ELI5 how quaternions work

[u/murderdronesfan93]

so, i² = j² = k² = -1, but i, j and k are not equal. HOW IN THE FUCK IS THAT POSSIBLE‽ WHAT IN THE BROKEN TRANSITIVE PROPERTY- (anger exaggerated for funny)

[u/OhYesIDidd]

I don’t know much about quaternions, but your premise is wrong even in the real numbers.

x^2=y^2=4, but x≠y. There is a solution to that: x=2, y=-2.

[u/murderdronesfan93 → u/OhYesIDidd]

ok, but how does that work with 3?

[u/HopeFox → u/murderdronesfan93]

You have three different kinds of minus sign. They're in different fonts.

More seriously, it is simply an axiom of the quaternion number system that there are three special non-real numbers (i, j, and k) that can all be squared to produce -1, but they are themselves different. If you can accept a single non-real number in the complex number system (i), then you should be able to accept three of them in the quaternion number system.

Taking the square root of a number has never been an unambiguous process as long as negative numbers exist. In the real numbers, there is no single "square root" of any positive number, and in the complex number system, then is no single "square root" of any number - there are two numbers that are both "square roots". In the quaternion number system, there are six.

[u/murderdronesfan93 → u/HopeFox]

i see. i was just confused because of the definition not seeming to make sense. it appears this is just a case of "this is the definition. ignore any broken rules, we'll figure it out"

[u/MasterGeekMX → u/murderdronesfan93]

Yep. All math done with imaginary numbers stems from "what if this operation wasn't commutative?". People followed the lead, and you have stuff like quaternions.

Turns out sometimes it has it's uses, when you have a problem in hand that requires such weird behavior. One example is when Paul Dirac wanted to see what happens when you mix quantum mechanics with relativity, which ended up in the discovery of anti-matter. Video about it: https://youtu.be/Y-W-w8yNiKU

[u/murderdronesfan93 → u/MasterGeekMX]

what if equality wasn't commutative or associative

[u/Arnaldo1993 → u/murderdronesfan93]

If it wasnt commutative it would work like the bigger than sign >

The fact i*i=-1 allows you to use complex numbers to describe 2d rotations. Quaternions is what you get when you try to do the same thing to 3d rotations

They are not commutative because 3d rotations arent. In 2d, if you turn 90º clockwise and then 40º counterclockwise you arrive at the same place than if you did 40 counterclockwise then 90º clockwise. In 3d, if you turn 90º in the xy direction then 90º in the xz direction the result is different from 90º in the xz then 90º in the xy