Editing Quaternion (section)

== Algebraic properties ==
[[File:Cayley graph Q8.svg|right|thumb|[[Cayley graph]] of {{math|Q<sub>8</sub>}}. The red arrows represent multiplication on the right by {{math|'''i'''}}, and the green arrows represent multiplication on the right by {{math|'''j'''}}.]]

The set <math>\mathbb H</math> of all quaternions is a vector space over the real numbers with [[dimension (vector space)|dimension]]&nbsp;4.{{efn|In comparison, the real numbers <math>\mathbb R</math> have dimension&nbsp;1, the complex numbers <math>\mathbb C</math> have dimension&nbsp;2, and the octonions <math>\mathbb O</math> have dimension&nbsp;8.}} Multiplication of quaternions is associative and distributes over vector addition, but with the exception of the scalar subset, it is not commutative. Therefore, the quaternions <math>\mathbb H</math> are a non-commutative, associative algebra over the real numbers. Even though <math>\mathbb H</math> contains copies of the complex numbers, it is not an associative algebra over the complex numbers.

Because it is possible to divide quaternions, they form a division algebra. This is a structure similar to a field except for the non-commutativity of multiplication. Finite-dimensional associative division algebras over the real numbers are very rare. The [[Frobenius theorem (real division algebras)|Frobenius theorem]] states that there are exactly three: <math>\mathbb R</math>, <math>\mathbb C</math>, and <math>\mathbb H</math>. The norm makes the quaternions into a [[composition algebra|normed algebra]], and normed division algebras over the real numbers are also very rare: [[Hurwitz's theorem (composition algebras)|Hurwitz's theorem]] says that there are only four: <math>\mathbb R</math>, <math>\mathbb C</math>, <math>\mathbb H</math>, and <math>\mathbb O</math> (the octonions). The quaternions are also an example of a [[composition algebra]] and of a unital [[Banach algebra]].

[[File:Quaternion-multiplication-cayley-3d-with-legend.png|thumb|Three-dimensional graph of Q<sub>8</sub>. Red, green and blue arrows represent multiplication by {{math|'''i'''}}, {{math|'''j'''}}, and {{math|'''k'''}}, respectively. Multiplication by negative numbers is omitted for clarity.]]

Because the product of any two basis vectors is plus or minus another basis vector, the set {{math|{{mset|±1, ±'''i''', ±'''j''', ±'''k'''}}}} forms a group under multiplication. This non-[[abelian group]] is called the quaternion group and is denoted {{math|Q<sub>8</sub>}}.<ref>{{cite web |title=quaternion group |website=Wolframalpha.com |url=http://www.wolframalpha.com/input/?i=quaternion+group}}</ref> The real [[group ring]] of {{math|Q<sub>8</sub>}} is a ring <math>\mathbb R[\mathrm Q_8]</math> which is also an eight-dimensional vector space over <math>\mathbb R.</math> It has one basis vector for each element of <math>\mathrm Q_8.</math> The quaternions are isomorphic to the [[quotient ring]] of <math>\mathbb R[\mathrm Q_8]</math> by the [[ideal (ring theory)|ideal]] generated by the elements {{math|1 + (−1)}}, {{math|'''i''' + (−'''i''')}}, {{math|'''j''' + (−'''j''')}}, and {{math|'''k''' + (−'''k''')}}. Here the first term in each of the differences is one of the basis elements {{math|1, '''i''', '''j'''}}, and {{math|'''k'''}}, and the second term is one of basis elements {{math|−1, −'''i''', −'''j'''}}, and {{math|−'''k'''}}, not the additive inverses of {{math|1, '''i''', '''j'''}}, and {{math|'''k'''}}.