From: Aaron Taylor Date: Wed, 5 May 2021 10:57:57 +0000 (-0700) Subject: Adding some more thoughts to my chapter-1 notes/speculations. X-Git-Url: http://git.subgeniuskitty.com/surreal-numbers/.git/commitdiff_plain/af665c3c5e87948b87cd492af528f545432fc118 Adding some more thoughts to my chapter-1 notes/speculations. --- diff --git a/notes/chapter-1.tex b/notes/chapter-1.tex index 9a1e41c..e340d75 100644 --- a/notes/chapter-1.tex +++ b/notes/chapter-1.tex @@ -30,19 +30,19 @@ 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 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. +\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 @@ -60,11 +60,8 @@ Two numbers $X$ and $Y$ are \emph{similar} iff, per Axiom \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. +no more than ten equivalence classes based on similarity, as shown below. $$\surreal{0}{} \similar \surreal{-1,0}{}$$ $$\surreal{}{}$$ @@ -80,7 +77,8 @@ $$\surreal{}{-1} \similar \surreal{}{-1,0} \similar \surreal{}{-1,1} \similar \s 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. +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 @@ -93,20 +91,42 @@ 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 $n-1$. +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. - +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. + +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}?