In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar to superreal numbers and hyperreal numbers. By limiting the construction to a Grothendieck universe, a set is obtained, rather than a class, with an honest field with the cardinality of some strongly inaccessible cardinal. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... Please refer to Real vs. ... In mathematics, an infinitesimal, or infinitely small number, is a number that is greater in absolute value than zero yet smaller than any positive real number. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... The superreal numbers compose a more inclusive category than hyperreal number. ... In mathematics, particularly in non-standard analysis and mathematical logic, hyperreal numbers or nonstandard reals (usually denoted as *R) denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle. ... In mathematics, a Grothendieck universe is a set with the following properties: If x ∈ U and if y ∈ x, then y ∈ U. If x,y ∈ U, then {x,y} ∈ U. If x ∈ U, then P(x) ∈ U. (P(x) is the power set of x. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a strongly inaccessible cardinal is an uncountable cardinal number κ that is regular and a strong limit cardinal. ...

The definition and construction of the surreals is due to John Horton Conway, and exemplifies Conway's characteristic notational cleverness and originality. They were introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. This book is a mathematical novelette, and is notable as one of the rare cases where a new mathematical idea was first presented in a work of fiction. In his book, which takes the form of a dialogue, Knuth coined the term surreal numbers for what Conway had simply called numbers originally. Conway liked the new name, and later adopted it himself. Conway then described the surreal numbers and used them for analyzing games in his 1976 book On Numbers and Games. John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. ... Donald Knuth at a reception for the Open Content Alliance. ... Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. ... On Numbers and Games is a mathematics book by John Conway, published by Academic Press Inc in 1976, ISBN 0121863506, and re-released by AK Peters in 2000 (ISBN 1568811276). ...

Constructing surreal numbers

The basic idea behind the construction of surreal numbers is similar to Dedekind cuts. We construct new numbers by representing them with two sets of numbers, L and R, that approximate the new number; the set L contains a set of numbers below the new number and the set R contains a set of numbers above the new number. We will write such an approximation as { L | R }. We will pose no restrictions upon L and R except that each of the numbers in L should be smaller than any number in R. For example, { {1, 2} | {5, 8} } is a valid construction of a certain number between 2 and 5. (Which number exactly and why will be explained later on.) The sets are explicitly allowed to be empty. The informal interpretation of a pair { L | {} } will be "a number higher than any number in L", and of { {} | R } "a number lower than any number in R". This leads to the following construction rule: In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards...

Construction Rule

If L and R are two sets of surreal numbers and no member of R is less than or equal to any member of L then { L | R } is a surreal number.

Given a surreal number x = { X_L | X_R } the sets X_L and X_R are called the left set of x and right set of x respectively. To avoid lots of brackets we will write { {a, b, ... } | { x, y, ... } } simply as { a, b, ... | x, y, ... } and { {a} | {} } as { a | } and { {} | {a} } as { | a }.

In order for the generated numbers to actually qualify as numbers there has to be a "less than or equal to" relation (here written as ≤) defined on them. This is supplied by the following rule:

Comparison Rule

For a surreal number x = { X_L | X_R } and y = { Y_L | Y_R } it holds that x ≤ y if and only if y is less than or equal to no member of X_L, and no member of Y_R is less than or equal to x.

The two rules are recursive, so we need some form of induction to put them to work. An obvious candidate would be finite induction, i.e., generate all numbers that can be constructed by applying the construction rule a finite number of times, but, as will be explained later on, things get really interesting if we also allow transfinite induction, i.e., apply the rule more often than that. If we want the generated numbers to represent numbers then the ordering that is defined upon them should be a total order. However, the relation ≤ defines only a total preorder, i.e., it is not antisymmetric. To remedy this we define the binary relation == over the generated surreal numbers such that In mathematics and computer science, recursion specifies (or constructs) a class of objects (or an object from a certain class) by defining a few very simple base cases (often just one), and then defining rules to break down complex cases into simpler cases. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence. ... Transfinite induction is the proof technique of mathematical induction when applied to (large) well-ordered sets, for instance to sets of ordinals or cardinals, or even to the class of all ordinals. ... In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In the mathematical area of order theory, a total preorder over a set X is a preorder ≤ over X that is total; that is, for all a and b in X, it holds that a ≤ b or b ≤ a. ... In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X, if a is related to b and b is related to a, then a = b. ... In mathematics, the concept of binary relation, sometimes called dyadic relation, is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ...

x == y iff x ≤ y and y ≤ x.

