What the fuck is this bullshit?

It doesn't intuitively make sense right off the bat but it works and gets us correct answers when we use it to find solutions to math problems.

[math]\mathbf R \left[ X \right] \,/\, \left( X^2 \,+\, 1 \right)[/math], or equivalently, [math]\mathrm{Vect}_{\mathbf R} \left[ \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix},\, \begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix} \right][/math].

>>9060697
You forgot about [math] \left( \mathbb{R}^2, +, \times \right) [/math] where

[math] +: [(a,b),(c,d)] \to (a+c,b+d) [/math]
[math] \times: [(a,b),(c,d)] \to (ac - bd,ad + bc) [/math]

>>9060730
I did not forget about it. I intentionally filtered anything that could make anyone puke.

>>9060736
But my version is the one commonly taught in classrooms. If anything, what makes people puke are the ones you posted.

>>9060755
You seem to forget that high school teachers have no taste. The only decent constructions of [math]\mathbf C[/math] are:
* quotient space to fit the historical approach as much as possible while staying rigorous;
* vector plane which trivially inherits its composition laws from [math]\mathscr M_2 \left( \mathbf R \right)[/math], giving you a complete construction in half a page.

>>9060766
>You seem to forget that high school teachers have no taste.

Kek, high school teachers do not teach that. High school teachers teact [math] i^2 = -1 [/math].

I meant that mine is taught in classrooms at the university level. Usually in any introduction to modern algebra.

Pros:
Clearly states the relation between the real numbers and the complex numbers
Naturally shows the geometric structure of the complex numbers by literally making it a field on top of the real plane.
Easy modern example to introduce students to the idea of fields and how to prove a structure forms a field.
Easy modern example to introduce students to algebras.

>>9060674
how's freshman year hs going anon?

>>9060674
It has to do with people pretending that space is time and vice versa.

>>9060697
hahahah take a load of this asshole

Has anyone tried doing the same thing by dividing by zero?

>>9061845
Complicated numbers: [math]j[/math] such that [math]0\, j \,=\, 1[/math]. They form a third linear dimension.
Hard numbers: [math]k[/math] such that [math]2^k \,=\, 0[/math]. They form a circular dimension.
Confusing numbers: [math]\ell[/math] such that [math]\int_{- \infty}^\infty \ell\, \mathrm dx \,=\, \ell[/math]. They form a dimension shaped like the 3D Hilbert curve.

>>9060674
A convention. It was defined that i would be equal to √-1. So by the definition of powers i^2 = -1

>>9061845
Actually, yes. What this guy said >>9061866 for [math]0j[/math]
https://en.wikipedia.org/wiki/Riemann_sphere

>>9061866
So doesn't that mean 0 is a "confusing" number?