View source for Surreal number

All readers are welcome to contribute to this resource.  Try to follow the notation of [[Wikipedia:Surreal number]]. To create an essay or article start with a title that not already on the list below. It is very easy to change a title.. If you need help, leave a message at [[User talk:Guy vandegrift]].
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[[Category:Open essay collections]]
==Subpage abstracts==
Each "abstract" begins with a and a link to a subpage.  The abstract is just a [[w:transclusion|transclusion]] of the first few lines.
===Abstracts for beginners===
{{/The dyadics}}<br>
{{/Root 2}}<br>
{{/Counting}}<br>[[file:Pinocchio paradox Large Print.svg|thumb|right|140px|]]
■ '''[[Russell's paradox]]''' resembles [[w:Pinocchio paradox|Pinocchio paradox]] in that it involves self-referencing. {{:Russell's paradox}}

===Advanced abstracts===
<!--{{/Why isn't half-omega the (left) child of omega?}}
----- -->
{{/Simplicity}}


[[Category:Open essay collections]]
----

==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)===
[https://www.m-a.org.uk/resources/downloads/4H-Jim-Simons-Meet-the-surreal-numbers.pdf '''Meet the surreal numbers'''] (by Jim Simons) is 39 pages long. The internet is full of introductions to surreal numbers.  Many contain the same essential insight can be found on Wikipedia's '''[[w:Surreal number|Surreal number]]'''.  But Jim Simon's article contains useful insights that most authors neglect to adequately cover.  His discussion of [[w:Ordinal number|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 ω<sup>2</sup> . Carrying on past things like ω <sup>2</sup>·7 + ω·42 + 1, we’ll come to ω<sup>3</sup> , ω<sup>4</sup> and so on to ω<sup>ω</sup> , 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.''

====Simplicity theorem====
:'''''The Simplicity Theorem. (If) x is a number, and z is a number with the earliest possible birthday that lies strictly between the left and right options of x, then x = z'''''

I find this easier to understand than the what appears in [[Wikipedia:Surreal number]].

===A Short Guide to Hackenbush (Padraic Bartlett)===
[https://www.math.uchicago.edu/~may/VIGRE/VIGRE2006/PAPERS/Bartlett.pdf '''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 [[w:Go (game)|go]] or [[w:Chess|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 [[w:Hackenbush|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".

[[File:Surreal number tree.svg|thumb|500px|Tree diagram shown at [[Wikipedia:Surreal number]]]]

===Beyond ω (with Andi Fugard)===
'''[https://andifugard.info/surreal-numbers/ numbers – Andi Fugard]''' presents a slightly different picture of &omega;/2. The image to the right is from the Wikipedia article, and I am confused by the tree diagram starting at &omega;. Compare the image to the right with [https://andifugard.info/wp-content/uploads/2022/07/surreal_birthdays.png this image]. All the dyadic (real) numbers are placed on a tree formation where each parent has two children.  They display two types of infinite series:
#The positive and negative integers are exterior elements and grow as <br>{{spaces|3}}{1, 2, 3,...}.
#Interior dyadics cut as fractions that get smaller, for example as <br>{{spaces|3}}{ &frac12;, &frac14;, &frac18;, ...}.

Look at the tree that begins at ω: Both sides match the tree that begins with 0, i.e., by adding or subtracting 1 from each previous element.  To the left we get smaller with  {ω, ω-1, ω-2,..., ω/2}. See also [https://andifugard.info/wp-content/uploads/2022/07/surreal_birthdays.png this image] and [https://andifugard.info/surreal-numbers/ this article]. I am quite confused by the fact that the left side of this tree doesn't continue with {ω, ω-1, ω-2,..., 0}.<ref>Here, 0 might not be a member of the set, but merely a point to which the series converges.</ref>  It is true that the square root of ω is smaller than every element in the series {ω, ω-1, ω-2,..., 0}. But how do we know that 2ω has the same birthday as ω/2?

