surreal number

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

=

⟨

N

L

∣

N

R

⟩

.

Not every number of this form is a surreal number. The surreal numbers

satisfy

two additional

properties

. First, if

x

∈

N

R

and

y

∈

N

L

then

x

≰

y

. Secondly, they must be well founded. These properties are both satisfied by the following construction of the surreal numbers and the

≤

relation

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

x

∈

R

and

y

∈

L

,

x

≰

y

,

⟨

L

∣

R

⟩

.

Define

N

≤

M

if there is no

x

∈

N

L

such that

M

≤

x

and no

y

∈

M

R

such that

y

≤

N

.

This process can be continued transfinitely, to define

infinite

and infinitesimal numbers. For

instance

if

ℤ

is the set of integers then

ω

=

⟨

ℤ

∣

⟩

. Note that this does not make equality the same as

identity

:

⟨

1

∣

1

⟩

=

⟨

∣

⟩

, 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

=

⟨

{

N

+

x

∣

x

∈

M

L

}

∪

{

M

+

x

∣

y

∈

N

L

}

∣

{

N

+

x

∣

x

∈

M

R

}

∪

{

M

+

x

∣

y

∈

N

R

}

⟩

It follows that

-

N

=

⟨

-

N

R

∣

-

N

L

⟩

.

The

definition

of

multiplication

can be written more easily by defining

M

⋅

N

L

=

{

M

⋅

x

∣

x

∈

N

L

}

and similarly for

N

R

.

Then

N

⋅

M

=

⟨

M

⋅

N

L

+

N

⋅

M

L

-

N

L

⋅

M

L

,

M

⋅

N

R

+

N

⋅

M

R

-

N

R

⋅

M

R

∣

M

⋅

N

L

+

N

⋅

M

R

-

N

L

⋅

M

R

,

M

⋅

N

R

+

N

⋅

M

L

-

N

R

⋅

M

L

⟩

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,

⟨

a

∣

b

⟩

is the simplest number between

a

and

b

. This can be easily used to define the dyadic fractions: for any integer

a

,

a

+

1

2

=

⟨

a

∣

a

+

1

⟩

. Then

1

2

=

⟨

0

∣

1

⟩

,

1

4

=

⟨

0

∣

1

2

⟩

, 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

⟨

0

∣

1

,

1

2

,

1

4

⁢

…

⟩

=

1

ω

, 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

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