| 1 | \newpage |
| 2 | \section{Notes: Chapter 1} |
| 3 | |
| 4 | \subsection{Review} |
| 5 | |
| 6 | In the first chapter, Knuth provides two axioms. |
| 7 | |
| 8 | \begin{axiom} \label{ax:number-definition} |
| 9 | Every number corresponds to two sets of previously created numbers, such that |
| 10 | no member of the left set is greater than or equal to any member of the right |
| 11 | set. |
| 12 | \end{axiom} |
| 13 | |
| 14 | For example, if $x = \surreal{X_L}{X_R}$, then $\forall x_L \in X_L$, it must |
| 15 | hold $\forall x_R \in X_R$ that $x_L \ngeq x_R$. |
| 16 | |
| 17 | \begin{axiom} \label{ax:leq-comparison} |
| 18 | One number is less than or equal to another number if and only if no member of |
| 19 | the first number's left set is greater than or equal to the second number, and |
| 20 | no member of the second number's right set is less than or equal to the first |
| 21 | number. |
| 22 | \end{axiom} |
| 23 | |
| 24 | For example, the relation $\surreal{X_L}{X_R} = x \leq y = \surreal{Y_L}{Y_R}$ |
| 25 | holds true if and only if both $X_L \ngeq y$ and $Y_R \nleq x$. |
| 26 | |
| 27 | With no surreal numbers yet in our possession, we construct the first surreal |
| 28 | number using the null set (or void set, as Knuth calls it) as both the left and |
| 29 | right set. Although we have not yet examined its properties, Knuth names this |
| 30 | number ``zero''. Thus, $\surreal{}{} = 0$. |
| 31 | |
| 32 | As his final trick, Knuth defines a second generation of surreal numbers using |
| 33 | $0$ in the left and right set, naming them $1$ and $-1$ and claiming the |
| 34 | following relation. |
| 35 | |
| 36 | $$-1 = \surreal{}{0} \leq 0 = \surreal{}{} \leq 1 = \surreal{0}{}$$ |
| 37 | |
| 38 | |
| 39 | \subsection{Exploration} |
| 40 | |
| 41 | \begin{defi} \label{defi:generation} |
| 42 | A \emph{generation} shall refer to the numbers generated by applying Axiom |
| 43 | \autoref{ax:number-definition} to all extant numbers. Generations are numbered |
| 44 | sequentially such that generation-0 consists of the number $0$, generation-1 |
| 45 | consists of the numbers $-1$ and $1$, etc. |
| 46 | \end{defi} |
| 47 | |
| 48 | Working by hand with Axiom \autoref{ax:number-definition}, generation-2 |
| 49 | contains the numbers shown below. |
| 50 | |
| 51 | $$\surreal{-1}{}, \surreal{1}{}, \surreal{-1,0}{}, \surreal{0,1}{},$$ |
| 52 | $$\surreal{-1,1}{}, \surreal{-1,0,1}{}, \surreal{-1}{1}, \surreal{0}{1},$$ |
| 53 | $$\surreal{-1,0}{1}, \surreal{}{1}, \surreal{-1}{0}, \surreal{}{-1},$$ |
| 54 | $$\surreal{-1}{0,1}, \surreal{}{0,1}, \surreal{}{-1,0}, \surreal{}{-1,1},$$ |
| 55 | $$\surreal{}{-1,0,1}$$ |
| 56 | |
| 57 | \begin{defi} \label{defi:similar} |
| 58 | Two numbers $X$ and $Y$ are \emph{similar} iff, per Axiom |
| 59 | \autoref{ax:leq-comparison}, $X \leq Y$ and $Y \leq X$. This is denoted by $X |
| 60 | \similar Y$. |
| 61 | \end{defi} |
| 62 | |
| 63 | From this point forward, we will refer to similar surreal numbers |
| 64 | interchangeably. |
| 65 | |
| 66 | Using this definition, the twenty numbers from generations 0-2 break down into |
| 67 | ten equivalence classes based on similarity, as shown below. |
| 68 | |
| 69 | $$\surreal{0}{} \similar \surreal{-1,0}{}$$ |
| 70 | $$\surreal{}{}$$ |
| 71 | $$\surreal{}{0} \similar \surreal{}{0,1}$$ |
| 72 | $$\surreal{-1}{}$$ |
| 73 | $$\surreal{1}{} \similar \surreal{0,1}{} \similar \surreal{-1,1}{} \similar \surreal{-1,0,1}{}$$ |
| 74 | $$\surreal{-1}{1}$$ |
| 75 | $$\surreal{0}{1} \similar \surreal{-1,0}{1}$$ |
| 76 | $$\surreal{}{1}$$ |
| 77 | $$\surreal{-1}{0} \similar \surreal{-1}{0,1}$$ |
| 78 | $$\surreal{}{-1} \similar \surreal{}{-1,0} \similar \surreal{}{-1,1} \similar \surreal{}{-1,0,1}$$ |
| 79 | |
| 80 | From this we see that, since Axiom \autoref{ax:leq-comparison} makes its |
| 81 | comparison element-wise, every surreal number generated by our current methods |
| 82 | must be similar to a surreal number containing one or zero elements in its left |
| 83 | and right sets. This motivates the following definition. |
| 84 | |
| 85 | \begin{defi} \label{defi:reduced-form} |
| 86 | The \emph{reduced form} for a surreal number $X = \surreal{X_L}{X_R}$ is |
| 87 | defined as $\surreal{x_l}{x_r}$ where $x_l$ is the largest element of $X_L$ and |
| 88 | $x_r$ is the smallest element of $X_R$. If either $X_R$ or $X_L$ are the empty |
| 89 | set, then the corresponding $x_r$ or $x_l$ also become the empty set. |
| 90 | \end{defi} |
| 91 | |
| 92 | Note that we are guaranteed largest and smallest elements of the corresponding |
| 93 | non-empty sets from Axiom \autoref{ax:leq-comparison} and our definition of |
| 94 | similarity. We are only building a one-dimensional number line. |
| 95 | |
| 96 | |
| 97 | \subsection{Conjecture} |
| 98 | |
| 99 | If we can build an addition operation which holds for $1 + (-1) = 0$, then we |
| 100 | could start trying to assign meaningful names to some of the elements from |
| 101 | generation-2. It appears that numbers of the form \surreal{n}{} behave like the |
| 102 | number $n+1$ and similarly, \surreal{}{n} behaves like $n-1$. |
| 103 | |
| 104 | If we write a program to generate a bunch of new surreal numbers and graph them |
| 105 | as ``generation vs magnitude'', perhaps we can assign some meaning to numbers |
| 106 | which don't fit the pattern mentioned in the previous paragraph. Maybe these |
| 107 | behave like $1/n$? |
| 108 | |
| 109 | It sort of feels like surreal numbers constructed via finite repetitions of our |
| 110 | current process will end up building something vaguely like the dyadic |
| 111 | rationals. |
| 112 | |