"}},"i":0}}]}" id="mwA3k">α = { α | } in the surreals; denotes the class of ordinal numbers, and because is cofinal in we have by extension.)
With a bit of set-theoretic care,^{ [lower-alpha 4] } can be equipped with a topology where the open sets are unions of open intervals (indexed by proper sets) and continuous functions can be defined.^{ [9] } An equivalent of Cauchy sequences can be defined as well, although they have to be indexed by the class of ordinals; these will always converge, but the limit may be either a number or a gap that can be expressed as with a_{α} decreasing and having no lower bound in . (All such gaps can be understood as Cauchy sequences themselves, but there are other types of gap that are not limits, such as ∞ and ).^{ [9] }
Based on unpublished work by Kruskal, a construction (by transfinite induction) that extends the real exponential function exp(x) (with base e) to the surreals was carried through by Gonshor.^{ [8] }^{: ch. 10 }
The powers of ω function is also an exponential function, but does not have the properties desired for an extension of the function on the reals. It will, however, be needed in the development of the base-e exponential, and it is this function that is meant whenever the notation ω^{x} is used in the following.
When y is a dyadic fraction, the power function , x ↦ x^{y} may be composed from multiplication, multiplicative inverse and square root, all of which can be defined inductively. Its values are completely determined by the basic relation x^{y+z} = x^{y} · x^{z}, and where defined it necessarily agrees with any other exponentiation that can exist.
The induction steps for the surreal exponential are based on the series expansion for the real exponential, more specifically those partial sums that can be shown by basic algebra to be positive but less than all later ones. For x positive these are denoted [x]_{n} and include all partial sums; for x negative but finite, [x]_{2n+1} denotes the odd steps in the series starting from the first one with a positive real part (which always exists). For x negative infinite the odd-numbered partial sums are strictly decreasing and the [x]_{2n+1} notation denotes the empty set, but it turns out that the corresponding elements are not needed in the induction.
The relations that hold for real x < y are then
and
and this can be extended to the surreals with the definition
This is well-defined for all surreal arguments (the value exists and does not depend on the choice of z_{L} and z_{R}).
Using this definition, the following hold:^{ [lower-alpha 5] }
The surreal exponential is essentially given by its behaviour on positive powers of ω, i.e., the function , combined with well-known behaviour on finite numbers. Only examples of the former will be given. In addition, holds for a large part of its range, for instance for any finite number with positive real part and any infinite number that is less than some iterated power of ω (ω^{ω··ω} for some number of levels).
A general exponentiation can be defined as x^{y} = exp(y · log x), giving an interpretation to expressions like 2^{ω} = exp(ω · log 2) = ω^{log 2 · ω}. Again it is essential to distinguish this definition from the "powers of ω" function, especially if ω may occur as the base.
A surcomplex number is a number of the form a + bi, where a and b are surreal numbers and i is the square root of −1.^{ [10] }^{ [11] } The surcomplex numbers form an algebraically closed field (except for being a proper class), isomorphic to the algebraic closure of the field generated by extending the rational numbers by a proper class of algebraically independent transcendental elements. Up to field isomorphism, this fact characterizes the field of surcomplex numbers within any fixed set theory.^{ [6] }^{: Th.27 }
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:
Addition, negation, 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}).
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.
If x, y, and z are surreals, and x = y, then xz = yz. However, if x, y, and z are games, and x = y, then it is not always true that xz = yz. Note that "=" here means equality, not identity.
The surreal numbers were originally motivated by studies of the game Go,^{ [2] } 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.
We consider games with these properties:
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 in which 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.
The zero Game (called 0) is the Game where L and R are both empty, so the player to move next (L or R) immediately loses. The sum of two Games G = { L1 | R1 } and H = { L2 | R2 } is defined as the Game G + H = { L1 + H, G + L2 | R1 + H, G + R2 } where the player to move chooses which of the Games to play in at each stage, and the loser is still the player who ends up with no legal move. One can imagine two chess boards between two players, with players making moves alternately, but with complete freedom as to which board to play on. If G is the Game {L | R}, −G is the Game {−R | −L}, i.e. with the role of the two players reversed. It is easy to show G – G = 0 for all Games G (where G – H is defined as G + (–H)).
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. We can classify all Games into four classes as follows:
More generally, we can define G > H as G – H > 0, and similarly for <, = and ||.
The notation G || H means that G and H are incomparable. G || H is equivalent to G − H || 0, i.e. that G > H, G < H and G = H are all false. 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 (*).
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, there might be two subgames where whoever moves first wins, but when they are combined into one big game, it is no longer the first player who wins. Fortunately, there is a way to do this analysis. The following theorem can be applied:
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.
