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.
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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:

Revision as of 12:31, 14 February 2007

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 $\{a + bi + cj + dk\}$, where $a, b, c, d$ are any real numbers and the behavior of $i, j, k$ is "as you would expect," with the properties:

  • $i^2 = j^2 = k^2 = ijk = -1$
  • $ij = k = -ji$, $jk = i = -kj$ and $ki = j = -ik$


Note in particular that multiplication of quaternions is not commutative. However, multiplication on certain subsets does behave well: the set $\{a + bi + 0j + 0k \mid a, b \in \mathbb{R}\}$ act exactly like the complex numbers.


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