Definition:Field Norm of Complex Number

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Let $z = a + i b$ be a complex number, where $a, b \in \R$.

Then the field norm of $z$ is written $\map N z$ and is defined as:

$\map N z := \cmod \alpha^2 = a^2 + b^2$

where $\cmod \alpha$ denotes the complex modulus of $\alpha$.

Also known as

Many sources refer to this concept as the norm of $z$.

However, it is important to note that the field norm of $z$ 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.

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

Also see

  • Results about the field norm of a complex number can be found here.