Editing Number (section)

==Extensions of the concept==

===''p''-adic numbers===
{{main|p-adic number|l1=''p''-adic number}}
The ''p''-adic numbers may have infinitely long expansions to the left of the decimal point, in the same way that real numbers may have infinitely long expansions to the right. The number system that results depends on what [[radix|base]] is used for the digits: any base is possible, but a [[prime number]] base provides the best mathematical properties. The set of the ''p''-adic numbers contains the rational numbers, but is not contained in the complex numbers.

The elements of an [[algebraic function field]] over a [[finite field]] and algebraic numbers have many similar properties (see [[Function field analogy]]). Therefore, they are often regarded as numbers by number theorists. The ''p''-adic numbers play an important role in this analogy.

===Hypercomplex numbers===
{{main|hypercomplex number}}
Some number systems that are not included in the complex numbers may be constructed from the real numbers <math>\mathbb{R}</math> in a way that generalize the construction of the complex numbers. They are sometimes called [[hypercomplex number]]s. They include the [[quaternion]]s <math>\mathbb{H}</math>, introduced by Sir [[William Rowan Hamilton]], in which multiplication is not [[commutative]], the [[octonion]]s <math>\mathbb{O}</math>, in which multiplication is not [[associative]] in addition to not being commutative, and the [[sedenion]]s <math>\mathbb{S}</math>, in which multiplication is not [[Alternative algebra|alternative]], neither associative nor commutative. The hypercomplex numbers include one real unit together with <math>2^n-1</math> imaginary units, for which ''n'' is a non-negative integer. For example, quaternions can generally represented using the form

<math display=block>a + b\,\mathbf i + c\,\mathbf j +d\,\mathbf k,</math>

where the coefficients {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|d}} are real numbers, and {{math|'''i''', '''j'''}}, {{math|'''k'''}} are 3 different imaginary units.

Each hypercomplex number system is a [[subset]] of the next hypercomplex number system of double dimensions obtained via the [[Cayley–Dickson construction]]. For example, the 4-dimensional quaternions <math>\mathbb{H}</math> are a subset of the 8-dimensional quaternions <math>\mathbb{O}</math>, which are in turn a subset of the 16-dimensional sedenions <math>\mathbb{S}</math>, in turn a subset of the 32-dimensional [[trigintaduonion]]s <math>\mathbb{T}</math>, and ''[[ad infinitum]]'' with <math>2^n</math> dimensions, with ''n'' being any non-negative integer. Including the complex and real numbers and their subsets, this can be expressed symbolically as:
:<math>\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S} \subset \mathbb{T} \subset \cdots</math>

Alternatively, starting from the real numbers <math>\mathbb{R}</math>, which have zero complex units, this can be expressed as

:<math>\mathcal C_0 \subset \mathcal C_1 \subset \mathcal C_2 \subset \mathcal C_3 \subset \mathcal C_4 \subset \mathcal C_5 \subset \cdots \subset C_n</math>

with <math>C_n</math> containing <math>2^n</math> dimensions.<ref name="Saniga">{{cite journal | last1=Saniga | first1=Metod | last2=Holweck | first2=Frédéric | last3=Pracna | first3=Petr | title=From Cayley-Dickson Algebras to Combinatorial Grassmannians | journal=Mathematics | publisher=MDPI AG | volume=3 | issue=4 | date=2015 | issn=2227-7390 | arxiv=1405.6888 | doi=10.3390/math3041192 | doi-access=free | pages=1192–1221}}</ref>

===Transfinite numbers===
{{main|transfinite number}}
For dealing with infinite [[set (mathematics)|sets]], the natural numbers have been generalized to the [[ordinal number]]s and to the [[cardinal number]]s. The former gives the ordering of the set, while the latter gives its size. For finite sets, both ordinal and cardinal numbers are identified with the natural numbers. In the infinite case, many ordinal numbers correspond to the same cardinal number.

===Nonstandard numbers===
[[Hyperreal number]]s are used in [[non-standard analysis]]. The hyperreals, or nonstandard reals (usually denoted as *'''R'''), denote an [[ordered field]] that is a proper [[Field extension|extension]] of the ordered field of [[real number]]s '''R''' and satisfies the [[transfer principle]]. This principle allows true [[first-order logic|first-order]] statements about '''R''' to be reinterpreted as true first-order statements about *'''R'''.

[[Superreal number|Superreal]] and [[surreal number]]s extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form [[field (mathematics)|fields]].