Editing Multiplication table (section)

==In abstract algebra==
Tables can also define binary operations on [[group (mathematics)|group]]s, [[field  (mathematics)|field]]s, [[ring (mathematics)|ring]]s, and other [[Abstract algebra|algebraic systems]]. In such contexts they are called [[Cayley table]]s.

For every natural number ''n'', addition and multiplication in '''Z<sub>''n''</sub>''', the ring of integers modulo ''n'', is described by an ''n'' by ''n'' table. (See [[Modular arithmetic]].) For example, the tables for '''Z<sub>5</sub>''' are:

{{col-begin|width=auto}}
{{col-break|gap=2em}}
{| class="wikitable" style="text-align: center; width:12em"
|-
! +
!0
!1
!2
!3
!4
|-
!0
| 0 || 1 || 2 || 3 || 4
|-
!1
| 1 || 2 || 3 || 4 || 0
|-
!2
| 2 || 3 || 4 || 0 || 1
|-
!3
| 3 || 4 || 0 || 1 || 2
|-
!4
| 4 || 0 || 1 || 2 || 3
|}
{{col-break|gap=2em}}
{| class="wikitable" style="text-align: center; width:12em"
|-
! ×
!0
!1
!2
!3
!4
|-
!0
| 0 || 0 || 0 || 0 || 0
|-
!1
| 0 || 1 || 2 || 3 || 4
|-
!2
| 0 || 2 || 4 || 1 || 3
|-
!3
| 0 || 3 || 1 || 4 || 2
|-
!4
| 0 || 4 || 3 || 2 || 1
|}
{{col-end}}

For other examples, see [[group (mathematics)|group]].

===Hypercomplex numbers===
[[Hypercomplex number]] multiplication tables show the non-[[commutative]] results of multiplying two hypercomplex imaginary units. The simplest example is that of the [[quaternion]] multiplication table.

:{|class="wikitable"
|+Quaternion multiplication table
|-
!width=15 nowrap|↓ × →
!width=15|{{math|1}}
!width=15|{{math|'''i'''}}
!width=15|{{math|'''j'''}}
!width=15|{{math|'''k'''}}
|-
!{{math|1}}
|{{math|1}}
|{{math|'''i'''}}
|{{math|'''j'''}}
|{{math|'''k'''}}
|-
!{{math|'''i'''}}
|{{math|'''i'''}}
|{{math|−1}}
|{{math|'''k'''}}
|{{math|−'''j'''}}
|-
!{{math|'''j'''}}
|{{math|'''j'''}}
|{{math|−'''k'''}}
|{{math|−1}}
|{{math|'''i'''}}
|-
!{{math|'''k'''}}
|{{math|'''k'''}}
|{{math|'''j'''}}
|{{math|−'''i'''}}
|{{math|−1}}
|}

For further examples, see {{section link|Octonion|Multiplication}}, {{section link|Sedenion|Multiplication}}, and {{section link|Trigintaduonion|Multiplication}}.