Since this defines an equivalence relation the ordering on the equivalence classes implied by ≤ will be a total order. The interpretation of this will be that if x and y are in the same equivalence class then they actually represent the same number. The equivalence classes to which x and y belong are denoted as [x] and [y] respectively. So if x and y belong to the same equivalence class then [x] = [y]. ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...

Let us now consider some examples and see how they behave under the ordering. The most simple example is of course

{ | } ie: { {} | {} }

which can be constructed without any induction at all. We will call this number 0 and the equivalence class [0] will be written as 0. By applying the construction rule we can consider the following three numbers

{ 0 | }, { | 0 } and { 0 | 0 }

The last number is however not a valid surreal number because 0 ≤ 0. If we now consider the ordering of the valid surreal numbers we will see that

{ | 0 } < 0 < { 0 | }

where x < y denotes that not(y ≤ x). We will refer to { | 0 } and { 0 | } as -1 and 1 respectively, and the corresponding equivalence classes as simply -1 and 1, respectively. Since every equivalence class contains only one element that has so far been defined, we can replace in statements about ordering the surreal numbers with their equivalence classes without the risk of ambiguity. For example, the statement above could also have been written as:

{ | 0 } < 0 < { 0 | }

or even

-1 < 0 < 1.

If we apply the construction rule once more we obtain the following ordered set:

{ | -1 } == { | -1, 0 } == { | -1, 1 } == { | -1, 0, 1 } <

{ | 0, 1 } == -1 <

{ -1 | 0 } == { -1 | 0, 1 } <

{ -1 | } == { | 1 } == { -1 | 1 } == 0 <

{ 0 | 1 } == { -1, 0 | 1 } <

{ -1, 0 | } == 1 <

{ 1 | } == { 0, 1 | } == { -1, 1 | } == { -1, 0, 1 | }

We can now make three observations:

1. We have found four new equivalence classes: [{ | -1 }], [{ -1 | 0 }], [{ 0 | 1 }], and [{ 1 | }].
2. All equivalence classes now contain more than one element.
3. The equivalence class of a number depends only on the maximal element of its left set and the minimal element of the right set.

The first observation raises the question of the interpretation of these new equivalence classes. Since the informal interpretation of { | -1 } is "the number just before -1" we will call it number -2 and denote its equivalence class as -2. For a similar reason we will call { 1 | } number 2 and its equivalence class 2. The number { -1 | 0 } is a number between -1 and 0 and we will call it -1/2 and its equivalence class -1/2. Finally we will call { 0 | 1 } the number 1/2 and its equivalence class 1/2. More justification for these names will be given once we have defined addition and multiplication.

The second observation raises the question if we can still replace the surreal numbers with their equivalence classes. Fortunately the answer is yes because it can be shown that

if [X_L] = [Y_L] and [X_R] = [Y_R] then [{ X_L | X_R }] = [{ Y_L | Y_R }]

where [X] denotes { [x] | x in X }. So the description of the ordered set that was found above can be rewritten to:

{ | -1 } == { | -1, 0 } == { | -1, 1 } == { | -1, 0, 1 } <

{ |0, 1 } == -1 <

{ -1 | 0 } == { -1| 0, 1 } <

{ -1 | } == { | 1 } == { -1 | 1 } == 0 <

{ 0 | 1 } == { -1, 0 | 1 } <

{ -1, 0 | } == 1 <

{ 1 | } == { 0, 1 | } == { -1, 1 | } == { -1, 0, 1 | }

which in turn can be rewritten as

-2 < -1 < -1/2 < 0 < 1/2 < 1 < 2.

