Surreal Numbers

Surreal numbers have been invented by John Conway and so named by Donald Knuth. There is much to justify the term.

The collection includes unheard of numbers as ω + π/(ω - 1)², where ω is the order-type of the natural numbers. The real numbers form a subset of the surreals, but only a minuscule part of the latter. The situation is reminiscent of the prevalence of the transcendental numbers among the reals, although it is incongruently worse. In the Zermelo-Frenkel axioms of Set Theory, the collection of surreal numbers is a proper class, too big to be a set.

Surreal numbers also form a field, in other words, commutative addition and subtraction are defined for any pair of surreal numbers; both operations are associative and addition is distributive with respect to multiplication; also, for any surreal number, there is an additive inverse and, for all, except 0, there is a multiplicative inverse. The field of surreal numbers is totally ordered while the operations of addition and multiplication (by a positive number) preserve the order.

On the down side, the field of surreal numbers is not Archimedean: in particular this means that some surreal numbers are infinitesimal, i.e., are less than any positive real number, e.g., 1/ω. ω itself is an example of a surreal number infinitely large.

Every surreal numbers is a game, although the converse is not true. (As games, the numbers are not interesting.) A very short introduction into surreal numbers that starts with Conway's definition of game is available elsewhere. A detailed introduction is available online.

Formally, surreal numbers are constructed inductively.

If L and R are two sets of (already constructed) numbers such that no element of L is ≥ any element of R, then {L|R} is a (surreal) number. All (surreal) numbers are constructed this way.

For the sake of convenience, we write, say, {a, b, c, ...|x, y, z, ...} instead of {{a, b, c, ...}|{x, y, z, ...}} spilling the elements of the left and right sets into the two-set notation. If x = {L|R}, xL stands for a generic element of L, while xR stands for a generic element of R so that x can also be written as x = {xL|xR}.

The theory of surreal numbers begins with just a few (inductive) definitions:

Definition of x ≥ y, x ≤ y

x ≥ y iff, for no xR, xR ≤ y, and, for no yL, x ≤ yL. x ≤ y iff y ≥ x.

Definition of x = y, x > y, x < y

x = y iff x ≥ y and x ≤ y,
x > y iff x ≥ y and x ≠ y,
x < y iff x ≤ y and x ≠ y.

(Note that equality is a defined relation.)

Definition of x + y

x + y = {xL + y, x + yL|xR + y, x + yR}.

Definition of -x

-x = {-xR|-xL}.

Definition of xy

xy = {xLy + xyL - xLyL, xRy + xyR - xRyR|xLy + xyR - xLyR, xRy + xyL - xRyL}.

(Strange as this definition appears at first, it is motivated by the requirement (x - xL)(y - yL) > 0 that a multiplication operation may be supposed to uphold.)

Now, all definitions are surely inductive as their right parts all include terms with L and R supposedly defined previously. However, induction has to start somewhere. And in all cases it does, vacuously! For, as Conway wrote, ... even before we have any numbers, we have a certain set of numbers, namely the empty set ø! So the earliest constructed number can only be {ø|ø}, or conventionally spilling its elements, {|}. This number is called 0!

0 = {|}.

Is it really a number? To verify that it is we need to check that for no 0L and 0R, 0L ≥ 0R. But this is of course true as there is neither 0L nor 0R!

If we apply the definitions a step a day, constructing the new numbers that only depend on the numbers constructed on previous days, each constructed number will get a birthday meaningfully assigned. 0 has been constructed on day 0! On day 1, in addition to 0 = {|}, we may form

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

The latter is not a number because 0 ≥ 0. Right? Well, this is true, as expected, but still has to be proved. The definitions in the theory of surreal numbers are few, but bear in mind that the numbers are being constructed from scratch, so that the the properties of theirs that get to be proved multiply.

So, why 0 ≥ 0? Because, in the absence of 0R and 0L, no inequality of the form 0R ≤ 0 or 0 ≤ 0L is possible.

Further, since 0 ≥ 0 we also have 0 = 0 as a consequence from the definitions.

