I don't think a vague but more precise mathematical explanation of the terms zero and pole are even that difficult to understand, x has a zero, 1/x has a pole, people kind of know what that means if you look at a graph of a pole, I don't think a rigorous definition of pole is that far off - a pole of f is just a zero of 1/f.

Instead we get waffle like:

> Again roughly speaking, zeros describe mathematically how a system reacts to some input in the short term, while poles describe how a system reacts in the long term.

I know it's "roughly" speaking, but isn't it too rough?