\underbrace{a\cdot a\cdot \ldots \cdot a}_{n\text{ times}})
=(\underbrace{a\cdot a\cdot \ldots \cdot a}_{m+n\text{ times}})=a^{m+n}$$
% The format is \underbrace{STUFF ENCLOSED IN THE BRACE}_{LABEL ON THE BRACE}.
% Similarly, one can use \overbrace{STUFF ENCLOSED IN THE BRACE}^{LABEL ON THE BRACE}.
% Note the switch from _ to ^.

We all know that $e^w=1+w+\frac{w^2}{2!}+\frac{w^3}{3!}+\dots$ for all $w\in\mathbb{R}$. 
But what happens if we're so reckless as to let $w=ix$, 
where $i=\sqrt{-1}\in\mathbb{C}$? Let's see.
\begin{align*}
e^{ix} &= 1 + ix +\frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} 
  + \frac{(ix)^5}{5!}  + \dots \\
&= 1 + ix -\frac{x^2}{2!} - \frac{ix^3}{3!} + \frac{x^4}{4!} 
  + \frac{ix^5}{5!} +  \dots \\
&= \underbrace{1 -\frac{x^2}{2!} + \frac{x^4}{4!} -  
  \dots}_{\text{the series for $\cos x$}} 
  + i\Bigl(\underbrace{x - \frac{x^3}{3!} +\frac{x^5}{5!} - 
  \dots}_{\text{the series for $\sin x$}} \Bigr)\\
&= \cos x + i\sin x
\end{align*}

% See Lesson Nine for the \overset and \underset commands.

\textbf{Boxes}

A consequence of the above formula is the following, which relates five 
of the most important numbers in mathematics and thus deserves its own box. 
$$\boxed{e^{i\pi}+1=0}$$

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