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# 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 $N=\langle N_{L}\mid N_{R}\rangle$.

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 $x\nleq y$. Secondly, they must be well founded. These properties are both satisfied by the following construction of the surreal numbers and the $\leq$ relation by mutual induction:

$\langle\mid\rangle$, 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 $y\in L$, $x\nleq y$, $\langle L\mid R\rangle$.

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

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

$N+M=\langle\{N+x\mid x\in M_{L}\}\cup\{M+x\mid y\in N_{L}\}\mid\{N+x\mid x\in M% _{R}\}\cup\{M+x\mid y\in N_{R}\}\rangle$ |

It follows that $-N=\langle-N_{R}\mid-N_{L}\rangle$.

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

$\displaystyle N\cdot M=$ | $\displaystyle\langle M\cdot N_{L}+N\cdot M_{L}-N_{L}\cdot M_{L},M\cdot N_{R}+N% \cdot M_{R}-N_{R}\cdot M_{R}\mid$ | ||

$\displaystyle M\cdot N_{L}+N\cdot M_{R}-N_{L}\cdot M_{R},M\cdot N_{R}+N\cdot M% _{L}-N_{R}\cdot M_{L}\rangle$ |

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 $\langle n-1\mid\rangle$ and so $-n=\langle\mid 1-n\rangle=\langle\mid(-n)+1\rangle$. So for instance $1=\langle 0\mid\rangle$.

In general, $\langle a\mid b\rangle$ is the simplest number between $a$ and $b$. This can be easily used to define the dyadic fractions: for any integer $a$, $a+\frac{1}{2}=\langle a\mid a+1\rangle$. Then $\frac{1}{2}=\langle 0\mid 1\rangle$, $\frac{1}{4}=\langle 0\mid\frac{1}{2}\rangle$, 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 $\langle\omega\mid\rangle=\omega+1$ and so on. Similarly, a starting infinitesimal can be found as $\langle 0\mid 1,\frac{1}{2},\frac{1}{4}\ldots\rangle=\frac{1}{\omega}$, and again more can be developed from there.

## Mathematics Subject Classification

00A05*no label found*

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