The third observation extends to all surreal numbers with finite left and right sets. For infinite left or right set, this is valid in an altered form, since infinite sets might not contain a maximal or minimal element. The number { {1, 2} | {5, 8} } therefore is equivalent to { 2 | 5 }, which will be exactly calculated later.

Computing with surreal numbers

The addition and multiplication of surreal numbers are defined by the following three rules:

Addition

x + y = { X_L + y ∪ x + Y_L | X_R + y ∪ x + Y_R }

where X + y = { x + y | x in X } and x + Y = { x + y | y in Y }.

Negation

-x = { -X_R | -X_L }

where -X = { -x | x in X }

Multiplication

xy = { (X_Ly + xY_L - X_LY_L) ∪ (X_Ry + xY_R - X_RY_R) | (X_Ly + xY_R - X_LY_R) ∪ (X_Ry + xY_L - X_RY_L) }

where XY = { xy | x in X and y in Y }, Xy = X{y} and xY = {x}Y.

These operations can be shown to be well-defined for surreal numbers, i.e., if they are applied to well-defined surreal numbers then the result will again be a well-defined surreal number, i.e., the left set of the result will be "smaller" than the right set.

With these rules we can now verify that the chosen names of the numbers we found so far are appropriate. It holds for instance that 0 + 0 = 0, 1 + 1 = 2, -(1) = -1 and 1/2 + 1/2 == 1. (Note the use of equality = and equivalence ==!)

The operations as defined above are defined for surreal numbers but we would like to generalize them for the equivalence classes we defined on them. This can be done without ambiguity because it holds that

if [x] = [x' ] and [y]=[y' ] then [x + y] = [x' + y' ] and [-x] = [-x' ] and [xy] = [x'y' ]

Finally it can be shown that the generalized operations on the equivalence classes have the desired algebraic properties, i.e., the equivalence classes plus their ordering and the algebraic operations constitute an ordered field, with the caveat that they do not form a set but a proper class, see below. In fact, it is a very special ordered field: the biggest one. Every other ordered field can be embedded in the surreals. (See also the definition of rational numbers and real numbers.) In mathematics, an ordered field is a field (F,+,*) together with a total order ≤ on F that is compatible with the algebraic operations in the following sense: if a ≤ b then a + c ≤ b + c if 0 ≤ a and 0 ≤ b then 0 ≤ a b It follows from these axioms... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

From now on we don't distinguish a surreal number from its equivalence class, and call the equivalence class itself a surreal number.

Generating surreal numbers using finite induction

Until now we have not really looked at what numbers we can and cannot create by applying the construction rule. We will first start with the numbers that can be created by applying the rule a finite number of times. We do this by inductively defining S_n with n a natural number as follows:

• S₀ = {0}
• S_{i + 1} is S_i plus the set of all surreal numbers that are generated by the construction rule from subsets of S_i.

The set of all surreal numbers that are generated in some S_i is denoted as S_ω. The first sets of equivalence classes we will find are as follows:

S₀ = { 0 }

S₁ = { -1 < 0 < 1 }

S₂ = { -2 < -1 < -1/2 < 0 < 1/2 < 1 < 2}

S₃ = { -3 < -2 < -1 1/2 < -1 < -3/4 < -1/2 < -1/4 < 0 < 1/4 < 1/2 < 3/4 < 1 < 1 1/2 < 2 < 3 }

S₄ = ...

This leads to the following observations:

1. In every step the maximum (minimum) is increased (decreased) by 1.
2. In every step we find the numbers that are in the middle of two consecutive numbers from the previous step.

