Calculating the norm of a complex number
In this quick post I'll dispel a common confusion in the basic math of complex numbers. It's often useful to calculate the norm-square (also known as absolute square) of a complex number z. This norm-square is denoted |z|^2. One could naively expect that:
Where z^* is the complex conjugate of z. But why? To understand the mechanics of this calculation, let's start by defining what a norm is.
Informally, a norm is a generalization of the concept of "length" or "size". For a real number r, the norm is its absolute value |r|. No matter if the number is positive or negative, the norm is its "length" - the distance from the origin.
A norm is defined similarly for complex numbers. Here's a standard geometrical interpretation of a complex number z, showing both cartesian and polar coordinates:
The "norm" of z is the length of the blue line, or the distance of its endpoint from the origin (in the polar representationz=re^{i\theta}, it's exactly r). The norm of a complex number uses the same notation as the absolute value: |z|, because it means the same thing. Another common name for the norm of complex numbers is modulus.
