4.2 What exactly is a negative number?

Let us take a digression and visit what exactly is a negative number. For example, what is -26?

Although we can explain it as “26 below zero”, “26 to the left of zero (on a number line)” or “negative 26”, we can simply call it “the negation of 26”, or -(26). Abstractly, what does (arithmetic) negation do?

In other words, given that x is a number, what exactly is -(x)?

We know some rules must hold true. For example, -(-(x)) = x. We also know that x + (-(x)) = 0.