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.
Alternative approaches to the surreal numbers complement Conway's exposition in terms of games.
In what is now called the sign-expansion or sign-sequence of a surreal number, a surreal number is a function whose domain is an ordinal and whose codomain is { −1, +1 }.^{ [8] }^{: ch. 2 } This is equivalent to Conway's L-R sequences.^{ [6] }
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).
For surreal numbers define the binary relation < to be lexicographic order (with the convention that "undefined values" are greater than −1 and less than 1). So x<y if one of the following holds:
Equivalently, let δ(x,y) = min({ dom(x), dom(y)} ∪ { α : α < dom(x) ∧ α < dom(y) ∧ x(α) ≠ y(α) }), so that x = y if and only if δ(x,y) = dom(x) = dom(y). Then, for numbers x and y, x<y if and only if one of the following holds:
For numbers x and y, x ≤ y if and only if x<y ∨ x = y, and x>y if and only if y<x. Also x ≥ y if and only if y ≤ x.
The relation < 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).
For sets of numbers, L and R such that ∀x ∈ L ∀y ∈ R (x<y), there exists a unique number z such that
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 denoted by σ(L,R).
For a number x, define its left set L(x) and right set R(x) by
then σ(L(x),R(x)) = x.
One advantage of this alternative realization is that equality is identity, not an inductively defined relation. Unlike Conway's realization of the surreal numbers, however, the sign-expansion requires a prior construction of the ordinals, while in Conway's realization, the ordinals are constructed as particular cases of surreals.
However, similar definitions can be made that eliminate the need for prior construction of the ordinals. For instance, we could let the surreals be the (recursively-defined) class of functions whose domain is a subset of the surreals satisfying the transitivity rule ∀g ∈ dom f (∀h ∈ dom g (h ∈ dom f )) and whose range is { −, + }. "Simpler than" is very simply defined now—x is simpler than y if x ∈ dom y. The total ordering is defined by considering x and y as sets of ordered pairs (as a function is normally defined): Either x = y, or else the surreal number z = x ∩ y is in the domain of x or the domain of y (or both, but in this case the signs must disagree). We then have x < y if x(z) = − or y(z) = + (or both). Converting these functions into sign sequences is a straightforward task; arrange the elements of dom f in order of simplicity (i.e., inclusion), and then write down the signs that f assigns to each of these elements in order. The ordinals then occur naturally as those surreal numbers whose range is { + }.
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
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 if and only if 0 < dom(x) and x(0) = +1, and x is negative if and only if 0 < dom(x) and x(0) = −1.
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
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.
The map from Conway's realization to sign expansions 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) }.
In another approach to the surreals, given by Alling,^{ [11] } explicit construction is bypassed altogether. Instead, a set of axioms is given that any particular approach to the surreals must satisfy. Much like the axiomatic approach to the reals, these axioms guarantee uniqueness up to isomorphism.
A triple is a surreal number system if and only if the following hold:
Both Conway's original construction and the sign-expansion construction of surreals satisfy these axioms.
Given these axioms, Alling^{ [11] } derives Conway's original definition of ≤ and develops surreal arithmetic.
A construction of the surreal numbers as a maximal binary pseudo-tree with simplicity (ancestor) and ordering relations is due to Philip Ehrlich.^{ [12] } The difference from the usual definition of a tree is that the set of ancestors of a vertex is well-ordered, but may not have a maximal element (immediate predecessor); in other words the order type of that set is a general ordinal number, not just a natural number. This construction fulfills Alling's axioms as well and can easily be mapped to the sign-sequence representation.
Alling^{ [11] }^{: th. 6.55, p. 246 } also proves that the field of surreal numbers is isomorphic (as an ordered field) to the field of Hahn series with real coefficients on the value group of surreal numbers themselves (the series representation corresponding to the normal form of a surreal number, as defined above). This provides a connection between surreal numbers and more conventional mathematical approaches to ordered field theory.
This isomorphism makes the surreal numbers into a valued field where the valuation is the additive inverse of the exponent of the leading term in the Conway normal form, e.g., ν(ω) = −1. The valuation ring then consists of the finite surreal numbers (numbers with a real and/or an infinitesimal part). The reason for the sign inversion is that the exponents in the Conway normal form constitute a reverse well-ordered set, whereas Hahn series are formulated in terms of (non-reversed) well-ordered subsets of the value group.
Philip Ehrlich has constructed an isomorphism between Conway's maximal surreal number field and the maximal hyperreals in von Neumann–Bernays–Gödel set theory.^{ [12] }
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation ; every complex number can be expressed in the form , where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number ,a is called the real part, and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world.
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Theorem 24.29. The surreal number system is the largest ordered field