As a consequence all generated numbers are dyadic fractions, i.e., can be written as an irreducible fraction In mathematics, a dyadic fraction or dyadic rational is a rational number that when written as a fraction has denominator a power of two, i. ... An irreducible fraction is a fraction a/b, where the numerator a is an integer and the denominator b is a positive integer, such that there is not another fraction c/d with c smaller in absolute value than a and 0<d<b, and c and d are integers...

a / 2^b

where a and b are integers and b ≥ 0. This means that fractions such as 1/3, 2/3, 4/3, 1/5, 5/3, 1/6 et cetera, will not be generated. Note that we can generate numbers that are arbitrarily close to them, but the numbers themselves are never generated. The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

"To Infinity and Beyond"

The next step consists of taking S_ω and continuing to apply the construction rule to it and thus constructing S_ω+1, S_ω+2 et cetera. Note that the left sets and right sets may now become infinite.

In fact, we can define a set S_a for any ordinal number a by transfinite induction. The first time a given surreal number appears in this process is called its birthday. Every surreal number has an ordinal number as its birthday. For example, the birthday of 0 is 0, and the birthday of 1/2 is 2. Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... Transfinite induction is the proof technique of mathematical induction when applied to (large) well-ordered sets, for instance to sets of ordinals or cardinals, or even to the class of all ordinals. ...

Already in S_ω+1 will we find the fractions that were missing in S_ω. For example, the fraction 1/3 can be defined as

1/3 = { { a / 2^b in S_ω | 3a < 2^b } | { a / 2^b in S_ω | 3a > 2^b } }.

The correctness of this definition follows from the fact that

3(1 / 3) == 1.

The birthday of 1/3 is ω+1.

Not only do all the rest of the rational numbers appear in S_ω+1; the remaining finite real numbers do too. For example In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

π = {3, 25/8, 201/64, ... | ..., 101/32, 51/16, 13/4, 7/2, 4}.

Another number that is already constructed in S_ω+1 is Lower-case pi The mathematical constant Ï€ is the ratio of a circles circumference (Greek περιφέρεια, periphery) to its diameter and is commonly used in mathematics, physics, and engineering. ...

ε = { 0 | ..., 1/16, 1/8, 1/4, 1/2, 1 }.

It is easy to see that this number is larger than zero but less than all positive fractions, and therefore an infinitesimal number. The name for its equivalence class is therefore ε. It is not the only positive infinitesimal because it holds for instance that In mathematics, an infinitesimal, or infinitely small number, is a number that is greater in absolute value than zero yet smaller than any positive real number. ...

2ε = { ε | ..., ε + 1/16, ε + 1/8, ε + 1/4, ε + 1/2, ε + 1 } and

ε / 2 = { 0 | ε }.

Note that these numbers are not yet generated in S_ω+1.

Next to infinitely small numbers also infinitely big numbers are generated such as

ω = { S_ω | }.

Its value is clearly bigger than any number in S_ω and its equivalence class is therefore called ω. This number is equivalent with the ordinal number with the same name. We also have the equality Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...

ω = [{ 1, 2, 3, 4, ... | }]

In fact, all ordinal numbers can be expressed as surreal numbers. Since addition and subtraction is defined for all surreal numbers we can use ω like any other number and show for example that

ω + 1 = { ω | } and

ω - 1 = { S_ω | ω }.

We can also do this for bigger numbers

ω + 2 = { ω + 1 | },

ω + 3 = { ω + 2 | },

ω - 2 = { S_ω | ω - 1 } and

ω - 3 = { S_ω | ω - 2 }

and even ω itself

ω + ω = { ω + S_ω | }

where x + Y = { x + y | y in Y }. Just as 2ω is bigger than ω it can also be shown that ω/2 is smaller than ω because

ω/2 = { S_ω | ω - S_ω }

where x - Y = { x - y | y in Y }. Finally, it can be shown that there is a close relationship between ω and ε because it holds that

1 / ε = ω

Note that addition of ordinals differs from addition of their surreal representations. The sum 1 + ω equals ω as ordinals, but as surreals 1 + ω = ω + 1 > ω.

