i is not a "trick" or a conceit to shortcut certain calculations like, say, the small angle approximation. i is a number and this must be true because of the fundamental theorem of algebra. Disbelieving in the imaginary numbers is no different from disbelieving in negative numbers.

"Imaginary" is an unfortunate name which gives makes this misunderstanding intuitive.

https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJ...

However, what's true about what you and GP have suggested is that both i and -1 are used as units. Writing -10 or 10i is similar to writing 10kg (more clearly, 10 × i, 10 × -1, 10 × 1kg). Units are not normally numbers, but they are for certain dimensionless quantities like % (1/100) or moles (6.02214076 × 10^23) and i and -1. That is another wrinkle which is genuinely confusing.

https://en.wikipedia.org/wiki/Dimensionless_quantity

Yes I agree, I was just arguing against "i=0+1*i so it's by definition a complex number" which is a self referential definition

If you take this tack, then 0 and 1 are not numbers either.

i is not a real number, is not an integer, is not a rational etc.

You need a base to define complex numbers, in that new space i=0+1*i and you could call that a complex number

0 and 1 help define integers, without {Empty, Something} (or empty, set of the empty, or whatever else base axioms you are using) there is no integers

The simple fact you wanted to write this:

i=0+1*i

Makes i a number. Since * is a binary operator in your space, i needs to be a number for 1*i to make any sense.

Similarly, if = is to be a binary relation in your space, i needs to be a number for i={anything} to make sense.

Comparing i with a unary operator like - shows the difference:

i*i=-1 makes perfect sense

-*-=???? does not make sense

i is a complex number, complex numbers are of the form real + i*real... Don't you see the recursive definition ? Same with 0 and 1 they are not numbers until you can actually define numbers, using 0 and 1

  i*i=-1 makes perfect sense

This is one definition of i. Or you could geometrically say i is the orthogonal unit vector in the (real,real) plane where you define multiplication as multiplying length and adding angles

There's no issue with recursive definitions. That's how arithmetic was original formalized by Peano's axioms [1].

[1] https://en.wikipedia.org/wiki/Peano_axioms

i is also a quaternion. So by this logic we could say complex numbers are made up of quaternions. But we don’t say such things because they wouldn’t be a good mental model of what we want to talk about.

> i is also a quaternion

Yeah, so ?

The fraction 1/2 is a rational and also a real and also a complex numbers and also a quaternion also an octonion ....

We use a number along with the minimal abstraction that is sufficient for our purpose.

i=0+1*i =0+1*(0+1*(0+1*(0+1*(0+1*(0+1*( ...*