The two constructs that remain - {0|} and {|0} - are easily proved to be numbers. They are given names:

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

Why? For, again, it is possible to prove that 0 ≤ {0|} and {|0} ≤ 0, whereas it is provably not true that either 0 ≥ {0|} and {|0} ≥ 0. It then follows that

-1 < 0 < 1.

Just two numbers have a birthday on day 1, but there is a plenty of facts that can be proved that involve only the three existing numbers. For example, one would think that

 1 + 0= 1,
 -1 + 1= 0,
 (-1) × 1= -1,

among other identities that come to mind. These are true, but have to be proved. Say, why -1 + 1= 0?

By the definition,

 -1 + 1= {|0} + {0|}
  = {(-1)L + 1, (-1) + 1L|(-1)R + 1, (-1) + 1R}
  = {-1 + 0|0 + 1}

because (-1)L and 1R do not exist. Assuming the identities -1 + 0 = -1 and 0 + 1 = 1 proved, we are now faced with proving that

0 = {-1|1}.

This follows from a general statement:

(1) {y, xL|xR} = x iff y ≥ x does not hold.

A similar assertion holds for {xL|y, xR}:

(2) {xL|y, xR} = x iff y ≤ x does not hold.

The first of these implies that 0 = {-1|} because -1 < 0 and then, from the second, 0 = {-1|1} because 0 < 1.

Conway proves (1) in a most elegant and concise form:

 

Let X = {y, xL|xR}.

Is X ≥ x? Yes, unless some XR ≤ x (no, since every XR is an xR) or X ≤ some xL (no, since every xL is an XL).

Is x ≥ X? Yes, unless xR ≤ X (no, since every xR is an XR) or x ≤ some XL (and so x ≤ y, since every XL is an xL.)

And (1) follows.

On day 2, we are going to create

 -2= {|-1}= {|-1, 0}= {|-1, 1}= {|-1, 0, 1},
 -1/2= {-1|0}= {-1|0, 1},  
 1/2= {0|1}= {-1, 0|1},  
 2= {1|}= {0, 1|}= {-1, 1|}= {-1, 0, 1|}

and verify that, say, 1/2 + 1/2 = 1. Nothing comes for granted.

On day 3, one creates numbers 1/4 = {0|1/2}, 3/4 = {1/2|1}, 3/2 = {1|2}, 3 = {2|} and their negatives. (These until some of their properties, like say 1/4 + 1/4 = 1/2, have been established remain names which are at best intelligent guesses.) In general, n = {n - 1|}, {n|n+1} = n + 1/2, {n|n + 1/2} = n + 1/4, and so on. All integers and dyadic fractions have thus finite birthdays. On day ω, we make up for the missing rationals, like 1/3 and all the irrational numbers and some. This is because of the association of Conway's definition with Dedekind's cuts. On day ω, we also define ω = {1, 2, ...|} or, as Conway writes, ω = {n|}, with generic n, and 1/ω = {0|2-n}.

{ω|} = {1, 2, ..., ω|} is naturally ω + 1, but what is x = {n|ω}? For one,

n < x < ω,

for every integer n. By definition, x + 1 = {n, x|ω + 1}. And since x < ω while ω + 1 is not, it must be that x + 1 = ω making x = ω - 1. Continuing in this manner, we construct ω - n which are all infinite. And then {n|ω - n} which is also infinite. We then verify that {n|ω - n} + {n|ω - n} = ω getting

ω/2 = {n|ω - n}.

But do not stop here. Define, ω/4, ω/8, ... and then ω = {n|ω/2n}. On the other hand,

 2ω = {ω + n|},
 3ω = {2ω + n|},
 4ω = {3ω + n|},
 ...
 ω² = {nω|},
 ...
 ωω = {ωn|},
 ...

Hmm, this may keep one busy for a long while.

References

  1. J. H. Conway, On Numbers And Games, A K Peters, 2001
  2. M. Gardner, Mathematical Carnival, Vintage Books, 1977
  3. M. Gardner, Penrose Tiles to Trapdoor Ciphers, W. H. Freeman and Company, 1989
  4. D. E. Knuth, Surreal Numbers, Addison-Wesley, 1974

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