Since every surreal number is constructed from surreal numbers "older" than itself, we can prove many theorems about surreals using transfinite induction: We show that a theorem holds for 0, and then show that it holds for x = { X_L | X_R } if it holds for all elements of X_L and X_R.

Lots of numbers can be generated this way; in fact so many that no set can hold them all. The surreal numbers, like the ordinal numbers, form a proper class. In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ...

Games

The definition of surreal numbers contained one restriction: each element of L must be strictly less than each element of R. If this restriction is dropped we can generate a more general class known as games. All games are constructed according to this rule:

Construction Rule

If L and R are two sets of games then { L | R } is a game.

Addition, negation, multiplication, and comparison are all defined the same way for both surreal numbers and games.

Every surreal number is a game, but not all games are surreal numbers, e.g. the game { 0 | 0 } is not a surreal number. The class of games is more general than the surreals, and has a simpler definition, but lacks some of the nicer properties of surreal numbers. The class of surreal numbers forms a field, but the class of games does not. The surreals have a total order: given any two surreals, they are either equal, or one is greater than the other. The games have only a partial order: there exist pairs of games that are neither equal, greater than, nor less than each other. Each surreal number is either positive, negative, or zero. Each game is either positive, negative, zero, or fuzzy (incomparable with zero, such as {1|-1}). In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... In combinatorial game theory, the zero game is the game where neither player has any legal options. ... This article may be too technical for most readers to understand. ...

A move in a game involves the player whose move it is choosing a game from those available in L (for the left player) or R (for the right player) and then passing this chosen game to the other player. A player who cannot move because the choice is from the empty set has lost. A positive game represents a win for the left player, a negative game for the right player, a zero game for the second player to move, and a fuzzy game for the first player to move. This article may be too technical for most readers to understand. ...

If x, y, and z are surreals, and x=y, then x z=y z. However, if x, y, and z are games, and x=y, then it is not always true that x z=y z. Note that "=" here means equality, not identity.

Surreal numbers and combinatorial game theory

The surreal numbers were originally motivated by studies of the game Go, and there are numerous connections between popular games and the surreals. In this section, we will use a capitalized Game for the mathematical object {L|R}, and the lowercase game for recreational games like Chess or Go. Combinatorial game theory (CGT) is a mathematical theory that studies a certain kind of game. ... Go is a strategic, two-player board game originating in ancient China between 2000 BC and 200 BC. Go is a popular game in East Asia. ... A chess table is a table with a chessboard painted or engraved on it. ...

We consider games with these properties:

• Two players (named Left and Right)
• Deterministic (no dice or shuffled cards)
• No hidden information (such as cards or tiles that a player hides)
• Players alternate taking turns
• Every game must end in a finite number of moves, even when the players don't alternate, and one player can move multiple times in a row
• As soon as there are no legal moves left for a player, the game ends, and that player loses

For most games, the initial board position gives no great advantage to either player. As the game progresses and one player starts to win, board positions will occur where that player has a clear advantage. For analyzing games, it is useful to associate a Game with every board position. The value of a given position will be the Game {L|R}, where L is the set of values of all the positions that can be reached in a single move by Left. Similarly, R is the set of values of all the positions that can be reached in a single move by Right. This simple way to associate Games with games yields a very interesting result. Suppose two perfect players play a game starting with a given position whose associated Game is x. The winner of the game is determined:

• If x > 0 then Left will win.
• If x < 0 then Right will win.
• If x = 0 then the player who goes second will win.
• If x || 0 then the player who goes first will win.

The notation G || H means that G and H are incomparable. G || H is equivalent to G-H || 0. Incomparable games are sometimes said to be confused with each other, because one or the other may be preferred by a player depending on what is added to it. A game confused with zero is said to be fuzzy, as opposed to positive, negative, or zero. An example of a fuzzy game is star (*). This article may be too technical for most readers to understand. ... A negative number is a number that is less than zero, such as −3. ... Star, written as * or *1, is the value given to the combinatorial game {0 | 0}, where zero is the zero game. ...

