Editing Number (section)

===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>