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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