[phailhaus]

Complex numbers are just a field over 2D vectors, no? When you find "complex solutions to an equation", you're not working with a real equation anymore, you're working in C. I hate when people talk about complex zeroes like they're a "secret solution", because you're literally not talking about the same equation anymore.

There's this lack of rigor where people casually move "between" R and C as if a complex number without an imaginary component suddenly becomes a real number, and it's all because of this terrible "a + bi" notation. It's more like (a, b). You can't ever discard that second component, it's always there.

[petters]

We identify the real number 2 with the rational number 2 with the integer 2 with the natural number 2. It does not seem so strange to also identify the complex number 2 with those.

[phailhaus]

If you say "this function f operates on the integers", you can't turn around and then go "ooh but it has solutions in the rationals!" No it doesn't, it doesn't exist in that space.

[lg5689]

You can't do this for general functions, but it's fine to do in cases where the definition of f naturally embeds into the rationals. For example, a polynomial over Z is also a polynomial over Q or C.

[meindnoch]

Sure, you can say that complex numbers are 2-tuples with a special rule for multiplication (a, b) * (c, d) = (ac - bd, ad + bc). Same way you can define rationals as 2-tuples with a special rule for addition (a, b) + (c, d) = ((ad + cb) / gcd(b, d), bd / gcd(b, d)).

But I think this way you'd lose insight as to where these rules really come from. The rule for complex multiplication is the way it is, precisely because it gives you an algebra that works as if you were manipulating a quantity that squared to -1.

[lg5689]

The movement from R to C can be done rigorously. It gets hand-waved away in more application-oriented math courses, but it's done properly in higher level theoretically-focused courses. Lifting from a smaller field (or other algebraic structure) to a larger one is a very powerful idea because it often reveals more structure that is not visible in the smaller field. Some good examples are using complex eigenvalues to understand real matrices, or using complex analysis to evaluate integrals over R.

[creata]

I hate when people casually move "between" Q and Z as if a rational number with unit denominator suddenly becomes an integer, and it's all because of this terrible "a/b" notation. It's more like (a, b). You can't ever discard that second component, it's always there. ;)

[phailhaus]

Yes, you're right. You can't say your function operates in Z "but has solutions in Q". That's what people are doing when they take a real function and go "ooh look, secret complex solutions!"