Phase of a real number Could someone please explain in what case the phase of a real number is equal to -pi (and not pi)?

I know that for positive numbers, the phase is zero. For zero, we define the phase as zero as well. And for negative numbers, the phase would be pi. But I was reading some script and there it says the phase of a real number is either 0, pi, or -pi.
Or $2\pi$, or $3\pi$, or any integer multiple of $\pi$.  Any odd multiple corresponds to -1 + 0i and any even multiple corresponds to 1 + 0i, aka -1 and 1.

"Phase of a real number" is a little bit of a misleading label.  What is required here is an understanding of the complex plane and what "phase" means in terms of a DFT bin value.

Your question is equivalent to "For what values of arg(z) is z a real number?"

If that is meaningless to you, I suggest you start by reading two blog articles of mine:

The Exponential Nature of the Complex Unit Circle

And the newest:

Angle Addition Formulas from Euler's Formula

There are of course many other searches.  Your terms should be "complex plane real values" for a start.

This is essential foundation material for a lot of DSP concepts.