Coherence (signal processing): Difference between revisions

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In [[mathematics]], the '''supernatural numbers''' (sometimes called '''generalized natural numbers''' or '''Steinitz numbers''') are a generalization of the [[natural number]]s. They were used by [[Ernst Steinitz]]<ref>{{cite journal | first=Ernst | last=Steinitz | authorlink=Ernst Steinitz | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002167042 | title=Algebraische Theorie der Körper | journal=Journal für die reine und angewandte Mathematik | year=1910 | pages=167–309 | jfm=41.0445.03 | issn=0075-4102 }}</ref> in 1910 as a part of his work on [[Field theory (mathematics)|field theory]].


A supernatural number <math>\omega</math> is a [[Formal calculation|formal]] [[Product (mathematics)|product]]:


: <math>\omega = \prod_p p^{n_p},</math>
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where <math>p</math> runs over all [[prime number]]s, and each <math>n_p</math> is zero, a natural number or [[infinity]]. Sometimes we write <math>v_p(\omega)</math> for <math>n_p</math>. If no <math>n_p = \infty</math> and there are only a finite number of non-zero <math>n_p</math> then we recover the positive integers. Slightly less intuitively, if all <math>n_p</math> are <math>\infty</math>, 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 <math>\omega</math> "infinitely often," by taking that prime's corresponding exponent to be the symbol <math>\infty</math>.
 
There is no natural way to add supernatural numbers, but they can be multiplied, with <math>\prod_p p^{n_p}\cdot\prod_p p^{m_p}=\prod_p p^{n_p+m_p}</math>. Similarly, the notion of divisibility extends to the supernaturals with <math>\omega_1\mid\omega_2</math> if <math>v_p(\omega_1)\leq v_p(\omega_2)</math> for all <math>p</math>. The notion of the [[least common multiple]] and [[greatest common divisor]] can also be generalized for supernatural numbers, by defining
 
: <math>\displaystyle \operatorname{lcm}(\{\omega_i\}) \displaystyle =\prod_p p^{\sup(v_p(\omega_i))}</math>
 
: <math>\displaystyle \operatorname{gcd}(\{\omega_i\}) \displaystyle =\prod_p p^{\inf(v_p(\omega_i))}</math>
 
With these definitions, the gcd or lcm of infinitely many natural numbers (or supernatural numbers) is supernatural number.
We can also extend the usual [[p-adic number|<math>p</math>-adic]] order functions to supernatural numbers by defining <math>v_p(\omega)=n_p</math> for each <math>p</math>
 
Supernatural numbers are used to define orders and indices of [[profinite group]]s and subgroups, in which case many of the theorems from [[finite group theory]] carry over exactly. They are also used implicitly in many [[Number theory|number-theoretical]] proofs, such as the density of the [[square-free integer]]s and bounds for odd [[perfect number]]s.
 
==References==
{{reflist}}
* {{cite book | last=Efrat | first=Ido | title=Valuations, orderings, and Milnor ''K''-theory | series=Mathematical Surveys and Monographs | volume=124 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=2006 | isbn=0-8218-4041-X | zbl=1103.12002 | page=125 }}
 
==External links==
* [http://planetmath.org/supernaturalnumber Planet Math: Supernatural number]
 
{{Number Systems}}
 
[[Category:Number theory]]
 
 
{{mathlogic-stub}}

Latest revision as of 22:02, 30 November 2014


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