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1 | \newpage |
2 | \section{Notes: Chapter 1} | |
3 | ||
420b0302 | 4 | \subsection{Review} |
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6 | In the first chapter, Knuth provides two axioms. |
7 | ||
8 | \begin{axiom} \label{ax:number-definition} | |
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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 | ||
420b0302 | 17 | \begin{axiom} \label{ax:leq-comparison} |
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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 | ||
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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 | $$\surreal{0}{} = \surreal{-1,0}{}$$ | |
70 | $$\surreal{}{}$$ | |
71 | $$\surreal{}{0}, \surreal{}{0,1}$$ | |
72 | $$\surreal{-1}{}$$ | |
73 | $$\surreal{1}{}, \surreal{0,1}{}, \surreal{-1,1}{}, \surreal{-1,0,1}{}$$ | |
74 | $$\surreal{-1}{1}$$ | |
75 | $$\surreal{0}{1}, \surreal{-1,0}{1}$$ | |
76 | $$\surreal{}{1}$$ | |
77 | $$\surreal{-1}{0}, \surreal{-1}{0,1}$$ | |
78 | $$\surreal{}{-1}, \surreal{}{-1,0}, \surreal{}{-1,1}, \surreal{}{-1,0,1}$$ | |
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 |