# Definition:Field Norm of Quaternion

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## Definition

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

## 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