Use of algebra

They are good for modelling rotations.
1+x^2 = 0 has no real solutions.
You could invent a number called i that solves it.
Or you could look at real 2x2 matrices and get a real solution. The matrices that solve it correspond to 90 degree rotations.

Gee guys, we have [math]\mathbb{R}^2[/math] where we very intuitively define
>(a,b) + (x,y) := (a+x,b+y)
Hmm... What if... No wait, listen to me guys, what if we also define a kind of multiplicative operation as follows so we can have a nifty field
>(a,b)(x,y) := (ax - by,ay + bx)

Jesus Christ how horrifying. Nothing worse than that could ever happen. Hold me

-2^2 = -4, brainlets don't know order of operations

I'm not saying its bad, I'm sayings it is not better than the complex numbers.

The product in the category of rings, implies RxR=R^2 is not a field.

Products are unique, so this new multiplication you want to define on R^2 is unnatural from a categorical perspective.
The whole purpose for the complex numbers existing, is to solve the equation x^2 +1 = 0.

So to construct such an object, it makes sense to look at the ring R[x]/.

The canonical structure on this ring, i.e. the quotient ring structure induced by the natural structure of R[x], is a field structure.

Not saying that you're wrong, or that abstract algebra is useless, but damn I hate you fucking algebraists so god damn much.

We were all having such a good time and then you just had to come in with your stupid algebra fucking shit. GET OUT

You can't rocket jump very effectively with them though, you'll need quake numbers for that.

When you say "natural" you really mean "naive".

whether it technically is R^2 or not, it doesn't change the fact that it's just a 2-dimensional vector with some fancy multiplication operator defined over it.

there's nothing "imaginary" or weird about it. just some weird notation that sometimes hides what's really going on or surprises people when "i" pops up seemingly out of nowhere

>then you are fucking wrong and the math is flawed.
Nope. Imaginary numbers are just a quirk of mathematics. Existing within the system, not outside of it, yet performing as well as any number from the outside. The fact that they absolutely work in accordance with mathematical logic is fascinating and because of this, they can be used to extrapolate truths applicable to the real world despite not being real themselves. It's as simple as that. This is basically the age old Rationalism vs. Empiricism debate. The fact is, if a tangible, real world truth can be derived using a purely abstract, logical tool (imaginary numbers), then they themselves must be true as well.

>It's just brainlets that insist on using the retarded a + ib form instead of (a,b).
lmao but it literally is a + bi
quantum mechanics