Added some more notes related to Chapter 1. Mostly misc explorations of the

[surreal-numbers] / notes / chapter-1.tex

diff --git a/notes/chapter-1.tex b/notes/chapter-1.tex
index 917a31a..aa7751c 100644 (file)
--- a/notes/chapter-1.tex
+++ b/notes/chapter-1.tex
@@ -1,9 +1,11 @@
 \newpage
 \section{Notes: Chapter 1}
 
 \newpage
 \section{Notes: Chapter 1}
 

-In the first chapter, Knuth provides two axioms, reproduced here.
+\subsection{Review}

 
 

-\begin{axiom}
+In the first chapter, Knuth provides two axioms.
+
+\begin{axiom} \label{ax:number-definition}

 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.
 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.


@@ -12,7 +14,7 @@ set.
 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 $x_L \ngeq x_R$.
 
 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}
+\begin{axiom} \label{ax:leq-comparison}

 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
 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


@@ -33,5 +35,78 @@ following relation.
 
 $$-1 = \surreal{}{0} \leq 0 = \surreal{}{} \leq 1 = \surreal{0}{}$$
 
 
 $$-1 = \surreal{}{0} \leq 0 = \surreal{}{} \leq 1 = \surreal{0}{}$$
 

-Before we try generating some more surreal numbers, we should look for useful
-equivalence classes.
+
+\subsection{Exploration}
+
+\begin{defi} \label{defi:generation}
+A \emph{generation} shall refer to the numbers generated by applying Axiom
+\autoref{ax:number-definition} to all extant numbers. Generations are numbered
+sequentially such that generation-0 consists of the number $0$, generation-1
+consists of the numbers $-1$ and $1$, etc.
+\end{defi}
+
+Working by hand with Axiom \autoref{ax:number-definition}, generation-2
+contains the numbers shown below.
+
+$$\surreal{-1}{}, \surreal{1}{}, \surreal{-1,0}{}, \surreal{0,1}{},$$
+$$\surreal{-1,1}{}, \surreal{-1,0,1}{}, \surreal{-1}{1}, \surreal{0}{1},$$
+$$\surreal{-1,0}{1}, \surreal{}{1}, \surreal{-1}{0}, \surreal{}{-1},$$
+$$\surreal{-1}{0,1}, \surreal{}{0,1}, \surreal{}{-1,0}, \surreal{}{-1,1},$$
+$$\surreal{}{-1,0,1}$$
+
+\begin{defi} \label{defi:similar}
+Two numbers $X$ and $Y$ are \emph{similar} iff, per Axiom
+\autoref{ax:leq-comparison}, $X \leq Y$ and $Y \leq X$. This is denoted by $X
+\similar Y$.
+\end{defi}
+
+From this point forward, we will refer to similar surreal numbers
+interchangeably.
+
+Using this definition, the twenty numbers from generations 0-2 break down into
+ten equivalence classes based on similarity, as shown below.
+
+$$\surreal{0}{} = \surreal{-1,0}{}$$
+$$\surreal{}{}$$
+$$\surreal{}{0}, \surreal{}{0,1}$$
+$$\surreal{-1}{}$$
+$$\surreal{1}{}, \surreal{0,1}{}, \surreal{-1,1}{}, \surreal{-1,0,1}{}$$
+$$\surreal{-1}{1}$$
+$$\surreal{0}{1}, \surreal{-1,0}{1}$$
+$$\surreal{}{1}$$
+$$\surreal{-1}{0}, \surreal{-1}{0,1}$$
+$$\surreal{}{-1}, \surreal{}{-1,0}, \surreal{}{-1,1}, \surreal{}{-1,0,1}$$
+
+From this we see that, since Axiom \autoref{ax:leq-comparison} makes its
+comparison element-wise, every surreal number generated by our current methods
+must be similar to a surreal number containing one or zero elements in its left
+and right sets. This motivates the following definition.
+
+\begin{defi} \label{defi:reduced-form}
+The \emph{reduced form} for a surreal number $X = \surreal{X_L}{X_R}$ is
+defined as $\surreal{x_l}{x_r}$ where $x_l$ is the largest element of $X_L$ and
+$x_r$ is the smallest element of $X_R$. If either $X_R$ or $X_L$ are the empty
+set, then the corresponding $x_r$ or $x_l$ also become the empty set.
+\end{defi}
+
+Note that we are guaranteed largest and smallest elements of the corresponding
+non-empty sets from Axiom \autoref{ax:leq-comparison} and our definition of
+similarity. We are only building a one-dimensional number line.
+
+
+\subsection{Conjecture}
+
+If we can build an addition operation which holds for $1 + (-1) = 0$, then we
+could start trying to assign meaningful names to some of the elements from
+generation-2. It appears that numbers of the form \surreal{n}{} behave like the
+number $n+1$ and similarly, \surreal{}{n} behaves like $n-1$.
+
+If we write a program to generate a bunch of new surreal numbers and graph them
+as ``generation vs magnitude'', perhaps we can assign some meaning to numbers
+which don't fit the pattern mentioned in the previous paragraph. Maybe these
+behave like $1/n$?
+
+It sort of feels like surreal numbers constructed via finite repetitions of our
+current process will end up building something vaguely like the dyadic
+rationals.
+