Surreal number

The following pages in this resource may be of interest to you

■ Surreal number/The dyadics is still under construction. It will be 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. Curiously, each set consists only of the null set $\{\,\},$ as well as sets that contain the null set in some way. Making matters even more complicated is each surreal can be associated with an endless variety of these pairs of sets.
■ 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.
■ 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 not attempt to the leave the reader with the rigorous understanding of this discussion required to fully understand surreal numbers.

More contributions are welcome! 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.

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.

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".

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.

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

External links

whitman.edu (Grimm.pdf) 30 pages. Excellent resource with proofs and connection to hackenbush.
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

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

Starting from zero

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