Quaternion Multplication is not Commutative
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Theorem
The operation of multplication on the quaternions $H$ is not commutative.
Proof
By definition of multplication:
| \(\ds \mathbf i \times \mathbf j\) | \(=\) | \(\ds \mathbf k\) | ||||||||||||
| \(\ds \mathbf j \times \mathbf i\) | \(=\) | \(\ds -\mathbf k\) |
$\blacksquare$
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem