Quaternions

Quaternions are a non-commutative extension of the complex numbers. They were first described by Sir William Rowan Hamilton in 1843. He famously inscribed their defining equation on Broom Bridge in Dublin when walking with his wife on 16 October 1843. They have many possible applications, including in computer graphics, but have during their history proved comparatively unpopular, with vectors being preferred instead.

Definition & basic operations

The quaternions, $\mathbb {H}$, are a four-dimensional normed division algebra over the real numbers.

$\mathbb {H} =\left\lbrace a+{\mathit {i}}b+{\mathit {j}}c+{\mathit {k}}d|a,b,c,d\in \mathbb {R} \right\rbrace$

${\mathit {i}}^{2}={\mathit {j}}^{2}={\mathit {k}}^{2}={\mathit {ijk}}=-1$