1. Introduction to James Gasper Mathias

2. What is TeX? TeX is a typesetting system that is used mostly for mathematical and scientific documents. It is highly customizable and has every possible character imaginable from every written language. TeX is open-source. It compiles the same resulting document from any computer.

3. \textbf{Prove the binomial theorem for complex numbers.} Base Case: Let $n = 1$, \begin{eqnarray*} \textmd{Then: }(z+w)^{1} &=& z+w \\&=& 1 \cdot z^{1} + 1 \cdot w^{1} \\&=& {{1}\choose{0}}z^{1} + {{1}\choose{1}}w^{1} \end{eqnarray*} Also we know that: $${{n}\choose{0}} = 1 = {{n+1}\choose{0}} \ \ \ \ \ \text{and} \ \ \ \ \ {{n}\choose{n}} = 1 = {{n+1}\choose{n+1}}$$ And we show that: \begin{eqnarray*} {{n}\choose{k}}+{{n}\choose{k-1}} &=& \frac{n!}{k!(n- k)!} + \frac{n!}{(k-1)!(n-(k-1))!} \\&=& \frac{n!}{k!(n-k)!} \cdot \frac{n-k+1}{n-k+1} + \frac{n!}{(k-1)!(n-k+1)!} \cdot \frac{k}{k} \\&=& \frac{n!(n-k+1)}{k!(n-k+1)!} + \frac{n!k}{k!(n- k+1)!} \\&=& \frac{n!(n-k+1)+n!k}{k!(n-k+1)!} \\&=& \frac{n!((n-k+1)+k)}{k!((n+1)-k)!} \\&=& \frac{n!(n+1)}{k!((n+1)-k)!} \\&=& \frac{(n+1)!}{k!((n+1)-k)!} \\&=& {{n+1}\choose{k}} \end{eqnarray*}

4. Inductive Step: \begin{eqnarray*} \textmd{Assume: }(z+w)^{n} &=& \sum_{k=0}^{n}{{n}\choose{k}}z^{n-k}w^{k} \\&=& {{n}\choose{0}}z^{n} + {{n}\choose{1}}z^{n-1}w + \cdots + {{n}\choose{n-1}}zw^{n-1} + {{n}\choose{n}}w^{n} \end{eqnarray*} \begin{eqnarray*} \textmd{Then: }(z+w)^{n+1} &=&(z+w) \cdot (z+w)^{n} \\&=& (z+w) \cdot \sum_{k=0}^{n}{{n}\choose{k}}z^{n-k}w^{k} \\&& \\&=& z \cdot \sum_{k=0}^{n}{{n}\choose{k}}z^{n-k}w^{k} + w \cdot \sum_{k=0}^{n}{{n}\choose{k}}z^{n-k}w^{k} \\&& \\&=& \sum_{k=0}^{n}{{n}\choose{k}}z^{n-k+1}w^{k} + \sum_{k=0}^{n}{{n}\choose{k}}z^{n-k}w^{k+1} \\&& \\&& \\&=& \left[ {{n}\choose{0}}z^{n+1} + {{n}\choose{1}}z^{n}w + \cdots + {{n}\choose{n-1}}z^{2}w^{n-1} + {{n}\choose{n}}zw^{n} \right] \\&& \ \ \ \ \ \ \ \ \ \ \ \ \ + \left[ {{n}\choose{0}}z^{n}w + {{n}\choose{1}}z^{n-1}w^{2} + \cdots + {{n}\choose{n-1}}zw^{n} + {{n}\choose{n}}w^{n+1} \right] \\&& \\&& \\&=& {{n}\choose{0}}z^{n+1} + \left[ {{n}\choose{1}}+{{n}\choose{0}} \right] z^{n}w + \left[ {{n}\choose{2}}+{{n}\choose{1}} \right] z^{n-1}w^{2} + \cdots \\&&+ \left[ {{n}\choose{n-1}}+{{n}\choose{n-2}} \right] z^{2}w^{n-1} + \left[ {{n}\choose{n}}+{{n}\choose{n-1}} \right] zw^{n} + {{n}\choose{n}}w^{n+1} \\&& \\&& \\&=& {{n+1}\choose{0}}z^{n+1} + {{n+1}\choose{1}}z^{n}w + {{n+1}\choose{2}}z^{n-1}w^{2} + \cdots \\&&+ {{n+1}\choose{n-1}}z^{2}w^{n-1} + {{n+1}\choose{n}}zw^{n} + {{n+1}\choose{n+1}}w^{n+1} \\&& \\&& \\&=& \sum_{k=0}^{n+1}{{n+1}\choose{k}}z^{n+1-k}w^{k} \end{eqnarray*} \begin{eqnarray*} \textmd{Thus: } (z+w)^{n} = \sum_{k=0}^{n}{{n}\choose{k}}z^{n-k}w^{k} \textmd{ for any positive integer $n$.} \end{eqnarray*}

