![]() |
This article may be too technical for most readers to understand. Please help improve ittomake it understandable to non-experts, without removing the technical details. (December 2022) (Learn how and when to remove this message)
|
Inmathematics, the supernatural numbers, sometimes called generalized natural numbersorSteinitz numbers, are a generalization of the natural numbers. They were used by Ernst Steinitz[1]: 249–251 in 1910 as a part of his work on field theory.
A supernatural number is a formal product:
where runs over all prime numbers, and each
is zero, a natural number or infinity. Sometimes
is used instead of
. If no
and there are only a finite number of non-zero
then we recover the positive integers. Slightly less intuitively, if all
are
, we get zero.[citation needed] Supernatural numbers extend beyond natural numbers by allowing the possibility of infinitely many prime factors, and by allowing any given prime to divide
"infinitely often," by taking that prime's corresponding exponent to be the symbol
.
There is no natural way to add supernatural numbers, but they can be multiplied, with . Similarly, the notion of divisibility extends to the supernaturals with
if
for all
. The notion of the least common multiple and greatest common divisor can also be generalized for supernatural numbers, by defining
and
With these definitions, the gcd or lcm of infinitely many natural numbers (or supernatural numbers) is a supernatural number.
We can also extend the usual -adic order functions to supernatural numbers by defining
for each
.
Supernatural numbers are used to define orders and indices of profinite groups and subgroups, in which case many of the theorems from finite group theory carry over exactly. They are used to encode the algebraic extensions of a finite field.[2]
Supernatural numbers also arise in the classification of uniformly hyperfinite algebras.
Number systems
| |
---|---|
Sets of definable numbers |
|
Composition algebras |
|
Split types |
|
Other hypercomplex |
|
Infinities and infinitesimals |
|
Other types |
|
|
![]() | This mathematical logic-related article is a stub. You can help Wikipedia by expanding it. |