SURREAL NUMBERS

"Surreal numbers are obtained from games. I felt guilty at Cambridge when I spent all day playing games, when I was supposed to be doing math. Then when I discovered surreal numbers I realized that playing is doing mathematics" (John Horton Conway).

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Conway's Theory
Introduction

Surreal numbers constitute a generalized number system:

• It includes the real numbers (integers, rationals and irrationals).
  
  
• Includes Robinson's hyperreal numbers.
  
  
• They have recursive structure, i.e., a surreal number includes surreals, thus overcoming the limitations in the representation of conventional numbers.
  
  
• It allows to represent infinitesimals and infinities (including Cantor's transfinites) and also to operate with them as with real numbers. For example, it is possible to calculate log(ω), 1/&ω, ω/2, −ω, ε/2, etc. (ω is the symbol used for Cantor's first transfinite ordinal, and ε is the symbol for infinitesimal).
  
  
• Every real number is surrounded by infinite surreals.

Structure

A surreal number is recursively defined as a number intermediate between two sets of numbers, which are also surreal.

A surreal number has the following structure: {L | R}, where L (left) and R (right ) two sets of surreal numbers, such that every number of L is less than every number of R. L and R can be the empty set ∅.

• If L and R are not both ∅, {L | R} represents the simplest intermediate number between the larger number of L and the smaller number of R.
  
  Example: {1, 3, 4 | 6 , 7}
  
  
• {∅ | R} = {| R} represents the simplest smaller number than every number in R.
  
  Example: {|6, 7}
  
  
• {L | ∅} = {L |} represents the simplest greater number than every number in L.
  
  Example: {−1, 0 |}
  
  
• {∅ | ∅} = {|} = 0 represents the first surreal number and "parent" of all surreals.

"Simplest" number is defined as follows:

• 0 is the simplest number that exists.
  
  
• The next simplest are −1 and 1. Then −2 and 2, and so on.
  
  
• If there is no intermediate integer between a and b, the simplest number is of the form m/2ⁿ, with the smallest possible power of 2 in the denominator. For example, the simplest number between 1 and 3 is 2, between −1 and −3 is −2, between 0 and 1 is 1/2, between 1 and 3 is 3/2, etc.

Characteristics

• Flexibility of representation is an important feature of surreal numbers. Indeed, in a number {L | R} one can eliminate all members of L except the largest, and one can also eliminate all members of R except the smallest, since they are all different representations of the same number. For example,
  
  {1, 2, 3 | 4, 5, 6} = {2, 3 | 4, 5} = {3 | 4}
  
  You could also add smaller numbers in L and larger numbers in R and the number would be the same. For example:
  
  0 = {|} = {−1|} = {−1 | 1} = {−2 | 1} = {|1, 2, 3, 4}
  
  
• In the set of surreals there is an order relation: for any pair of surreal numbers a and b, the law of trichotomy is satisfied (< i>a<b or a=b or a>b).
  
  A surreal number a = {A_L | A_R} is less than another b = {B_L | B_R} if every number of A_L is less than b and a is less than every number of B_R. The definition, in this case, is also recursive.

Generation of surreals

Surreal numbers are generated from 0 = {|}, in successive generations:

1. Generation 1 (the children of 0) is:
   
   {0|} = 1 {|0} = −1 ({0 | 0} is not a valid surreal)
   
   and the (ordered) population is 3 elements: −1 < 0 < 1
   
   
2. In generation 2 there are 16 possible configurations of surreal numbers (by combining on the left and right sides ∅, −1, 0 and 1), but there are only 4 new ones:
   
   2 = {1|} −2 = {|−1} 1/2 = {0 | 1} −1/2 = {−1 | 0}
   
   and the (ordered) population is 6 elements:
   
   −2 < −1 < −1/2 < 0 < 1/2 < 1
   
   
3. In generation 3 there are 8 new ones:
   
   3 = {2|} −3 = {|&−2} 3/2 = {1 | 2} −3/2 = {−2 | −1}
   
   1/4 = {0 | 1/2} −1/4 = {−1/2 | 0} 3/4 = {1/2 | 1} −3/4 = {−1 | | −1/2}
   
   and the (ordered) population is 15 elements:
   
   −3 < −2 < −3/2 < −1 < −3/4 < −1/2 < −1/4 < 0 < 1/4 < 1/2 < 3/4 < 1 < 3/2 < 2 < 3

In general, in the generation n 2ⁿ new numbers are created and the total population contains 2ⁿ⁺¹ − 1 members. All numbers are dyadic fractions, that is, of the type m/2ⁿ, with n ≥ 0. This means that fractions such as 1/3, 4/3, 1/5, etc. cannot be generated, but can be represented by numbers as close as desired, but never exact numbers.


