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
The theory of surreal numbers begins with just a few (inductive) definitions:
Definition of x ≥ y, x ≤ y
x ≥ y iff, for no x^{R}, x^{R} ≤ y, and, for no y^{L},
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 = {x^{L} + y, x + y^{L}|x^{R} + y, x + y^{R}}.
Definition of -x
-x = {-x^{R}|-x^{L}}.
Definition of xy
xy = {x^{L}y + xy^{L} - x^{L}y^{L}, x^{R}y + xy^{R} - x^{R}y^{R}|x^{L}y + xy^{R} - x^{L}y^{R}, x^{R}y + xy^{L} - x^{R}y^{L}}.
(Strange as this definition appears at first, it is motivated by the requirement
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
0 = {|}.
Is it really a number? To verify that it is we need to check that for no 0^{L} and 0^{R},
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|}, {|0}, and {0|0}.
The latter is not a number because
So, why 0 ≥ 0? Because, in the absence of 0^{R} and 0^{L}, no inequality of the form
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
-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
By the definition,
-1 + 1 | = {|0} + {0|} | |
= {(-1)^{L} + 1, (-1) + 1^{L}|(-1)^{R} + 1, (-1) + 1^{R}} | ||
= {-1 + 0|0 + 1} |
because (-1)^{L} and 1^{R} do not exist. Assuming the identities
0 = {-1|1}.
This follows from a general statement:
(1) | {y, x^{L}|x^{R}} = x iff y ≥ x does not hold. |
A similar assertion holds for {x^{L}|y, x^{R}}:
(2) | {x^{L}|y, x^{R}} = x iff y ≤ x does not hold. |
The first of these implies that 0 = {-1|} because
Conway proves (1) in a most elegant and concise form:
Let X = {y, x^{L}|x^{R}}. Is X ≥ x? Yes, unless some X^{R} ≤ x (no, since every X^{R} is an x^{R}) or Is x ≥ X? Yes, unless x^{R} ≤ X (no, since every x^{R} is an X^{R}) or x ≤ some X^{L} (and so |
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
{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/2 = {n|ω - n}.
But do not stop here. Define, ω/4, ω/8, ... and then
2ω = {ω + n|}, | |
3ω = {2ω + n|}, | |
4ω = {3ω + n|}, | |
... | |
ω² = {nω|}, | |
... | |
ω^{ω} = {ω^{n}|}, | |
... |
Hmm, this may keep one busy for a long while.
References
- J. H. Conway, On Numbers And Games, A K Peters, 2001
- M. Gardner, Mathematical Carnival, Vintage Books, 1977
- M. Gardner, Penrose Tiles to Trapdoor Ciphers, W. H. Freeman and Company, 1989
- D. E. Knuth, Surreal Numbers, Addison-Wesley, 1974
- What Is Infinity?
- What Is Finite?
- Infinity As a Limit
- Cardinal Numbers
- Ordinal Numbers
- Surreal Numbers
- Infinitesimals. Non-standard Analysis
- Various Geometric Infinities
- Paradoxes of Infinity
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