'''Answer:''' Quoting page 30 of  [https://www.m-a.org.uk/resources/downloads/4H-Jim-Simons-Meet-the-surreal-numbers.pdf '''Meet the surreal numbers''']:

x = {ℕ | ω − ℕ} can hardly be ω − ω because that is equal to 0.<ref>Here, ℕ refers to <math>\mathbb{N}</math>, or the [[wikipedia:natural numbers|natural numbers]].</ref> So what it it? x = {ℕ | ω − ℕ}, and x + x = {x + ℕ | x + ω − ℕ}. Now x < ω − n for any n ∈ ℕ, so x + n < ω, ie so all the left option of x + x are less than ω, but are infinite. Similarly, x > n, so x + ω − n > ω, ie all the right options of x + x are bigger than ω. Therefore x + x = ω, and x = ω/2.

=== Surreal Numbers –  An Introduction (Claus Tøndering) ===
[https://www.tondering.dk/download/sur.pdf '''Surreal Numbers – An Introduction'''] is 51 pages of careful rigor. It employs the notation of logic and set theory.  Each surreal number can be expressed in a  variety of "forms" (each consisting of a pair of sets.)  On page 25 Tøndering proves that if x is the oldest surreal number between a and b, then {a | b} = x.

===Math576 Combinatorial Game Theory===
[https://people.math.sc.edu/lu/teaching/2020fall_576/note1.pdf Lecture note for Math576 Combinatorial Game Theory.pdf ] is an 84 page document that thoroughly connects surreal numbers to hackenbush.
===Surreal Numbers and Transseries===
[https://www.fields.utoronto.ca/talk-media/1/50/58/slides.pdf www.fields.utoronto.ca/talk-media/1/50/58/slides] concisely presents 12 pages of insight into how both real numbers, as well as von Neumann's ordinal numbers, embedded in the surreals.

===Video===
[https://www.youtube.com/watch?v=ZYj4NkeGPdM HACKENBUSH: a window to a new world of math] is an hour-long video that begins with a solid introduction to hackenbush, but also goes far beyond.

=== Other useful (but informal) discussions ===

* Mathematical Musings https://www.youtube.com/watch?v=QaK81xaIb-U
* https://www.quora.com/What%E2%80%99s-the-difference-between-surreal-and-hyperreal-numbers-in-layman%E2%80%99s-terms

===Wikiversity/Wikipedia/Wikibooks===
[[File:Conway names, the simplicity hierarchy and the surreal number tree.pdf|thumb|Conway names, the simplicity hierarchy and the surreal number tree (PHILIP EHRLICH)]]
{{colbegin}}
*[[w:Surreal numbers]]
*[[w:Ordinal arithmetic]]
*[[w:Infinity#Real_analysis]]
*[[w:Transfinite number#Examples]]
*[[w:Hyperreal number]]
*[[w:Von Neumann universe]]
*[[Surreal_number/Simple_hackenbush#References]]
*[[Introduction to Category Theory/Products and Coproducts of Sets]]
*[[Introduction to Category Theory/Sets and Functions]]
*[[Set theory]]
*[[b:Surreal_Numbers_and_Games]]

