X-Git-Url: http://git.subgeniuskitty.com/surreal-numbers/.git/blobdiff_plain/cc0282e4a0b88cf3605bfd5b211ab0a28949bf16..af665c3c5e87948b87cd492af528f545432fc118:/notes/chapter-1.tex?ds=inline

diff --git a/notes/chapter-1.tex b/notes/chapter-1.tex
index 917a31a..e340d75 100644
--- a/notes/chapter-1.tex
+++ b/notes/chapter-1.tex
@@ -1,9 +1,11 @@
 \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.
@@ -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 $x_L \ngeq x_R$.
 
-\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
@@ -28,10 +30,103 @@ 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
+$0$ in the left and right set, naming them $1$ and $-1$ and establishing the
 following relation.
 
-$$-1 = \surreal{}{0} \leq 0 = \surreal{}{} \leq 1 = \surreal{0}{}$$
+$$-1 \equiv \surreal{}{0} \leq 0 \equiv \surreal{}{} \leq 1 \equiv \surreal{0}{}$$
+
+
+\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 possible combinations of 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}
+
+Using this definition, the twenty numbers from generations 0-2 break down into
+no more than ten equivalence classes based on similarity, as shown below.
+
+$$\surreal{0}{} \similar \surreal{-1,0}{}$$
+$$\surreal{}{}$$
+$$\surreal{}{0} \similar \surreal{}{0,1}$$
+$$\surreal{-1}{}$$
+$$\surreal{1}{} \similar \surreal{0,1}{} \similar \surreal{-1,1}{} \similar \surreal{-1,0,1}{}$$
+$$\surreal{-1}{1}$$
+$$\surreal{0}{1} \similar \surreal{-1,0}{1}$$
+$$\surreal{}{1}$$
+$$\surreal{-1}{0} \similar \surreal{-1}{0,1}$$
+$$\surreal{}{-1} \similar \surreal{}{-1,0} \similar \surreal{}{-1,1} \similar \surreal{}{-1,0,1}$$
+
+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 since only the largest member of the left set and smallest
+member of the right set are important. 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.
+
+Note that the reduced form is not unique since $\surreal{-1}{1} \similar
+\surreal{}{}$.
+
+
+\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 $n-1$. That seems to
+build the integers.
+
+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.
+
+I think some of our equivalence classes are similar. I suspect that the number
+line maintains symmetry with each generation. Since we have ten equivalence
+classes, an even number, symmetry is broken if none of the equivalence classes
+are similar. In fact, since it appears that $\surreal{-1}{1} \similar
+\surreal{}{}$, I'm sure that our equivalence classes can be collapsed further.
+
+I think I can use Definition \autoref{defi:generation} to start proving some
+surreal number properties inductively.
 
-Before we try generating some more surreal numbers, we should look for useful
-equivalence classes.
+I'm having a lot of problems inserting generation-n into the numberline
+containing generation-0 to generation-(n-1). Since Axiom
+\autoref{ax:leq-comparison} requires comparing against the number itself rather
+than just comparing the sets which define the number, it's hard to slot a new
+generation's number in. For example, how do I test \surreal{}{-1} and
+\surreal{}{1} without working the existing numberline from both ends? I'm
+tempted to note that no number can contain itself, and that the two sets must
+be `less than on the left' and `greater than on the right' to allow just
+comparing sets and finding the right spot on the numberline by working from
+both directions inward, rather than just left to right. Can I make that both
+rigorous and equivalent to Axiom \autoref{ax:leq-comparison}?
