The spatial metaphor is the number line. Positive operations and numbers move you to the right. Negative operations and numbers move you to the left. So a neg

The spatial metaphor is the number line. Positive operations and numbers move you to the right. Negative operations and numbers move you to the left. So a negative number is simply a count of a change in direction. This works very literally. Ten miles west followed by five miles east is five miles west. There's nothing mysterious or weird about the "absence of westness" when you've turned around and started off in the opposite direction. It generalises neatly to complex numbers where i is a rotation in the complex plane - instead of being pointlessly-weird-for-the-sake-of-it: "the square root of -1 which we've spent years telling you can't exist, and now we're telling you it can." Technically this is a form of basis vector. Beyond that things get complicated. But the idea is still valid - you define your direction markers (even if they're functions instead of constants) and then you can work out where you are in the space you're exploring, and what "movement", "counting" and "position" mean in that space. Multiplying by zero is always a null movement of no distance.