{{colend}}

===External links===
*[https://eprints.whiterose.ac.uk/123644/1/1608.03413v2.pdf SURREAL NUMBERS WITH DERIVATION, HARDY FIELDS AND TRANSSERIES: A SURVEY, (MANTOVA & MATUSINSKI)] ([https://arxiv.org/abs/1608.03413 alternative link])This '''might''' show a simple way to represent surreal numbers as a series expansion.
* [https://arxiv.org/pdf/1210.5675 <small>arxiv.org/pdf/1210.5675</small> CONVERGENCE ON SURREALS <small>(ISTVAN MEZO)</small>]: Points out that the limits used with real numbers do not apply to surreal numbers.  For example, despite that <math>\tfrac 1 2 \omega</math> is obtained by subtracting <math>1</math> form <math>\omega</math> and infinite number of times, it is '''not true''' that <math display="inline">\sum_{n\to\infty} 1=\tfrac 1 2 \omega.</math> See also  [https://arxiv.org/pdf/1603.09289 <small>arxiv.org/pdf/1603.09289</small> SURREAL LIMITS <small>(PAOLO LIPPARINI AND ISTVAN MEZO)</small>] Also helpful is https://math.stackexchange.com/questions/1237974/decoding-the-sign-expansion-of-surreal-numbers&nbsp;. 
*'''[https://hiaatcnd.github.io/2019/11/09/Hackenbush/ Chitanda'a Blog]''' is a must read! I don't know why Google never found it.  I found it using Explorer (images).  Shows an amazing connection between hackenbush and the tree of surreal numbers.
*I haven't digested '''[https://math.stackexchange.com/questions/816540/proof-of-conways-simplicity-rule-for-surreal-numbers math.stackexchange question 816540]''' on the proof of the simplicity rule.  But it seems sufficiently well written to be worth reading ([[W:Surreal numbers|Wikipedia]] states the rule without proof, or even a claim that it can be proven.)
*[https://www.whitman.edu/documents/Academics/Mathematics/Grimm.pdf whitman.edu (Grimm.pdf)]  30 pages. Excellent resource with proofs and connection to hackenbush. A curious feature of this resource is that he claims that all red-blue hackenbush games are numbers.  His argument that any non-zero hackenbush position can be used to create a zero position by subtracting a number that has been created by a collection of simple trees.  Since every number can be represented by a simple hackenbush game, then every position can be "zeroed" by a simple game.  Unfortunately, Grimm doesn't explain how omega can be represented by a hackenbush position.
*[https://drive.google.com/file/d/1iVqRJJ4D0prkPboquCmGEVzOC0cdS5j-/view Steven Charlton - An Very Brief Introduction to Surreal Numbers] [https://guests.mpim-bonn.mpg.de/spc/talks/7_gandalf_surrealnumbers_notes.pdf (alternate link)] 6 pages.
*[https://arxiv.org/pdf/1810.10373 Surreal Birthdays and Their Arithmetic (Matthew Roughan)] 13 pages. Directed Acyclic Graphs (DAG); 2*3 & 3/4 + 3/4/ (a bit complicated).   
*[https://medium.com/badiou-and-science/badiou-and-science-1-4-1-the-surreal-numbers-part-1-95dadcf4c554 badiou-and-science part-1] Nice graphs. 8 minute read.  Seems identical to article at [https://www.scientificamerican.com/article/surreal-numbers-are-a-real-thing-heres-how-to-make-them/ scietificamerican.com], except the former has links to more game theory articles by same author.
*[https://web.mit.edu/sp.268/www/surreal.pdf what is a game? (mit)] 13 pages.*
*[https://math.stackexchange.com/questions/816540/proof-of-conways-simplicity-rule-for-surreal-numbers stackexchange definition of all dyadics] 
*[https://math.stackexchange.com/questions/926581/in-the-surreal-numbers-is-it-fair-to-say-0-9-repeating-is-not-equal-to-1 stackexchange on .9999]
*See also [[Draft:Surreal_number#Collection_of_online_resources_saved_on_disk]]
*[https://web.archive.org/web/20150429210249/http://euclid.colorado.edu/~monkd/m8714.html course notes on set theory]






=====proof of conways simplicity rule=====

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

<math>0=\{|\}</math>

<math>n+1=\{n|\}</math>

<math>-n-1=\{|-n\}</math>

<math>\tfrac{2p+1}{2^{q+1}}=\left\{
\tfrac{p}{2q}
|
\tfrac{p+1}{2q}
\right\}</math>

====Limits and analysis====
Infinity is easy to imagine, but difficult to incorporate into rigorous mathematics.  The following "logic" certainly violates the rules of mathematics and is therefore nonsensical:
:<math>\lim_{\epsilon\to 0} \frac 1 \epsilon =\infty\;</math> and <math>\;\lim_{\epsilon\to 0} \frac 2 \epsilon =\infty\;</math> implies <math>\;\frac\infty\infty =2.</math>

This is why expressions like <math>\infty/\infty</math> and <math>0/0</math> of often called [[wikt:indeterminant|indeterminant]].  Most of the time, mistakes like this can be avoided by utilizing concepts taught in a course on [[w:Mathematical analysis|mathematical analysis]].

