From cc0282e4a0b88cf3605bfd5b211ab0a28949bf16 Mon Sep 17 00:00:00 2001
From: Aaron Taylor <ataylor@subgeniuskitty.com>
Date: Sun, 2 May 2021 23:24:31 -0700
Subject: [PATCH] Initial commit of some notes related to chapter 1.

---
 notes/chapter-1.tex    | 35 ++++++++++++++++++++++++++++++++++-
 notes/config.tex       |  3 ++-
 notes/introduction.tex | 15 ++++++++++++++-
 3 files changed, 50 insertions(+), 3 deletions(-)

diff --git a/notes/chapter-1.tex b/notes/chapter-1.tex
index 265fd80..917a31a 100644
--- a/notes/chapter-1.tex
+++ b/notes/chapter-1.tex
@@ -1,4 +1,37 @@
 \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.
diff --git a/notes/config.tex b/notes/config.tex
index 619418f..1587fd3 100644
--- a/notes/config.tex
+++ b/notes/config.tex
@@ -33,6 +33,7 @@
 \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}
@@ -42,7 +43,6 @@
 	\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}}
@@ -56,6 +56,7 @@
 	}    
 \theoremstyle{definition}
 \newtheorem{defi}{Definition}
+\newtheorem{axiom}{Axiom}
 \newtheorem{theorem}{Theorem}
 \newtheorem{lemma}[theorem]{Lemma}
 \newtheorem{problem}{Problem}
diff --git a/notes/introduction.tex b/notes/introduction.tex
index a96df0d..7f1e7b6 100644
--- a/notes/introduction.tex
+++ b/notes/introduction.tex
@@ -1,4 +1,17 @@
 \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
+$x \leq y$.
-- 
2.20.1