Sometimes when a game nears the end, it will decompose into several smaller games that do not interact, except in that each player's turn allows moving in only one of them. For example, in Go, the board will slowly fill up with pieces until there are just a few small islands of empty space where a player can move. Each island is like a separate game of Go, played on a very small board. It would be useful if each subgame could be analyzed separately, and then the results combined to give an analysis of the entire game. This doesn't appear to be easy to do. For example, you might have two subgames where whoever moves first wins, but when they are combined into one big game, it's no longer the first player who wins. Fortunately, there is a way to do this analysis. Just use the following remarkable theorem:

If a big game decomposes into two smaller games, and the small games have associated Games of x and y, then the big game will have an associated Game of x+y.

A game composed of smaller games is called the disjunctive sum of those smaller games, and the theorem states that the method of addition we defined is equivalent to taking the disjunctive sum of the addends. The disjunctive sum of two games is a game in which the two games are played in parallel, with each player being allowed to move in just one of the games per turn. ...

Historically, Conway developed the theory of surreal numbers in the reverse order of how it has been presented here. He was analyzing Go endgames, and realized that it would be useful to have some way to combine the analyses of non-interacting subgames into an analysis of their disjunctive sum. From this he invented the concept of a Game and the addition operator for it. From there he moved on to developing a definition of negation and comparison. Then he noticed that a certain class of Games had interesting properties; this class became the surreal numbers. Finally, he developed the multiplication operator, and proved that the surreals are actually a field, and that it includes both the reals and ordinals. Yose is a Japanese language term used in the board game go in connection with go endgame plays. ... The disjunctive sum of two games is a game in which the two games are played in parallel, with each player being allowed to move in just one of the games per turn. ...

Alternative realization

Definitions

In an alternative realization, a surreal number is a function whose domain is an ordinal and range is a subset of { - 1, + 1 }. In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, the domain of a function is the set of all input values to the function. ... Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... In mathematics, the range of a function is the set of all output values produced by that function. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ...

Define the binary predicate "simpler than" on numbers by x is simpler than y if x is a proper subset of y, i.e. if dom(x) < dom(y) and x(α) = y(α) for all α < dom(x). A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ...

For numbers x and y, define x < y if one of the following holds:

• x is simpler than y and y(dom(x)) = + 1;
• y is simpler than x and x(dom(y)) = - 1;
• there exists a number z such that z is simpler than x, z is simpler than y, x(dom(z)) = - 1 and y(dom(z)) = + 1.

Equavalently, let δ(x,y) = min({ dom(x), dom(y)} ∪ { α : α < dom(x) ∧ α < dom(y) ∧ x(α) ≠ y(α) }), so that x = y iff δ(x,y) = dom(x) = dom(y). Then, for numbers x and y, x < y iff one of the following holds:

• δ(x,y) = dom(x) ∧ δ(x,y) < dom(y) ∧ y(δ(x,y)) = + 1;
• δ(x,y) < dom(x) ∧ δ(x,y) = dom(y) ∧ x(δ(x,y)) = - 1;
• δ(x,y) < dom(x) ∧ δ(x,y) < dom(y) ∧ x(δ(x,y)) = - 1 ∧ y(δ(x,y)) = + 1.

For numbers x and y, x ≤ y iff x < y ∨ x = y, x > y iff y < x, and x ≥ y iff y ≤ x.

< is transitive, and for all numbers x and y, exactly one of x < y, x = y, x > y, holds (law of trichotomy). This means that < is a linear order (except that < is a proper class). In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ... Generally, a trichotomy is a splitting into three disjoint parts. ... In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ...

For sets of numbers, L and R such that ∀x ∈ L ∀y ∈ R (x < y), there exists a unique number z such that

• ∀x ∈ L (x < z) ∧ ∀y ∈ R (z < y),
• For any number w such that ∀x ∈ L (x < w) ∧ ∀y ∈ R (w < y), w = z or z is simpler than w.

