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MISC COMMENTS

• See also this Youtube video, Surreal number/See also, Chatbot math/Gemini/24.03/Surreal numbers, and other WMF resources: W:Surreal number, W:Set theory, W:Infinity, B:Surreal Numbers and Games.... and maybe see https://ncatlab.org/nlab/show/surreal+number

It is nearly impossible to introduce surreal numbers without either trivializing difficult concepts, or describing these same concepts in a way no novice could possibly understand. What I can do is locate points on the real number line that correspond to some of the surreal numbers. And, I can briefly introduce a few concepts that while peripheral to the topic of surreal numbers, appear to be similar to related topics that are essential to far more difficult concepts that are essential to the topic of surreal numbers. I emphasize "appear to be", because I myself to not fully grasp the more difficult topics. This begs the question of why a non-expert would be attempting to explain surreal numbers: First, we are allowed to do things that the other WMF wikis forbid. Also, in applied mathematics (especially physics) one is permitted to judge a mathematical on the problems it solves and the insight it yields, without worrying too much about rigor.

Finally, a conversation with chatbot Gemini (14 May 2024) informed me that I should study: Set Theory (especially with well-ordered sets like ordinals and proper classes). Ordinal Arithmetic (defines addition and multiplication for infinite numbers). Axiomatic Set Theory (Zermelo-Fraenkel set theory with Choice, and Alternative Set Theory (von Neumann-Bernays-Gödel.)

Wikipedia links

• Surreal number:

Surreal number, Infinitesimal, Epsilon number, Hackenbush, Infinity plus one, w:,

• Set Theory:

Set theory, Class, Ordinal number, Von Neumann–Bernays–Gödel set theory, Zermelo–Fraenkel set theory, Ordinal arithmetic, Axiom of global choice, Paradoxes of set theory, Glossary of set theory, w:, w:,

• This page introduce fundamentals: Dyadic rational numbers are easy to grasp, and will be introduced sequentially in a way that superficially seem to "fill the number line".

This discussion mixes prerequisite knowledge required by the Wikipedia article with insights that might make surreal numbers more interesting. Our focus is on concepts involving to countability and infinity that are easy to explain. To this end, we employ the language of naive set theory in a way that cannot fully explain or describe surreal numbers. The current author of this resource is not an expert on this subject, and for that reason, corrections and elaborations are welcome. While this might seem like a chaotic way to start a resource, it will generate a number of projects for students that take the form, "Is this really true?".

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Latex

$\begin{array}{ccccccccccccccccccccccccccccccc}& & & & & & & & & & & & & & & 1& & & & & & & & & & & & & & & \\ & & & & & & & \frac{1}{2}& & & & & & & & & & & & & & & 2& & & & & & & \\ & & & \frac{1}{4}& & & & & & & & \frac{3}{4}& & & & & & & & 1\frac{1}{2}& & & & & & & & 3& & & \\ & \frac{1}{8}& & & & \frac{3}{8}& & & & \frac{5}{8}& & & & \frac{7}{8}& & & & 1\frac{1}{4}& & & & 1\frac{3}{4}& & & & 2\frac{1}{2}& & & & 4& \\ \frac{1}{16}& & \frac{3}{16}& & \frac{5}{16}& & \frac{7}{16}& & \frac{9}{16}& & \frac{11}{16}& & \frac{13}{16}& & \frac{15}{16}& & 1\frac{1}{8}& & 1\frac{3}{8}& & 1\frac{5}{8}& & 1\frac{7}{8}& & 2\frac{1}{4}& & 2\frac{3}{4}& & 3\frac{1}{2}& & 5\end{array}$

Bigger

The first 31 positive surreal numbers are created on days 1-5, and are all rational dyadic fractions. They range from 1 through 5, and almost half of them are less than 1.

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