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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.

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 representation, it's exactly r). It doesn't seem to fit the requirement of being a "length" or distance from the origin, which we'd expect to be a real quantity.

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