Surreal number

All readers are welcome to contribute to this resource. Try to follow the notation of Wikipedia:Surreal number, but don't forget that on on Wikiversity, anything goes! 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.

Subpage abstracts

Each "abstract" begins with a and a link to a subpage. The abstract is just a transclusion of the first few lines.

Abstracts for beginners

Dyadic fractions subdivide an inch

■ Surreal number/The dyadics is a very simple introduction that might make surreal numbers seem easy to understand (if not a bit silly.) The figure to the right illustrates how rulers often subdivide the inch into dyadic fractions. The construction of surreal numbers begins with these dyadics (ranging from −∞ to +∞.) Each is assigned "birthday" and is associated with a pair of sets. Remarkably, all elements involve only the empty set, $\{\,\}.$ (Visit Surreal number/The dyadics for more details.)
■ Surreal number/Root 2 begins by discussing how to construct a pair of infinite series that converge to ${\sqrt {2}}$ (one from above and one from below). One derivation of this series is presented, but we also hint at how Newton's method might be used to construct the same pair of sequences. Later, we discuss how to generalize this method to include other roots $(x^{1/n})$ . These pairs of sequences can be used to express surreal forms for irrational numbers that are the n-th root of an integer. z (Visit Surreal number/Root 2 for more details.)

■ Surreal number/Counting introduces concepts involving infinity that are necessary for any student wishing to learn about surreal numbers. No claim is made that this prerequisite material is sufficient. This introduction begins with Cantor's diagonal argument. This introduction makes no attempt to the leave the reader with the rigorous understanding of surreal numbers.(Visit Surreal number/Counting for more details.)

■ Russell's paradox resembles Pinocchio paradox in that it involves self-referencing. Pinocchio's words are in conflict with his nose: If his words are true, his nose would not grow. But since his nose is growing, we know his words are false. In short:

His words are false if his words are true.
His words are true if his words are false.

With Russell's paradox, the self-referencing conflict is between a set's definition and the items that are in the set. The paradox is important when sets are allowed be contain sets as members, as is the case with the pair of sets used to define surreal numbers.

Advanced abstracts

\{2{\tfrac {1}{2}}|4{\tfrac {1}{2}}\}=3

, not

{\color {Red}3{\tfrac {1}{2}}}

(which is the average of

2{\tfrac {1}{2}}

and

4{\tfrac {1}{2}}

)

■ Surreal number/Simplicity illustrates that illustrates a case where the surreal number $x$ is not midway bwetten $X_{L}$ and $X_{R}$ (as it is for ${ 0 | 1 } = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1/2$ , ${ 0 | 1 / 2} = 1 / 4$ , and ${1 / 2 | 1 } = 3 / 4$ .)

Instead, the simplicity rule yields:

\{2{\tfrac {1}{2}}|4{\tfrac {1}{2}}\}=\{2|\;\}=3,

which is the oldest number between $2{\tfrac {1}{2}}$ and $4{\tfrac {1}{2}}.$

...the rest is under construction...💧

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 (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 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 ω^ω , 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)

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".

Tree diagram shown at Wikipedia:Surreal number

Beyond ω (with Andi Fugard)

numbers – Andi Fugard presents a slightly different picture of ω/2. The image to the right is from the Wikipedia article, and I am confused by the tree diagram starting at ω. Compare the image to the right with 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
{1, 2, 3,...}.
Interior dyadics cut as fractions that get smaller, for example as
{ ½, ¼, ⅛, ...}.

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 this image and 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}.^[1] 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 Meet the surreal numbers:

x = {ℕ | ω − ℕ} can hardly be ω − ω because that is equal to 0.^[2] 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)

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

Lecture note for Math576 Combinatorial Game Theory.pdf is an 84 page document that thoroughly connects surreal numbers to hackenbush.

Surreal Numbers and Transseries

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

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

Wikiversity/Wikipedia/Wikibooks

Conway names, the simplicity hierarchy and the surreal number tree (PHILIP EHRLICH)

External links

SURREAL NUMBERS WITH DERIVATION, HARDY FIELDS AND TRANSSERIES: A SURVEY, (MANTOVA & MATUSINSKI) (alternative link)This might show a simple way to represent surreal numbers as a series expansion.
arxiv.org/pdf/1210.5675 CONVERGENCE ON SURREALS (ISTVAN MEZO): Points out that the limits used with real numbers do not apply to surreal numbers. For example, despite that ${\tfrac {1}{2}}\omega$ is obtained by subtracting $1$ form $\omega$ and infinite number of times, it is not true that ${\textstyle \sum _{n\to \infty }1={\tfrac {1}{2}}\omega .}$ See also arxiv.org/pdf/1603.09289 SURREAL LIMITS (PAOLO LIPPARINI AND ISTVAN MEZO) Also helpful is https://math.stackexchange.com/questions/1237974/decoding-the-sign-expansion-of-surreal-numbers .
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 math.stackexchange question 816540 on the proof of the simplicity rule. But it seems sufficiently well written to be worth reading (Wikipedia states the rule without proof, or even a claim that it can be proven.)
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.
Steven Charlton - An Very Brief Introduction to Surreal Numbers (alternate link) 6 pages.
Surreal Birthdays and Their Arithmetic (Matthew Roughan) 13 pages. Directed Acyclic Graphs (DAG); 2*3 & 3/4 + 3/4/ (a bit complicated).
badiou-and-science part-1 Nice graphs. 8 minute read. Seems identical to article at scietificamerican.com, except the former has links to more game theory articles by same author.
what is a game? (mit) 13 pages.*
stackexchange definition of all dyadics
stackexchange on .9999
See also Draft:Surreal_number#Collection_of_online_resources_saved_on_disk
course notes on set theory

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 into rigorous mathematics. The following "logic" certainly violates the rules of mathematics and is therefore nonsensical:

\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.

