Surreal number

In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field.[a] If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers (including the hyperreal numbers) can be realized as subfields of the surreals.[1] The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been shown (in von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorphic to the maximal class surreal field.

A visualization of the surreal number tree.

1 History of the concept

2 Description

3 Construction

3.1 Forms

3.2 Numeric forms and their equivalence classes

3.3 Order

3.4 Induction

3.4.1 Induction rule

3.4.2 Birthday property

4 Arithmetic

4.1 Negation

4.2 Addition

4.3 Multiplication

4.4 Division

4.5 Consistency

4.6 Arithmetic closure

5 Infinity

5.1 Contents of Sω

6 Transfinite induction

7 Powers of ω and the Conway normal form

8 Gaps and continuity

9 Exponential function

9.1 Other exponentials

9.2 Basic induction

9.3 Results

9.4 Examples

9.5 Exponentiation

10 Surcomplex numbers

11 Games

12 Application to combinatorial game theory

13 Alternative realizations

13.1 Sign expansion

13.1.1 Definitions

13.1.2 Addition and multiplication

13.1.3 Correspondence with Conway’s realization

13.2 Axiomatic approach

13.3 Simplicity hierarchy

13.4 Hahn series

14 Relation to hyperreals

15 See also