\newpage
\section{Notes: Chapter 1}
-Hello from chapter 1.
+In the first chapter, Knuth provides two axioms, reproduced here.
+
+\begin{axiom}
+Every number corresponds to two sets of previously created numbers, such that
+no member of the left set is greater than or equal to any member of the right
+set.
+\end{axiom}
+
+For example, if $x = \surreal{X_L}{X_R}$, then $\forall x_L \in X_L$, it must
+hold $\forall x_R \in X_R$ that $x_L \ngeq x_R$.
+
+\begin{axiom}
+One number is less than or equal to another number if and only if no member of
+the first number's left set is greater than or equal to the second number, and
+no member of the second number's right set is less than or equal to the first
+number.
+\end{axiom}
+
+For example, the relation $\surreal{X_L}{X_R} = x \leq y = \surreal{Y_L}{Y_R}$
+holds true if and only if both $X_L \ngeq y$ and $Y_R \nleq x$.
+
+With no surreal numbers yet in our possession, we construct the first surreal
+number using the null set (or void set, as Knuth calls it) as both the left and
+right set. Although we have not yet examined its properties, Knuth names this
+number ``zero''. Thus, $\surreal{}{} = 0$.
+
+As his final trick, Knuth defines a second generation of surreal numbers using
+$0$ in the left and right set, naming them $1$ and $-1$ and claiming the
+following relation.
+
+$$-1 = \surreal{}{0} \leq 0 = \surreal{}{} \leq 1 = \surreal{0}{}$$
+
+Before we try generating some more surreal numbers, we should look for useful
+equivalence classes.
\newcommand{\sups}[1]{\ensuremath{^{\textrm{#1}}}}
\newcommand{\subs}[1]{\ensuremath{_{\textrm{#1}}}}
\newcommand{\abs}[1]{\left|#1\right|}
+\newcommand{\surreal}[2]{\ensuremath{\left\langle #1 \vert #2 \right\rangle}}
\newcommand{\horzline}[1][350]{\begin{center} \line(1,0){#1} \end{center}}
\renewcommand\maketitle{
\thispagestyle{plain}
\end{center}
\horzline
}
-% TODO: What do I want to define for things like bra-ket notation?
\renewcommand*{\vec}[1]{\ensuremath{{\bf#1}}}
\newcommand*{\norm}[1]{\ensuremath{\left\lVert#1 \right\rVert}}
\newcommand*{\ip}[1]{\ensuremath{\langle#1\rangle}}
}
\theoremstyle{definition}
\newtheorem{defi}{Definition}
+\newtheorem{axiom}{Axiom}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{problem}{Problem}
\newpage
\section{Introduction}
-Hello, World!
+These notes accompany the book Surreal Numbers by Donald Knuth, specifically
+the 1974 edition. They are only intended to further my own understanding; no
+guarantees of accuracy, relevance, or significance are extended.
+
+\section{Notation}
+
+A surreal number $X$ consisting of left set $X_L$ and right set $X_R$ is
+represented as \surreal{X_L}{X_R}. The void set, as Knuth named it, is
+represented by leaving the appropriate left or right set empty, as in
+\surreal{}{}, the first surreal number defined.
+
+When applying binary relations like less-than-or-equal to sets, the notation $X
+\leq Y$ means that, $\forall x \in X$ and $\forall y \in Y$, it holds true that
+$x \leq y$.