Definition:Field Norm of Complex Number

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 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 Equals Field Norm


 * Field Norm of Complex Number is Positive Definite
 * Field Norm of Complex Number is Multiplicative Function
 * Field Norm of Complex Number is not Norm


 * Definition:Complex Modulus


 * Definition:Field Norm
 * Definition:Norm