| 1 | \newpage |
| 2 | \section{Notes: Chapter 2} |
| 3 | |
| 4 | \subsection{Review} |
| 5 | |
| 6 | In the second chapter, Knuth doesn't add much new information. The only |
| 7 | interesting bits I see are the following two quotes: |
| 8 | |
| 9 | \begin{framed} |
| 10 | ``The Stone's version is a little different, but $x \leq y$ must mean the same |
| 11 | thing as $y \leq x$.'' |
| 12 | \end{framed} |
| 13 | |
| 14 | \begin{framed} |
| 15 | ``On the first day of creation, Conway ``proves'' that $0 \leq 0$. Why should |
| 16 | he bother to prove that something is less than or equal to itself, since it's |
| 17 | obviously equal to itself.'' |
| 18 | \end{framed} |
| 19 | |
| 20 | Why, indeed? It seems our prior efforts are headed in a useful direction. |
| 21 | |
| 22 | |
| 23 | \subsection{Exploration} |
| 24 | |
| 25 | This week I'm interested in defining an addition operation on the surreal |
| 26 | numbers generated during my Chapter 1 explorations. |
| 27 | |
| 28 | When crafting generation-2, we noted that $\surreal{}{} \similar |
| 29 | \surreal{-1}{1}$, leading us to conjecture that each number takes on a value |
| 30 | directly between the value of its left and right sets, ignoring the null set |
| 31 | for now, and assuming the number is in reduced form. |
| 32 | |
| 33 | Let's begin by attempting to create an addition rule satisfying |
| 34 | |
| 35 | $$-1 + 1 = 0.$$ |
| 36 | |
| 37 | Since $-1 = \surreal{}{0}$ and $1 = \surreal{0}{}$, and since $\surreal{0}{0}$ |
| 38 | is not a valid number per Axiom \autoref{ax:number-definition}, we're pushed |
| 39 | toward a definition for |
| 40 | |
| 41 | $$x + y = z$$ |
| 42 | |
| 43 | which discards $X_R$ and $Y_L$. This rules out simple definitions like $Z_L = |
| 44 | X_L \cup Y_L$ and $Z_R = X_R \cup Y_R$. |
| 45 | |
| 46 | We could try something like $Z_L = X_L \cup y$ and $Z_R = Y_R \cup x$, but then |
| 47 | addition is non-commutative and non-closed, creating non-numbers if $y > x$. |
| 48 | But, if we could rework that definition to function regardless of the relative |
| 49 | magnitude of $x$ and $y$, then we might be on to something useful. |
| 50 | |
| 51 | |
| 52 | \subsection{Conjecture} |
| 53 | |
| 54 | Geometrically, and allowing myself a recursive definition, I want something |
| 55 | like $Z_L = x$ and $Z_R = x + y + y$, but expressed in terms of set operations. |
| 56 | |