Difference between revisions of "Quaternion"
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| − | The '''quaternions''' are a [[division ring]] (that is, a [[ring]] in which each element has a multiplicative [[inverse]]; alternatively, a non[[commutative]] [[field]]) which generalize the [[complex number]]s. | + | The '''quaternions''' are a [[division ring]] (that is, a [[ring]] in which each element has a multiplicative [[inverse with respect to an operation | inverse]]; alternatively, a non[[commutative]] [[field]]) which generalize the [[complex number]]s. |
Formally, the quaternions are the set <math>\{a + bi + cj + dk\}</math>, where <math>a, b, c, d</math> are any [[real number]]s and the behavior of <math>i, j, k</math> is "as you would expect," with the properties: | Formally, the quaternions are the set <math>\{a + bi + cj + dk\}</math>, where <math>a, b, c, d</math> are any [[real number]]s and the behavior of <math>i, j, k</math> is "as you would expect," with the properties: | ||
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| − | Note in particular that multiplication of quaternions is not commutative. However, multiplication on certain [[subset]]s does behave well: the set <math>\{a + bi + 0j + 0k \mid a, b \in \mathbb{R}\}</math> act exactly like the complex | + | Note in particular that multiplication of quaternions is not commutative. However, multiplication on certain [[subset]]s does behave well: the set <math>\{a + bi + 0j + 0k \mid a, b \in \mathbb{R}\}</math> act exactly like the [[complex number]]s. |
| + | ==See Also== | ||
| + | *[[Real number|Real numbers]] | ||
| + | *[[Complex numbers]] | ||
| + | *[[Rational numbers]] | ||
| + | *[[Integers]] | ||
| + | *[[Irrational number]] | ||
| + | *[[Transcendental number]] | ||
{{stub}} | {{stub}} | ||
| + | [[category:Definition]] | ||
Latest revision as of 11:29, 27 September 2024
The quaternions are a division ring (that is, a ring in which each element has a multiplicative inverse; alternatively, a noncommutative field) which generalize the complex numbers.
Formally, the quaternions are the set 
, where 
are any real numbers and the behavior of 
is "as you would expect," with the properties:
, 
and 
, 
and 
Note in particular that multiplication of quaternions is not commutative. However, multiplication on certain subsets does behave well: the set 
act exactly like the complex numbers.
See Also
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