Definition:Field Norm of Quaternion

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Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion.

Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$.

The norm of $\mathbf x$ is the real number defined as:

$n \left({\mathbf x}\right) := \left\vert{\mathbf x \overline {\mathbf x} }\right\vert = \left\vert{\overline {\mathbf x} \mathbf x }\right\vert = a^2 + b^2 + c^2 + d^2$

Also known as

Many sources refer to this concept as the norm of $\mathbf x$.

However, it is important to note that the field norm of $\mathbf x$ is not actually a norm as is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ for a general ring or vector space, as it does not satisfy the triangle inequality.

It also needs to be pointed out that not even field norm is a good name, because the quaternions $\mathbb H$ do not even form a field.

This confusing piece of anomalous nomenclature just has to be lived with.

Also see