==Images==
{{wide image|Positive surreals to seventh birthday.svg|2000px}}
[[File:Números hiperreales.png|400px|thumb|Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1). This image appears in the Wikipedia articles [[w:Hyperreal number]], [[w:Infinitesimal]], and [[w:Infinity]].]]
{{clear}}

===Using Latex {matrix}===
With Latex, it is easier to use dots instead of arrows to highlight how parents give rise to children. The following two trees (starting from 1 and 0, respectively) are written as latex matrices:

<math display="block"> a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad
a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad
a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad
a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad
a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad
a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad
</math>

<math display="block">\begin{matrix}
&&&&&&&&&&&&&&& 1 &&&&&&&&&&&&&&& \\
&&&&&&& \tfrac 1 2&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&2 &&&&&&& \\

&&& \tfrac 1 4 &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot& \tfrac 3 4 &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot& 1\tfrac 1 2 &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot& 3 &&&  \\

&\tfrac 1 8 &\cdot&\cdot &\cdot& \tfrac 3 8 &\cdot&\cdot &\cdot& \tfrac 5 8 &\cdot&\cdot &\cdot& \tfrac 7 8 &\cdot&\cdot &\cdot& 1\tfrac 1 4 &\cdot&\cdot &\cdot& 1\tfrac 3 4 &\cdot&\cdot &\cdot& 2\tfrac 1 2 &\cdot&\cdot &\cdot& 4 &  \\

\tfrac{1}{16} &\cdot& \tfrac{3}{16} &\cdot& \tfrac{5}{16} &\cdot& \tfrac{7}{16} &\cdot&\tfrac{9}{16} &\cdot& \tfrac{11}{16} &\cdot& \tfrac{13}{16} &\cdot& \tfrac{15}{16} &\cdot& 1\tfrac 1 8 &\cdot& 1\tfrac 3 8 &\cdot& 1\tfrac 5 8 &\cdot& 1\tfrac 7 8 &\cdot& 2\tfrac 1 4 &\cdot& 2\tfrac 3 4 &\cdot& 3\tfrac 1 2  &\cdot& 5 &
\end{matrix}</math>

===Starting from zero===
<small><math>\begin{matrix}
&&&&&&&&&&&&&&& 0 &&&&&&&&&&&&&&& \\

&&&&&&&{\color{Red}\overline{1}}&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot& 1 &&&&&&&\\

&&&{\color{Red}\overline{2}}&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&{\color{Red}\overline{\tfrac 1 2}}&\cdot&\cdot &\cdot&\cdot&\cdot&\cdot&\cdot&\tfrac 1 2 &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot & 2 &&&\\

&{\color{Red}\overline{3}} &\cdot&\cdot &\cdot& {\color{Red}\overline{1\tfrac 1 2}} &\cdot&\cdot &\cdot& {\color{Red}\overline{\tfrac 3 4}} &\cdot&\cdot &\cdot& {\color{Red}\overline{\tfrac 1 4}} &\cdot&\cdot &\cdot& \tfrac 1 4 &\cdot&\cdot &\cdot& \tfrac 3 4 &\cdot&\cdot &\cdot& 1\tfrac 1 2 &\cdot&\cdot &\cdot& 3 &\\

{\color{Red}\overline{4}} &\cdot& {\color{Red}\overline{2\tfrac 1 2}} &\cdot& {\color{Red}\overline{1\tfrac 3 4}} &\cdot& {\color{Red}\overline{1\tfrac 1 4}} &\cdot&{\color{Red}\overline{\tfrac 7 8}} &\cdot&{\color{Red}\overline{\tfrac 5 8}} &\cdot&{\color{Red}\overline{\tfrac 3 8}} &\cdot& {\color{Red}\overline{\tfrac 1 8}} &\cdot& \tfrac 1 8 &\cdot& \tfrac 3 8 &\cdot& \tfrac 5 8 &\cdot& \tfrac 7 8 &\cdot& 1\tfrac 1 4 &\cdot& 1\tfrac 3 4 &\cdot& 2\tfrac 1 2  &\cdot& 4
\end{matrix}</math></small>

====With sets====
{{cot|under construction}}
<math>
{\color{Red}\begin{matrix}\{ \overline{L} | \overline{R} \} \\[-.5 ex]\text{-}a\tfrac b c\end{matrix}}
</math>
-----


<small><math display=block>\begin{matrix}
&&&&&&&&&&&&&&& 0 &&&&&&&&&&&&&&& \\

&&&&&&&

\begin{matrix}
{\color{red}\{ \; | }0 
{\color{red}\}} \\[-.5 ex]
{\color{red}\text{-}1 }
\end{matrix}

&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot& 

1 

&&&&&&&\\

&&&
{\color{Red}\begin{matrix}\{ \; | \overline{1} \} \\[-.5 ex] \text{-}2\end{matrix}}

&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&

\begin{matrix}
{\color{red}\{ \overline 1 | }0 
{\color{red}\}} \\[-.5 ex]
{\color{red}\text{-}\tfrac 1 2 }
\end{matrix}

&\cdot&\cdot &\cdot&\cdot&\cdot&\cdot&\cdot&

\tfrac 1 2 

&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot & 

2 

&&&\\&{\color{Red}\begin{matrix}\{\;|\overline{2}\}\\[-.5 ex]  \text{-}3  \end{matrix}}&\cdot&\cdot &\cdot& 

{\color{Red}
\begin{matrix}
\{ \overline 2 | \overline 1 \} \\[-.5 ex]
\text{-1}\tfrac 1 2 
\end{matrix}}

&\cdot&\cdot &\cdot& 

{\color{Red}\begin{matrix}\{ \overline{1} | \overline{\tfrac 1 2} \} \\[-.5 ex]\text{-}\tfrac 3 4 \end{matrix}}

&\cdot&\cdot &\cdot& 

\begin{matrix}
{\color{red}\{ \overline{\tfrac 1 2} | }0 
{\color{red}\}} \\[-.5 ex]
{\color{red}\text{-}\tfrac 1 4 }
\end{matrix}

&\cdot&\cdot &\cdot& 

\tfrac 1 4 &\cdot&\cdot &\cdot& 

\tfrac 3 4 

&\cdot&\cdot &\cdot& 

1\tfrac 1 2 

&\cdot&\cdot &\cdot& 3 &\\

{\color{Red}\begin{matrix}\{ \; | \overline{3} \} \\[-.5 ex]\text{-4} \end{matrix}} &\cdot& 

{\color{Red}\begin{matrix}\{ \overline 3 | \overline{2} \} \\[-.5 ex]\text{-}2\tfrac 1 2\end{matrix}}

&\cdot& 

{\color{Red}\begin{matrix}\{\overline{2}  |\overline{1\tfrac 1 2}  \} \\[-.5 ex]\text{-}1\tfrac 3 4\end{matrix}}

&\cdot& 

{\color{Red}\begin{matrix}\{ \overline{1\frac 1 2} | \overline{1} \} \\[-.5 ex]\text{-}1\tfrac 1 4\end{matrix}}

&\cdot&

{\color{Red}\begin{matrix}\{ \overline 1 | \overline{\tfrac 3 4} \} \\[-.5 ex]\text{-}\tfrac 7 8\end{matrix}}

&\cdot&

{\color{Red}\overline{\tfrac 5 8}} 

&\cdot&{

\color{Red}\overline{\tfrac 3 8}} 

&\cdot& 

\begin{matrix}
{\color{red}\{ \overline{\tfrac 1 4} | }0 
{\color{red}\}} \\[-.5 ex]
{\color{red}\text{-}\tfrac 1 8 }
\end{matrix} 


&\cdot& \tfrac 1 8 &\cdot& 

\tfrac 3 8 

&\cdot& 

\tfrac 5 8 

&\cdot& 

\tfrac 7 8

 &\cdot& 

1\tfrac 1 4 &\cdot& 

1\tfrac 3 4 

&\cdot& 2

\tfrac 1 2 
 &\cdot& 

4

\end{matrix}</math></small>

{{cob}}

==Categories==
[[Draft:Surreal number]]
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[[Category:Numbers]]