5. How to Install LaTeX Website https://latex-project.org/ Download Wait for download Install Update old packages Find and install new packages Download supplemental TeX programs http://www.xm1math.net/texmaker/ http://texstudio.sourceforge.net/

6. The Preamble Preamble: – \documentclass[options]{class} – \usepackage[options]{package1,package2, …} – \newcommand{newcommand}{oldcommand} – \title{title} – \author{me} – \date{today?} \documentclass[12pt]{article} \topmargin -15mm \textheight 24truecm \textwidth 16truecm \oddsidemargin 5mm \evensidemargin 5mm \setlength\parskip{10pt} \pagestyle{empty} \usepackage{boxedminipage,amsfonts,amsmath,amss ymb,graphicx,amsthm,t1enc,subfig} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newtheorem{thm}{Theorem} \newtheorem{lem}{Lemma} \newtheorem{prop}{Proposition} \newtheorem{corly}{Corollary} \newtheorem{defn}{Definition} \newtheorem{expl}{Example}

7. The Document Remember: \begin{} is followed by \end{}. LaTeX is case-sensitive. Don’t worry about text formatting; the preamble takes care of that. Make use of different structures. Use * to disable automatic numbering. \begin{document} \maketitle \begin{abstract} Write something here \end{abstract} \begin{problem}[1] Problem 1. \end{problem} \begin{solution} Solution 1 \end{solution} \section*{A} \subsection*{A.1} \section*{B} \end{document}

8. Important Characters $math$ for in-line math formulas $$math$$ for out-of-line formulas { and } for invisible parentheses \command for specific command & for alignment in special structures \\ for next line \frac{}{} and \dfrac{}{} for fractions % for invisible comments and notes In-line math formula: $\sum_{i=1}^{\infty} \sqrt[i]{i} = \infty$ vs. out-of-line math formula: $$\sum_{i=1}^{\infty} \sqrt[i]{i} = \infty$$ \\ Using (): $e^(x+1)$ vs. using \{\}: $e^{x+1}$ \\ %This sentence does not exist. Regular fraction: $\frac{1}{\pi}$ vs. display fraction: $\dfrac{1}{\pi}$

9. Other Structures \newtheorem{thm}{Theorem} \newtheorem{prop}{Proposition} \begin{thm}[$1^{\text{st}}$ Bible Theorem] God wrote the Bible. \end{thm} \begin{thm}[$2^{\text{st}}$ Bible Theorem] Everything God writes must be true. \end{thm} \begin{prop}[$1^{\text{st}}$ Bible Proposition] The Bible is true. \end{prop} \begin{proof} By \textbf{Theorems 1 \textmd{and} 2}, since everything God writes must be true, and since God wrote the Bible, so the Bible must be true. \end{proof} Theorems Propositions Proofs

10. \textbf{Ordinal Numbers} \begin{align} 0 =& \ \emptyset \\1 =& \ \{ 0 \} = \{ \emptyset \} \\2 =& \ \{ 0, 1 \} = \{ \emptyset, \{ \emptyset \} \} \\3 =& \ \{ 0, 1, 2 \} = \{ \emptyset, \{ \emptyset \}, \{ \emptyset, \{ \emptyset \} \} \} \\ \vdots& \\n+1 =& \ \{ 0, \ldots, n\} = n \cup \{n\} \\ \vdots& \\\aleph_{0} = \omega =& \ \{ 0, 1, \ldots\} = \mathbb{N} \\\omega +1 =& \ \omega \cup \{\omega\} \\ \vdots& \end{align} \\ \textbf{Famous Quotes Throughout the Ages} \begin{itemize} \item ``Expecto Patronum!" \subitem -Harry Potter \item ``I Choose You!" \subitem -Ash Ketchum \item ``Autobots, Roll Out!" \subitem -Optimus Prime \end{itemize} Align Itemize Enumerate Other Structures

11. Interecting Examples Look at Examples in “Introduction to LaTeX.tex”

12. Websites https://latex-project.org/ http://detexify.kirelabs.org/classify.html http://texample.net/ http://latex.wikia.com/wiki/List_of_LaTeX_symbols http://artofproblemsolving.com/wiki/index.php/LaTeX https://www.overleaf.com/latex/learn/free-online- introduction-to-latex-part-1 https://www.overleaf.com/latex/learn/free-online- introduction-to-latex-part-1 https://www.overleaf.com/latex/templates