Supernatural number
In mathematics, the supernatural numbers, sometimes called generalized natural numbers or Steinitz numbers, are a generalization of the natural numbers. They were used by Ernst Steinitz 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 either zero, a natural number or infinity. This can be thought of as the prime factorization of the supernatural number. In the special case where only finitely many values are nonzero and no index is, we get the positive natural numbers; for example, can be represented as
There are also many supernatural numbers with no natural counterpart, such as
and
.
Sometimes is used instead of. can be seen as the p-adic valuation of, and it agrees with the standard p-adic valuation on the natural numbers.
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 is a supernatural number.
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.
Supernatural numbers also arise in the classification of uniformly hyperfinite algebras.