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1% LESSON ONE
2% Text following a percentage sign (%) is ignored by the typesetter.
3% In this manual, we will use such text for comments and notes.
4% You do not need to type out these comments.
5
6\documentclass{amsart}
7% Almost all commands start with a backslash (\).
8% amsart is short for American Mathematical Society article.
9
10\thispagestyle{empty}
11%We use this to prevent a page number from appearing.
12
13\begin{document}
14\begin{center}
15 \textbf{Lesson One: The Basics}
16\end{center}
17
18\textbf{Font Families}
19
20% One or more blank lines end a paragraph and put the typesetter on a new line.
21I \textbf{love} math!
22
23\LaTeX\ does \emph{not} rhyme with ``paychecks''!\\
24% The \ after \LaTeX creates a small horizontal space. Try leaving it out.
25% The \\ at the end creates additional vertical space between lines.
26% See Lesson Six for more on spacing.
27% In most cases, \emph and \textit look the same.
28
29\textbf{Math Enviroments, Superscripts, and Subscripts}
30
31The numbers 3, 4, and 5 are a Pythagorean triple because $3^2+4^2=5^2$.\\
32% The dollar signs ($) open and close the inline math environment.
33
34We all know that $(x^n)'=nx^{n-1}$, but what is $(x^x)'$?\\
35% Note that the typesetter ignores multiple spaces as above.
36
37The recursive definition for the Fibonacci numbers is
38
39$F_1=1$, $F_2=1$, and $F_n=F_{n-1}+F_{n-2}$ for all integers $n>2$.\\
40
41Compare $\frac{1}{a+b}\neq\frac{1}{a}+\frac{1}{b}$ to the following.
42
43$$\frac{1}{a+b}\neq\frac{1}{a}+\frac{1}{b}$$
44%The double dollar signs ($$) open and close the displayed math environment.
45
46We can also display it midline by using ``display-style'':
47$\displaystyle\frac{1}{a+b}\neq\frac{1}{a}+\frac{1}{b}$.\\
48% Make sure your typeset quotation marks look right!
49
50% To force a page break at any place in your document, you can use
51% \pagebreak or \newpage.
52
53
54\begin{center}
55 \textbf{Exercise One: The Basics}
56\end{center}
57
58A certain math professor will \textbf{scream} and may even \textbf{cry} if you
59confuse the Latin terms \emph{id est}, meaning ``that is to say'' and
60\emph{exempli gratia}, meaning ``for instance.''
61Compare the following.\\
62
63I adore polynomials, \emph{e.g.}, $x^{4}+x^{2}+1$.
64
65
66I adore polynomials, \emph{i.e.}, expressions of the form
67$a_{0} + a_{1}x + a_{2}x^{2} + \dots + a_{n}x^{n}$,where $a_{0},a_{1},...,a_{n}$ are
68constants and $n$ is a non-negative integer.\\
69
70Do you lie awake at night wondering what is the smallest positive integer that
71can be written as the sum of the two perfect cubes in two distinct ways? Well,
72wonder no more:
73
74$$ 1729 = 10^{3} + 9^{3} = 12^{3} + 1^{3} $$
75
76We all learned in first grade that $(a + b)^{3} \neq a^{3} + b^{3}$, but when we
77study modular arithmetic we'll find that the two \emph{are} equal in ``mod 3.''
78
79The famous mathematician Euler (rhymes with ``boiler'' \textbf{NOT} with
80``ruler'') used the geometric series formula
81$a + ar + ar^{2} + ar^{3} + ... = a / (1 - r)$ to conclude the following
82
83$$ 1 - 1 + 1 - 1 + 1 - \dots = \frac{1}{2} $$
84
85But we know the formula only applies if -1 $<$ $r$ $<$ 1, so this result is not
86valid.
87
88
89\end{document}