Supernatural number
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the supernatural numbers, sometimes called generalized natural numbers or Steinitz numbers, are a generalization of the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s. They were used by
Ernst Steinitz Ernst Steinitz (13 June 1871 – 29 September 1928) was a German mathematician. Biography Steinitz was born in Laurahütte (Siemianowice Śląskie), Silesia, Germany (now in Poland), the son of Sigismund Steinitz, a Jewish coal merchant, and ...
in 1910 as a part of his work on field theory. A supernatural number \omega is a
formal Formal, formality, informal or informality imply the complying with, or not complying with, some set of requirements (forms, in Ancient Greek). They may refer to: Dress code and events * Formal wear, attire for formal events * Semi-formal attire ...
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
: : \omega = \prod_p p^, where p runs over all
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s, and each n_p is zero, a natural number or
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
. Sometimes v_p(\omega) is used instead of n_p. If no n_p = \infty and there are only a finite number of non-zero n_p then we recover the positive integers. Slightly less intuitively, if all n_p are \infty, we get zero. Supernatural numbers extend beyond natural numbers by allowing the possibility of infinitely many prime factors, and by allowing any given prime to divide \omega "infinitely often," by taking that prime's corresponding exponent to be the symbol \infty. There is no natural way to add supernatural numbers, but they can be multiplied, with \prod_p p^\cdot\prod_p p^=\prod_p p^. Similarly, the notion of divisibility extends to the supernaturals with \omega_1\mid\omega_2 if v_p(\omega_1)\leq v_p(\omega_2) for all p. The notion of the
least common multiple In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by bot ...
and
greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
can also be generalized for supernatural numbers, by defining : \displaystyle \operatorname(\) \displaystyle =\prod_p p^ and : \displaystyle \operatorname(\) \displaystyle =\prod_p p^. 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 p-adic order functions to supernatural numbers by defining v_p(\omega)=n_p for each p. Supernatural numbers are used to define orders and indices of
profinite group In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. ...
s and subgroups, in which case many of the theorems from
finite group theory Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...
carry over exactly. They are used to encode the algebraic extensions of a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
.Brawley & Schnibben (1989) pp.25-26 Supernatural numbers also arise in the classification of uniformly hyperfinite algebras.


See also

*
Profinite integer In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat) :\widehat = \varprojlim \mathbb/n\mathbb = \prod_p \mathbb_p where :\varprojlim \mathbb/n\mathbb indicates the profinite completion of \mathbb ...


References

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External links


Planet Math: Supernatural number
Number theory Infinity {{mathlogic-stub