You are right, negative numbers are a useful concept which can be mapped to a number of real world operations, but that in some fundamental way not the same thing as positive integers. The same can be said of zero, of course.

This is probably why it took humans ages to come up with such concepts, the jump is bigger than it seems after you think of this as "normal".

On a related note, real numbers are far weirder than most people think....

You can go five miles east instead of west (more of a vector, but still makes sense if your movements are limited to a number line instead of a number plane).

Imaginary numbers are more of the same with a twist. i is just a 𝛑/2 rotation on a plane instead of a negative number, which can be thought of as rotation by 𝛑 on a line.

The big mistake with imaginary numbers is calling them imaginary. There's nothing imaginary about them. They're a very specific kind of operation which can be expanded with very little thought or effort to complex numbers, which have incredibly useful properties in engineering.

Calling them "imaginary" is cripplingly confusing for almost everyone, and many never get over it.

You can also go five miles south instead of west

> You can't have -5 rocks

You can have 5 rocks destroyed in the future. That to me, is what -5 physically means: a guarantee that the object associated with the number will be eliminated out of existence in the future and decrement the negative number by 1.

Some people call it debt, but to me, emotionally that word feels too financial. So I prefer “a guarantee to be eliminated out of existence in the future “, or something like that.

Conversely, 5 cows means: 5 cows currently in existence.

5 cows - 5 cows means: I see 5 cows and now they don’t exist any more and there is nothing.

I wish my math skills were better, I am optimistic that I’d find a similar thing for imaginary numbers and maybe even complex numbers.

With that said, I do get where you’re coming from and I find it a compelling perspective as well. It’s simply that I feel the perspective I described as well.

Exactly, you use it to store some information that has no real quantity but may be converted to a real quantity in the future through some other process.

I'm a polytechnic university student, we use imaginary numbers extensively in all sorts of places, especially whenever there is any oscillatory behaviour, such as an electrical signal or a light wave. A complex number is just a two-dimensional vector with real/imaginary components, whcih provides an amplitude and a phase (angle). An oscillating sinusoidal signal/wave may appear to be zero and completely static if you freeze time at the right moment, but as time progresses, it will continue oscillating, like a swing in a park.

In a way, the magnitude represents the built up "momentum" of the system, whilst the real quantity is the immediate physical value at any given point in time (given by the phase). The amplitude is always the same at any given moment, even when the swing is vertical, it has momentum which will help it reach its maximum height.

Personally, I still think they are just "invented", but I think the vast majority of engineers much prefer them to the alternative, manipulating trigonometric functions (every engineer's nightmare). They're a neat way to represent the exchange of potential and mechanical/electrical energy with a single value and some simplified mathematics (this is an engineer's, not a mathematician's, point of view). Like negative numbers, we could have chosen to have two positive quantities, balance and debt, instead we find use in merging these definitions, whether negative values make sense or not. We have become used to to negative numbers representing the "inverse" action, which makes sense when representing a phyiscal quantity such as velocity.

Note that, while the complex numbers are isomorphic to two-dimensional vectors, they are not used the same way. In particular, there is no equivalent to complex multiplication or division that is normally used with two dimensional vectors (though you could define them of course, they are not normally used).

The difference is also reinforced by the fact that there is no equivalent of the complex numbers for 3-dimensional vectors, or really any other n-dimensional vectors. That is, you can't define a multiplication and addition operation for n-tuples of real numbers for n>2, with the usual semantics of multiplication and addition (associativity, distributivity, inverse, neutral element).

Also thinking about a number line is useful when talking about both negative numbers AND complex ones: negative numbers are to the left of 0, but complex numbers are up and down from the number line.

The best mental model of negative numbers that works for me is to treat it as direction.

In your example, if I have "-5" rocks it means I owe +5 rocks to someone else, let's say John. If, after a few days Rachel were to give me +5 rocks and then you square it off with John you are left with no rocks. Directionally Rachel → (5 rocks) You → (5 rocks) John;

So +/- stand for things flowing into and away from you respectively.

Numbers in laymans terms imply something that can be quantified and compared, which in the end I think leads to a lot of confusion when introducing the term.

That's why I'd prefer to relabel them like "2-d numbers" for instance, to make it clear that some properties are affected like going from points on a line to points on the plane.