Quaternions

$\mathbb{H}$, the quaternions — which IS $\mathrm{Cl}(0,2)$: two anticommuting generators squaring to $-1$, with $k = f_1f_2$ as the third unit.

WikipediaQuaternion↗MathWorldQuaternion↗WikidataQ173853↗Encyclopedia of MathematicsQuaternion↗nLabquaternion↗Britannicatopic/quaternion↗

Quaternions$\mathbb{H}$

\mathbb{H}the same algebra, written as a set

Typed signatures

Type	Package	Applies
untyped	hypercomplex	always

Domain: Hypercomplex algebra

Details

• $i = f_1$, $j = f_2$, $k = f_1f_2$, and $i^2 = j^2 = k^2 = ijk = -1$
• Multiplication is NOT commutative: juxtaposition keeps the order, but an explicit $\times$ does not -- use NonCommutativeMultiply (or $\otimes$) there
• Usable as a set: $f_1f_2 \in \mathbb{H}$ is true, $f_3 \in \mathbb{H}$ is false
• The other named algebras are BicomplexNumbers, TricomplexNumbers, SplitComplexNumbers and DualNumbers

Examples

Basic #

$ijk = -1$

live ↗

In$\mathrm{NonCommutativeMultiply}(f_1, f_2, f_1f_2)$Out$-1$

Properties #

$k \in \mathbb{H}$

live ↗

In$f_1f_2∈\mathrm{Quaternions}$Out$\mathrm{True}$

a multicomplex unit is not a quaternion

live ↗

In$i_1∈\mathrm{Quaternions}$Out$\mathrm{False}$

About

Examples	3
Checked against	Wolfram Language 1 differ
Crosswalk	WikipediaMathWorldWikidataEncyclopedia of MathematicsnLabBritannica

See also: Basis, NonCommutativeMultiply, AlgebraSignature