# surreal number

The surreal numbers^{} are a generalization^{} 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 ${N}_{L}$ and ${N}_{R}$. (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 theory^{}.) A surreal number is written $N=\u27e8{N}_{L}\mid {N}_{R}\u27e9$.

Not every number of this form is a surreal number. The surreal numbers satisfy two additional properties. First, if $x\in {N}_{R}$ and $y\in {N}_{L}$ then $x\nleqq y$. Secondly, they must be well founded. These properties are both satisfied by the following construction of the surreal numbers and the $\le $ relation^{} by mutual induction:

$\u27e8\mid \u27e9$, 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 $x\in R$ and $y\in L$, $x\nleqq y$, $\u27e8L\mid R\u27e9$.

Define $N\le M$ if there is no $x\in {N}_{L}$ such that $M\le x$ and no $y\in {M}_{R}$ such that $y\le N$.

This process can be continued transfinitely, to define infinite^{} and infinitesimal numbers. For instance if $\mathbb{Z}$ is the set of integers then $\omega =\u27e8\mathbb{Z}\mid \u27e9$. Note that this does not make equality the same as identity: $\u27e81\mid 1\u27e9=\u27e8\mid \u27e9$, for instance.

It can be shown that $N$ is “sandwiched” between the elements of ${N}_{L}$ and ${N}_{R}$: it is larger than any element of ${N}_{L}$ and smaller than any element of ${N}_{R}$.

Addition^{} of surreal numbers is defined by

$$N+M=\u27e8\{N+x\mid x\in {M}_{L}\}\cup \{M+x\mid y\in {N}_{L}\}\mid \{N+x\mid x\in {M}_{R}\}\cup \{M+x\mid y\in {N}_{R}\}\u27e9$$ |

It follows that $-N=\u27e8-{N}_{R}\mid -{N}_{L}\u27e9$.

The definition of multiplication can be written more easily by defining $M\cdot {N}_{L}=\{M\cdot x\mid x\in {N}_{L}\}$ and similarly for ${N}_{R}$.

Then

$N\cdot M=$ | $\u27e8M\cdot {N}_{L}+N\cdot {M}_{L}-{N}_{L}\cdot {M}_{L},M\cdot {N}_{R}+N\cdot {M}_{R}-{N}_{R}\cdot {M}_{R}\mid $ | ||

$M\cdot {N}_{L}+N\cdot {M}_{R}-{N}_{L}\cdot {M}_{R},M\cdot {N}_{R}+N\cdot {M}_{L}-{N}_{R}\cdot {M}_{L}\u27e9$ |

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 $\u27e8n-1\mid \u27e9$ and so $-n=\u27e8\mid 1-n\u27e9=\u27e8\mid (-n)+1\u27e9$. So for instance $1=\u27e80\mid \u27e9$.

In general, $\u27e8a\mid b\u27e9$ is the simplest number between $a$ and $b$. This can be easily used to define the dyadic fractions: for any integer $a$, $a+\frac{1}{2}=\u27e8a\mid a+1\u27e9$. Then $\frac{1}{2}=\u27e80\mid 1\u27e9$, $\frac{1}{4}=\u27e80\mid \frac{1}{2}\u27e9$, 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 $\omega $ as defined above and adding numbers such as $\u27e8\omega \mid \u27e9=\omega +1$ and so on. Similarly, a starting infinitesimal can be found as $\u27e80\mid 1,\frac{1}{2},\frac{1}{4}\mathrm{\dots}\u27e9=\frac{1}{\omega}$, 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 |