Sum of surreals

The sum of a = {A_L | | A_R} and b = {B_L | B_R} is defined recursively:

a+b = {A_L+b, a+B _L | A_R+b, a+B_R}

with the following rules:

1. The sum of an element a a set L or R of a surreal number is performed by distributing the sum: x+{a, b, c} = {x+a, x+b, x+c}
   
   
2. The sum of two sets is performed by complete distribution. For example:
   
   {1, 2}+{10, 20, 30} = {11, 12, 21, 22, 31, 32}
   
   
3. Addition and subtraction of the empty set:
   
   x+∅ = ∅ and x−∅ = ∅ (for all x)

The negative of a surreal number x (−x) is defined as the result of multiplying −1 to each of the components of x. Subtraction is defined as x−y = x+(−y).

Examples:

1+1/2 = {0 | ∅}+{0 | 1} = {0+1/2, 1+0 | ∅+1/2,1+1} = {0+1/2, 1+0 | 1+1}

0+1/2 = {∅ | ∅}+{0 | 1} = {∅+1/2, 0+0 | ∅+1/2, 0+1} = {0+0 | 0+1}

Analogously, 0+0 = 0 and 0+1 = 1. Then 0+1/2 = {0 | 1} = 1/2

Therefore, 1+1/2 = {0+1/2, 1+0 | 1+1} = {1/2,1 | 2} = {1 | 2} = 3/2


Product of surreals

The product of a = {A_L | | A_R} and b = {B_L | B_R} is also defined recursively:

ab = {A_Lb + a + B_L − A_LB_L, A_Rb + a + B_R − A_RB_R | A_Lb + a + B_R − A_LB_R, A_Rb + a + B_L − A_RB_L}

Properties

It is shown that:

xy = yx
x(y+z) = xy+xz
1x = x1 = x
0x = x0 = 0

Every number x (except 0) has its inverse, so that x(1/x) = 1

The surreal numbers form a commutative (abelian) group with respect to the sum and also with respect to the product. With both operations the class of surreal numbers forms a body.


Nonadic rational numbers and irrational numbers

Non-diadic rational numbers are those that do not have the form m/2ⁿ. Nonadiadic rational numbers and irrational numbers are specified by sets of infinite elements. For example:

1/3 = {1/4, 5/16, 21/64, 85/256, ... | 1/2, 3/8, 11/32, 43/128, ...}

π = {3/1, 25/8, 201/64, ... | 13/4, 101/32, 3217/1024, ...}

The construction of surreal numbers bears certain similarities to Dedekind's cuts (cuts), which defines an irrational number from two sequences of rational numbers.


Specification of infinities and infintegers

This is the most salient aspect of surreals. Here are some examples:

1. ω = {N|} indicates a number greater than any integer and corresponds to ω, Cantor's first transfinite ordinal (N is the set of natural numbers).
   
   
2. ω+1 = {N|} + {0|} = {N +1, ω+0 | ∅+1, ω+∅} = {N, ω|} = {ω|} since N+1 = N and ω is greater than all integers.
   
   
3. ω+2 = {N |} + {1|} = {N +2, ω+1 | ∅+2, ω+∅} = {N +2, ω+1|} =. {N, ω+1|} = {ω+1|} since N +2 = N and ω is greater than all integers.
   
   
4. ω+3 = {ω+2|}
   
   
5. ω−1 = {N |} + {|0} = { N −1,ω+∅ | ∅−1, ω+0} = {N −1 | ω+0} = {N | ω}
   
   
6. ω−2 = {N | ω−1}
   
   
7. ω−3 = {N | ω−2}
   
   
8. ω+ω = {ω+N |}
   
   
9. 2ω = {ω+N |}
   
   
10. 3ω = {2ω+N |}
    
    
11. 4ω = {3ω+N |}
    
    
12. ω/2 = {N |ω−N}
    
    
13. ω² = {ω, 2ω, 3ω, 4ω, ...|}
    
    li)ω^ω = {ω, ω² , ω³ , ω⁴ , ...|}
    
    
14. −ω = {|N}
    
    
15. √ω= {N | ω, ω/2, ω/3, ω/4, ...}
    
    
16. ε = {0 | 1, 1/2 , 1/4 , 1/8 , 1/16 , ...} (infinitesimal number)
    
    
17. ε+1 = {1 | 2, 3/2, 5/4, 9/8, 17/16, ...}
    
    
18. 2ε = {ε | | 1+ε, 1/2+ε, 1/4+ε, 1/8+ε, 1/16+ε, ...}
    
    
19. ε/2= {0 | ε}
    
    
20. √ε= {ε, 2ε, 3ε, 4ε,...| 1, 1/2, 1/4, 1/8, 1/16, ...}
    
    
21. 1/ε = ω

Specification in MENTAL
Surreal numbers have great expressive power because they harmonize opposites. Without pretending to be exhaustive, here are some specifications of surreals in MENTAL:


Representation of surreal numbers

A surreal number can be represented by a sequence of 3 elements: a set L, a symbol | and a set R: (L | R) or simply L|R. If L or R does not exist, the empty set ∅ or its equivalent symbol must be specified: {}.