Images

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.

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:

$a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad a\quad$

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

Starting from zero

${\begin{matrix}&&&&&&&&&&&&&&&0&&&&&&&&&&&&&&&\\&&&&&&&{\color {Red}{\overline {1}}}&\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &1&&&&&&&\\&&&{\color {Red}{\overline {2}}}&\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &{\color {Red}{\overline {\tfrac {1}{2}}}}&\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &{\tfrac {1}{2}}&\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &2&&&\\&{\color {Red}{\overline {3}}}&\cdot &\cdot &\cdot &{\color {Red}{\overline {1{\tfrac {1}{2}}}}}&\cdot &\cdot &\cdot &{\color {Red}{\overline {\tfrac {3}{4}}}}&\cdot &\cdot &\cdot &{\color {Red}{\overline {\tfrac {1}{4}}}}&\cdot &\cdot &\cdot &{\tfrac {1}{4}}&\cdot &\cdot &\cdot &{\tfrac {3}{4}}&\cdot &\cdot &\cdot &1{\tfrac {1}{2}}&\cdot &\cdot &\cdot &3&\\{\color {Red}{\overline {4}}}&\cdot &{\color {Red}{\overline {2{\tfrac {1}{2}}}}}&\cdot &{\color {Red}{\overline {1{\tfrac {3}{4}}}}}&\cdot &{\color {Red}{\overline {1{\tfrac {1}{4}}}}}&\cdot &{\color {Red}{\overline {\tfrac {7}{8}}}}&\cdot &{\color {Red}{\overline {\tfrac {5}{8}}}}&\cdot &{\color {Red}{\overline {\tfrac {3}{8}}}}&\cdot &{\color {Red}{\overline {\tfrac {1}{8}}}}&\cdot &{\tfrac {1}{8}}&\cdot &{\tfrac {3}{8}}&\cdot &{\tfrac {5}{8}}&\cdot &{\tfrac {7}{8}}&\cdot &1{\tfrac {1}{4}}&\cdot &1{\tfrac {3}{4}}&\cdot &2{\tfrac {1}{2}}&\cdot &4\end{matrix}}$

With sets

under construction

${\color {Red}{\begin{matrix}\{{\overline {L}}|{\overline {R}}\}\\[-.5ex]{\text{-}}a{\tfrac {b}{c}}\end{matrix}}}$

${\begin{matrix}&&&&&&&&&&&&&&&0&&&&&&&&&&&&&&&\\&&&&&&&{\begin{matrix}{\color {red}\{\;|}0{\color {red}\}}\\[-.5ex]{\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}}\}\\[-.5ex]{\text{-}}2\end{matrix}}}&\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &{\begin{matrix}{\color {red}\{{\overline {1}}|}0{\color {red}\}}\\[-.5ex]{\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}}\}\\[-.5ex]{\text{-}}3\end{matrix}}}&\cdot &\cdot &\cdot &{\color {Red}{\begin{matrix}\{{\overline {2}}|{\overline {1}}\}\\[-.5ex]{\text{-1}}{\tfrac {1}{2}}\end{matrix}}}&\cdot &\cdot &\cdot &{\color {Red}{\begin{matrix}\{{\overline {1}}|{\overline {\tfrac {1}{2}}}\}\\[-.5ex]{\text{-}}{\tfrac {3}{4}}\end{matrix}}}&\cdot &\cdot &\cdot &{\begin{matrix}{\color {red}\{{\overline {\tfrac {1}{2}}}|}0{\color {red}\}}\\[-.5ex]{\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}}\}\\[-.5ex]{\text{-4}}\end{matrix}}}&\cdot &{\color {Red}{\begin{matrix}\{{\overline {3}}|{\overline {2}}\}\\[-.5ex]{\text{-}}2{\tfrac {1}{2}}\end{matrix}}}&\cdot &{\color {Red}{\begin{matrix}\{{\overline {2}}|{\overline {1{\tfrac {1}{2}}}}\}\\[-.5ex]{\text{-}}1{\tfrac {3}{4}}\end{matrix}}}&\cdot &{\color {Red}{\begin{matrix}\{{\overline {1{\frac {1}{2}}}}|{\overline {1}}\}\\[-.5ex]{\text{-}}1{\tfrac {1}{4}}\end{matrix}}}&\cdot &{\color {Red}{\begin{matrix}\{{\overline {1}}|{\overline {\tfrac {3}{4}}}\}\\[-.5ex]{\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}\}}\\[-.5ex]{\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}}$