PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (198)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Medium Entry average rating: No information on entry rating
surreal number (Definition)

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

$\displaystyle N+M=\langle \{N+x\mid x\in M_L\}\cup\{M+x\mid y\in N_L\}\mid \{N+x\mid x\in M_R\}\cup\{M+x\mid y\in N_R\}\rangle$

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.



"surreal number" is owned by Henry.
(view preamble)

View style:

Also defines:  omnific integers
Log in to rate this entry.
(view current ratings)

Cross-references: ordinal arithmetic, fractions, dyadic fractions, positive, field, axioms, multiplication, addition, identity, equality, integers, infinitesimal, infinite, induction, relation, properties, number, set theory, basis, ordered pair, right, reals
There are 5 references to this entry.

This is version 6 of surreal number, born on 2002-08-24, modified 2005-03-03.
Object id is 3352, canonical name is SurrealNumber.
Accessed 10499 times total.

Classification:
AMS MSC: 00A05 (General :: General and miscellaneous specific topics :: General mathematics)
Pending Errata and Addenda
None.
[ View all 5 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)