Definition of finite surreals

(∅|∅ = 0) // surreal parent

Generic expression that replaces each set L and R of every surreal number with a set of a single element: the maximum and minimum, respectively:

⟨( x|y =: ( (u = max(x)) (v = min(y))) ( u<v → ( ¡(u+v)÷2) ←' (v−u > 1) → ¡(u+1) ) ) )! )⟩

Examples:

{-1 0}|{2 3} // ev. 1
{-1 0 1}|∅ // ev. 2
{∅|(2 3} // ev. 1
{1 2}|∅ // ev. 3
{12}|&∅ // ev. 13
{1 2}|{1 3} // is self-evaluating (not a valid surreal)

Definition of infinite surreals

A series of axioms would have to be defined beforehand. Examples:

⟨( N+n = N )⟩
( ω = N|∅ )

From these axioms, theorems can be deduced. For example:

⟨( ω+n = ω+n−1|∅ )⟩

Sum of surreal numbers

We first define the laws governing the addition of surreals and then define the sum, according to these laws:

⟨( x+∅ = ∅ )⟩ ⟨( x−∅ = ∅ )⟩ // addition and subtraction of empty set

⟨( (n+x = {[n+[x↓]]}) ← {x↓}=x )⟩ // sum distribution

⟨( (x+y = {[[x↓]+[y↓]] }) ← {x↓}=x ← {y↓}=x )⟩ // full distribution

⟨( x1|x2 + y1|y2 = {(x1 + y1|y2) (x1|x2 + y1)}||{{(x2 + y1|y2) (x1|x2 + y2)} )⟩

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Addenda
Origin of surreal numbers

Surreal numbers were invented (or discovered) by John Conway in 1969, inspired by the game of Go, a very popular game in China and Japan. Go is called "Fencing" in English [Wang An-Po, 1970]. The game has very simple rules, but produces highly complex recursive structures.

The structure of a surreal number is also recursive. Its representation consists of two parts, in analogy with the two players. The valid surreal numbers would correspond to the allowed moves of the game. A particular surreal number g={{a, b, c, ...|d, e, f, . ..} can be considered as a game situation between two players, L and R. where a, b, c, ... are the possible (legal) moves for L and d, e, f, ... are the legal moves for R. Assuming L choose b, for example, we arrive at another configuration b={A, B, C, . ...|D, E, F, ...}. As it is now your turn to play R, you can choose. e.g., E, leading to another configuration. And so on.

In 1972, Conway told Donald Knuth about this new number system. Knuth was fascinated by its potential and originality, and publicized it in a short story entitled "Surreal Numbers," subtitled "How Two Former Students Turned to Pure Mathematics and Found Total Happiness" [Knuth, 1974]. Knuth himself baptized these numbers as "surreal" (from the French "sur", over), to indicate that they are numbers that are "above" the real ones.

Conway is also the author of the "game of life", a system that takes place on a board, which is also based on very simple rules and produces complex dynamic configurations.

Surreal numbers constitute a mathematical universe of great possibilities, but still little investigated. They have application in game theory. Having fractal structure, they constitute a representation system suitable for the interpretation of the "many worlds" of quantum physics.


Surcomplex numbers

A surcomplex number is a complex number a+b*i, where a and b are surreal numbers, and i is the imaginary unit. Surreal numbers form a body, like surreal numbers.


Bibliography

• Conway, John H. On Numbers and Games. Academic Press, 1976. [El libro principal de los números surreales].
  
  
• Conway, J.H.; Guy, R.K. The Book of Numbers. Springer-Verlag, 1996.
  
  
• Gardner, Martin. Los números surreales de Conway. Capítulo 4 de Mosaicos de Penrose y escotillas cifradas. Editorial Labor, S.A., 1990.
  
  
• Gonshor, Harry. An Introduction to Surreal Numbers. Cambridge University Press, 1986. [Amplía la teoría de los números surreales. Utiliza una notación diferente].
  
  
• Knuth, Donald E. Surreal Numbers. Addisson-Wesley, 1974. Números surreales. Reverté, Barcelona, 1979.
  
  
• Knuth, Donald E. Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. Addisson-Wesley, 1974. Disponible online.
  
  
• Peterson, Ivars. Mathematical Treks. From Surreal Numbers to Magic Circles. The Mathematical Association of America, Spectrum Series, 2002.
  
  
• Tondering, Clams. Surreal Numbers – An Introduction. Internet.
  
  
• Wang An-Po, Ambrosio. El Cercado. Un milenario y fascinante juego chino. Editado por el autor, Madrid, 1970.