surreal number

The surreal numbersMathworldPlanetmath are a generalizationPlanetmathPlanetmath of the reals. Each surreal number consists of two parts (called the left and right), each of which is a set of surreal numbers. For any surreal number N, these parts can be called NL and NR. (This could be viewed as an ordered pair of sets, however the surreal numbers were intended to be a basis for mathematics, not something to be embedded in set theoryMathworldPlanetmath.) A surreal number is written N=NLNR.

Not every number of this form is a surreal number. The surreal numbers satisfy two additional properties. First, if xNR and yNL then xy. Secondly, they must be well founded. These properties are both satisfied by the following construction of the surreal numbers and the relationMathworldPlanetmath by mutual induction:

, which has both left and right parts empty, is 0.

Given two (possibly empty) sets of surreal numbers R and L such that for any xR and yL, xy, LR.

Define NM if there is no xNL such that Mx and no yMR such that yN.

This process can be continued transfinitely, to define infiniteMathworldPlanetmath and infinitesimal numbers. For instance if is the set of integers then ω=. Note that this does not make equality the same as identity: 11=, for instance.

It can be shown that N is “sandwiched” between the elements of NL and NR: it is larger than any element of NL and smaller than any element of NR.

AdditionPlanetmathPlanetmath of surreal numbers is defined by


It follows that -N=-NR-NL.

The definition of multiplication can be written more easily by defining MNL={MxxNL} and similarly for NR.



The surreal numbers satisfy the axioms for a field under addition and multiplication (whether they really are a field is complicated by the fact that they are too large to be a set).

The integers of surreal mathematics are called the omnific integers. In general positive integers n can always be written n-1 and so -n=1-n=(-n)+1. So for instance 1=0.

In general, ab is the simplest number between a and b. This can be easily used to define the dyadic fractions: for any integer a, a+12=aa+1. Then 12=01, 14=012, and so on. This can then be used to locate non-dyadic fractions by pinning them between a left part which gets infinitely close from below and a right part which gets infinitely close from above.

Ordinal arithmetic can be defined starting with ω as defined above and adding numbers such as ω=ω+1 and so on. Similarly, a starting infinitesimal can be found as 01,12,14=1ω, and again more can be developed from there.

Title surreal number
Canonical name SurrealNumber
Date of creation 2013-03-22 12:58:49
Last modified on 2013-03-22 12:58:49
Owner Henry (455)
Last modified by Henry (455)
Numerical id 9
Author Henry (455)
Entry type Definition
Classification msc 00A05
Defines omnific integers