Definition:Field Norm of Complex Number
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Definition
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
- Field Norm of Complex Number is Positive Definite
- Field Norm of Complex Number is Multiplicative Function
- Field Norm of Complex Number is not Norm
- Results about the field norm of a complex number can be found here.
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $9$: Rings: Exercise $19$