Absolutely agree.

What's your preferred terminology? I like "Forward" and "Reverse" for positive and negative number and "Lateral" for the imaginary unit, since it is just perpendicular to the Forward and Reverse numbers.

Do you have a suggestion for an alternative to "real number"?

In quantum physics complex numbers are used to represent circular symmetry, so they are more like modulus and phase.

> They are terrible choices of name, coined by people who didn't really understand them, and didn't want you to understand them either.

Sorry to be blunt, but your comment is horrible. There is far more going on here than you realize. Real numbers have an isomorphism with our reality. We thought we didn't need imaginary numbers to describe reality. They were inteded to be used more as a mathematical tool.

Another good example is euclidean vs non-euclidean geometry. They are both incredibly useful, but the former one was though to be "real" and the latter "imaginary". But it turns, physics is okay with us living in a non-euclidean universe (https://en.wikipedia.org/wiki/Shape_of_the_universe).

Mathematicians have largely abandoned the term "imaginary number" and use "complex number" instead. I never used the term "imaginary number" when I was teaching (except to point out that it wasn't very good terminology).

But the old term still survives in some areas. For instance, the real and imaginary parts of a complex number:

  re (3 + 4 i) = 3  and  im (3 + 4 i) = 4.

I think about it this way: the complex-ness of the number is really just a representational trick. What it means, intuitively, depends on what the question is really asking and why the behaviors of math on complex numbers is well-suited to representing the answer, and that reason might change based on the context.

You can (almost). You can do the same thing again to get quaternions, and again to get octonions, and so on.

https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_constru...

That's different from extending 1d to 2d, 3d, etc as vectors.

Representing complex numbers as 2-dim. real vectors is just one way of constructing complex numbers. It is convenient because you can draw 2-dim. vectors, and it makes complex numbers feel more tangible.

Something like the original motivation (which you might find less "arbitrary") goes like this. We know how to solve some quadratic equations like x^2 = 9: We get x = 3 and x = -3 (though it was a nontrivial thing to realize those negative solutions are there). What about x^2 = 2? We can "see" the solution should be the length of the hypotenuse of a right triangle whose legs are both 1, so maybe encouraged by the geometry we don't feel too bad about having solutions (+/- sqrt(2)) which are, however, not rational numbers.

(Notice that we keep "extending" our idea of "number". Well, if we didn't have feature creep no one would buy the new version.)

Since we now have negative numbers, we ask about solutions to x^2 = -1. The first response is there are "no solutions", because there are no real numbers that square to -1. Rather than give up, we decide to "extend" the real numbers to provide solutions to this equation.

The idea is actually pretty easy if you know some modular arithmetic. For instance, mod 6 arithmetic means that you take the integers and consider things that differ by multiples of 6 to be "the same". As a consequence, 17 = 5, 4 = -14, and (importantly) 6 = 0.

Thinking in the same way, to get solutions to x^2 = -1, we want x^2 + 1 = 0. That's like "6 = 0", so "x^2 + 1" should be our "modulus", but what "number system" do we "mod out of" (instead of the integers, with the integers mod 6)? Since x^2 + 1 is a real polynomial, it looks like our "base number system" should be R[x], the (ring of) polynomials with real coefficients (stuff like -17 x^3 + 1.374 x^2 - 31/42 and so on).

So we should take real polynomials, but agree that two polynomials differing by a multiple of x^2 + 1 are "the same". We conclude that the complex numbers "should be"

  R[x]  mod  <x^2 + 1>

where the "<x^2 + 1>" means "all (polynomial) multiples of x^2 + 1".

For instance,

  x^3 + 4 x + 1 = (x^3 + x) + (3 x + 1) = x(x^2 + 1) + (3 x + 1)
    = 3 x + 1 (mod x^2 + 1)  (the multiple of x^2 + 1 is 0)

(To agree that some things should be regarded as "the same" is to construct an equivalence relation, and the "bags" of "same things" are equivalence classes. I'm omitting the details from abstract algebra, but this construction is called "forming the quotient ring R[x]/<x^2 + 1>".)

By this construction, "3 + 4 i" appears as

  3 + 4 x (mod x^2 + 1)  or  (3 + 4 x) + <x^2 + 1>.

In the construction of the complex numbers via 2-dim. real vectors, "3 + 4 i" appears as (3, 4).

You can also construct the complex numbers as a subring of the 2 x 2 real matrices, so "3 + 4 i" appears as

  3 [1 0]  +  4 [0  1]
    [0 1]       [-1 0] .

Notice that

  [0  1]^2 = [-1 0] = - [1 0]   (like i^2 = -1)
  [-1 0]     [0 -1]     [0 1]

So those are three ways of constructing the complex numbers, and each has pros and cons. No one of them "is" the complex numbers. "The" complex numbers are really an isomorphism class of rings, and we're just giving different representatives from that class.

There are 2d values that aren't imaginary numbers.

And real numbers are ∞-d, over rationals.