Surreal number


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Essential reading for advanced students
So far, the subpages to this resource are target the absolute beginner. I hope that changes, but for now people already familiar with the basics might want to look at the following websites:

Meet the surreal numbers (Jim Simons)
Meet the surreal numbers is 29 pages long. The internet is full of introductions to surreal numbers. Many contain the same essential insight can be found on Wikipedia's Surreal number. But Jim Simon's article contains useful insights that most authors neglect to adequately cover. His discussion of ordinal numbers begins with this introduction:
 * Just as we don’t need much set theory, we don’t need to know much about ordinals, but it is helpful to know a little. Ordinals extend the idea of counting into the infinite in the simplest way imaginable: just keep on counting. So we start with the natural numbers: 0, 1, 2, 3, 4, . . ., but we don’t stop there, we keep on with a new number called ω, then ω + 1, ω + 2, ω + 3 and so on. After all those we come to ω + ω = ω·2. Carrying on we come to ω·3, ω·4 etc and so on to ω2 . Carrying on past things like ω 2·7 + ω·42 + 1, we’ll come to ω3, ω4 and so on to ωω , and this is just the beginning. To see a bit more clearly where this is heading, we’ll look at von Neumann’s construction of the ordinals.

A Short Guide to Hackenbush (Padraic Bartlett)
A Short Guide to Hackenbush is 25 pages long. It will give insight as to what inspired the invention of surreal numbers. It turns out that the surreal numbers are inspired by an effort to attach a value to a game, not unlike go or chess, but some peculiarities that make it easy to attach a rational number that predicts how the game will end if both sides play flawlessly. Non-negative values correspond to games where the person who moves first will lose, and the magnitude (absolute value) tells us something about the margin or victory. A value of zero corresponds to a position where the loser is the person whose turn it is to move (ties are impossible in red-blue hackenbush.)  For those who love math as a beautiful tool for solving real-life problems, this application to game theory is the most likely way to make surreal numbers seem "useful".

Surreal Numbers – An Introduction (Claus Tøndering)
Surreal Numbers – An Introduction is 51 pages long. each surreal number can be expressed in a bewildering variety of "forms". Each form consists of two sets. On page 25 Tøndering proves that if x is the oldest surreal number between a and b, then {a | b} = x.

Wikiversity/Wikipedia/Wikibooks

 * Surreal numbers
 * Infinity
 * Transfinite number
 * Hyperreal number
 * Von Neumann universe
 * Surreal_number/Simple_hackenbush
 * Introduction to Category Theory/Products and Coproducts of Sets
 * Introduction to Category Theory/Sets and Functions
 * Set theory
 * Surreal_Numbers_and_Games

proof of conways simplicity rule
https://math.stackexchange.com/questions/816540/proof-of-conways-simplicity-rule-for-surreal-numbers

$$0=\{|\}$$

$$n+1=\{n|\}$$

$$-n-1=\{|-n\}$$

$$\tfrac{2p+1}{2^{q+1}}=\left\{ \tfrac{p}{2q} \tfrac{p+1}{2q} \right\}$$

Limits and analysis
Infinity is easy to imagine, but difficult to incorporate}}

Infinity is easy to imagine, but difficult to incorporate into rigorous mathematics. The following calculation certainly violates the rules of mathematics:
 * $$\lim_{\epsilon\to 0} \frac 1 \epsilon =\infty\;$$ and $$\;\lim_{\epsilon\to 0} \frac 2 \epsilon =\infty\;$$ implies $$\;\frac\infty\infty =2.$$

This is why expressions like $$\infty/\infty$$ and $$0/0$$ of often called indeterminant. Most of the time, mistakes like this can be avoided by utilizing concepts taught in a course on mathematical analysis. Both the Wikipedia article and a query sent to an online chatbot suggest that no widely known problems in this field have been solved using surreal numbers. For that reason, any forrey into surreal numbers should probably be viewed as recreational.

Images


$$\begin{matrix} &&&&& &&&&& &&&&& 1 &&&&& &&&&& &&&&& \\ &&&&& && \tfrac 1 2 &&&&& &&&&& &&&&& 2 &&&&& && \\ &&& \tfrac 1 4 &&&&& &&& \tfrac 3 4 &&&&& &&& 1\tfrac 1 2 &&&&& &&& 3 &&& \\ &\tfrac 1 8 &&&& \tfrac 3 8 &&&& \tfrac 5 8 &&&& \tfrac 7 8 &&&& 1\tfrac 1 4 &&&& 1\tfrac 3 4 &&&& 2\tfrac 1 2 &&&& 4 & \\ \tfrac{1}{16} && \tfrac{3}{16} && \tfrac{5}{16} && \tfrac{7}{16} &&\tfrac{9}{16} && \tfrac{11}{16} && \tfrac{13}{16} && \tfrac{15}{16} && 1\tfrac 1 8 && 1\tfrac 3 8 && 1\tfrac 5 8 && 1\tfrac 7 8 && 2\tfrac 1 4 && 2\tfrac 3 4 && 3\tfrac 1 2 && 5 \end{matrix}$$

Categories
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