by tsimionescu | 2026-02-11 20:21:06+00:00
> And in defining the operation of multiplication, you essentially define a way to move around the 2 dimensional set now. And moving around 2 dimensions is exactly the same thing as rotation+scaling.

Again, this is not how complex numbers were defined. The only original goal was to come up with a number that can solve the equation x^2 + 1 = 0, so that (R + {this number}, + , *), becomes an algebraically closed field. Once you've set this goal, there is really a single simple choice for how the operations will work, because everything else is already constrained. If x^2 + 1 = 0, we already know that:

(a + bx) (c + dx) = [polynomial multiplication]
= ac + (bc + ad)x + (bd)x^2 =
= ac + (bc + ad)x + bd(x^2 + 1 - 1) =
= ac + (bc + ad)x + bd(x^2 + 1) - bd = [x^2 + 1 = 0 by definition]
= ac + (bc + ad)x + bd * 0 - bd =
= (ac - bd) + x(bc + ad).

So the formula for complex number multiplication comes out of the arithmetic of real numbers, extended with this extra entity defined simply by being a root of x^2 + 1. The fact that this operation happens to represent a rotation in the RxR plane is "an accident" (I'm sure there are deep ties that make this necessary, probably related to the structure of polynomials themselves).

And while you can define other algebraically closed fields that include the reals as a subfield, the complex numbers are the simplest such set. R^n for n>2 is clearly more complex, for example. So there is a clear reason to prefer sqrt(-1) = i, and thus ending up with a 2d vector space.

> What Im trying to say is that all the definitions sort of assume that if you have some operation that you can do on real numbers, and have a result, if you just plug in a complex value, it all works out, so people think that i behaves like a number. I personally don't think that this is the case, specifically about i behaving like a number just because those results work out.

Again, complex multiplication is not some intentional construction, it is baked into how we defined the complex numbers the first time we did. Again, our goal was to find (well, define) the solution(s) of the equation x^2 + 1 = 0. The fact that we can plug in this x to any formula that involves other numbers just falls out of this goal, it's not an additional assumption. In the case of e(x), this is simpler to see with the power series formula:

e(nx) = sum(1 + nx + (nx)^2/2! + (nx)^3/3! + (nx)^4/4! + ...)
but, by definition, x^2 + 1 = 0, so x^2 = -1, so the formula becomes:
e(nx) = sum(1 + nx - n^2/2! - xn^3/3! + n^4/4! + ...) =
= sum(1 - n^2/2! + n^4/4! ...) + sum(xn - xn^3/3! + xn^5/5! ...) =
= sum(1 - n^2/2! + n^4/4! ...) + x sum(n - n^3/3! + n^5/5! ...) =
= cos n + x sin n

Note that this falls out of the properties of e(x), cos(x), and sin(x) for real numbers, and the single property of i that it is a solution of x^2 + 1 = 0.

I also think that the definition that e(x) is "take e x times in multiplication and then compute the limit" is any more intuitive. I certainly don't think that `e^2 = e(2) = lim (1 + 2/n) ^ n, with n-> infinity` is any more intuitive than the definition of `e(2i) = lim (1 + 2i/n)^n, with n -> infinity`.

> which is somewhat pointless because like I mentioned - the value of e (i.e 2.7) is never really used to compute anything in regards to complex numbers in polar form of re^ix, all you care about is r and the x which is the angle

This is also not really true, because the value of e is deeply tied to the values of cos x and sin x. This also becomes visible if you want to compute 2^(ix). 2^(ix) = e ^ ix (log_e 2) = cos (x log_e 2) + i sin (x log_e 2), using log_e to denote the natural logarithm to make it clearer that it is related to e. So the value of e itself is still there in the formula, even if we discount the relationship between e and cos and sin.