From: Aaron Taylor <ataylor@subgeniuskitty.com>
Date: Wed, 12 May 2021 08:19:31 +0000 (-0700)
Subject: Added notes related to Chapter 2.
X-Git-Url: http://git.subgeniuskitty.com/surreal-numbers/.git/commitdiff_plain/37beaed00bac5ed57a7e5ac1a4b51b1080392dc4

Added notes related to Chapter 2.
---

diff --git a/notes/chapter-2.tex b/notes/chapter-2.tex
new file mode 100644
index 0000000..448be37
--- /dev/null
+++ b/notes/chapter-2.tex
@@ -0,0 +1,56 @@
+\newpage
+\section{Notes: Chapter 2}
+
+\subsection{Review}
+
+In the second chapter, Knuth doesn't add much new information. The only
+interesting bits I see are the following two quotes:
+
+\begin{framed}
+``The Stone's version is a little different, but $x \leq y$ must mean the same
+thing as $y \leq x$.''
+\end{framed}
+
+\begin{framed}
+``On the first day of creation, Conway ``proves'' that $0 \leq 0$. Why should
+he bother to prove that something is less than or equal to itself, since it's
+obviously equal to itself.''
+\end{framed}
+
+Why, indeed? It seems our prior efforts are headed in a useful direction.
+
+
+\subsection{Exploration}
+
+This week I'm interested in defining an addition operation on the surreal
+numbers generated during my Chapter 1 explorations.
+
+When crafting generation-2, we noted that $\surreal{}{} \similar
+\surreal{-1}{1}$, leading us to conjecture that each number takes on a value
+directly between the value of its left and right sets, ignoring the null set
+for now, and assuming the number is in reduced form.
+
+Let's begin by attempting to create an addition rule satisfying
+
+$$-1 + 1 = 0.$$
+
+Since $-1 = \surreal{}{0}$ and $1 = \surreal{0}{}$, and since $\surreal{0}{0}$
+is not a valid number per Axiom \autoref{ax:number-definition}, we're pushed
+toward a definition for
+
+$$x + y = z$$
+
+which discards $X_R$ and $Y_L$. This rules out simple definitions like $Z_L =
+X_L \cup Y_L$ and $Z_R = X_R \cup Y_R$.
+
+We could try something like $Z_L = X_L \cup y$ and $Z_R = Y_R \cup x$, but then
+addition is non-commutative and non-closed, creating non-numbers if $y > x$.
+But, if we could rework that definition to function regardless of the relative
+magnitude of $x$ and $y$, then we might be on to something useful.
+
+
+\subsection{Conjecture}
+
+Geometrically, and allowing myself a recursive definition, I want something
+like $Z_L = x$ and $Z_R = x + y + y$, but expressed in terms of set operations.
+
diff --git a/notes/notes.tex b/notes/notes.tex
index cc76d96..1457fe2 100644
--- a/notes/notes.tex
+++ b/notes/notes.tex
@@ -10,6 +10,7 @@
 
 \include{introduction}
 \include{chapter-1}
+\include{chapter-2}
 
 %===============================================================================