Furthermore, z is constructible from L and R by transfinite induction. z is the simplest number between L and R. Let the unique number z be deonted by σ(L,R). In mathematics, there are several notions of constructibility: a point in the Euclidean plane that can be constructed with unruled straightedge and compass. ...

For a number x, define its left set L(x) and right set R(x) by

• L(x) = { x|_α : α < dom(x) ∧ x(α) = + 1 };
• R(x) = { x|_α : α < dom(x) ∧ x(α) = - 1 },

then σ(L(x),R(x)) = x.

Addition and Multiplication

The sum x + y of two numbers, x and y, is defined by induction on dom(x) and dom(y) by x + y = σ(L,R), where

• L = { u + y : u ∈ L(x) } ∪{ x + v : v ∈ L(y) },
• R = { u + y : u ∈ R(x) } ∪{ x + v : v ∈ R(y) }.

The additive identity is given by the number 0 = { }, i.e. the number 0 is the unique function whose domain is the ordinal 0, and the additive inverse of the number x is the number - x, given by dom(- x) = dom(x), and, for α < dom(x), (- x)(α) = - 1 if x(α) = + 1, and (- x)(α) = + 1 if x(α) = - 1.

It follows that a number x is positive iff 0 < dom(x) and x(0) = + 1, and x is negative iff 0 < dom(x) and x(0) = - 1. In common usage positive is sometimes used in affirmation, as a synonym for yes or to express certainty. Look up Positive on Wiktionary, the free dictionary In mathematics, a number is called positive if it is bigger than zero. ... Negative has meaning in several contexts: Look up negative in Wiktionary, the free dictionary Negative and non-negative numbers Negative (photography) In optics, diverging lenses are also called negative lenses. ...

The product xy of two numbers, x and y, is defined by induction on dom(x) and dom(y) by xy = σ(L,R), where

• L = { uy + xv - uv : u ∈ L(x), v ∈ L(y) } ∪ { uy + xv - uv : u ∈ R(x), v ∈ R(y) },
• R = { uy + xv - uv : u ∈ L(x), v ∈ R(y) } ∪ { uy + xv - uv : u ∈ R(x), v ∈ L(y) }.

The multiplicative identity is given by the number 1 = { (0,+ 1) }, i.e. the number 1 has domain equal to the ordinal 1, and 1(0) = + 1.

Correspondence between realizations

The map from Conway's realization to the alternative realization is given by f({ L | R }) = σ(M,S), where M = { f(x) : x ∈ L } and S = { f(x) : x ∈ R }.

The inverse map from the alternative realization to Conway's realization is given by g(x) = { L | R }, where L = { g(y) : y ∈ L(x) } and R = { g(y) : y ∈ R(x) }.

Further reading

• Donald Knuth's original exposition: Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. 1974, ISBN 0201038129. More information can be found at the book's official homepage
• An update of the classic 1976 book defining the surreal numbers, and exploring their connections to games: On Numbers And Games, 2nd ed., John Conway, 2001, ISBN 1568811276.
• An update of the first part of the 1981 book that presented surreal numbers and the analysis of games to a broader audience: Winning Ways for Your Mathematical Plays, vol. 1, 2nd ed., Berlekamp, Conway, and Guy, 2001, ISBN 1568811306.
• Martin Gardner Penrose Tiles to Trapdoor Ciphers chapter 4 — not especially technical overview; reprints the 1976 Scientific American article

Donald Knuth at a reception for the Open Content Alliance. ... Martin Gardner (born October 21, 1914) is an American recreational mathematician, skeptic, and author of the long-running but now discontinued Mathematical Games column in Scientific American. ... 1976 (MCMLXXVI) is a leap year starting on Thursday (link will take you to calendar). ...

External links

• A gentle yet thorough introduction by Claus Tøndering
• Surreal number on PlanetMath.