The video sounded too complicated. My own “proof” is imagining that we already have a number system with independent units 1, i, and j, and asking how much would be i×j. Plausible answers are 1, −1, i, -i, j, or -j, and each of them quickly results in a problem. For example, if i×j=1, then j is actually 1/i, which already exists as a complex number, so it is not an independent third dimension. But if i×j=i, then j=1. Etc.
To put j outside the plane defined by 1 and i, the result of i×j must be yet another dimension, let’s call it k… and we have reinvented quaternions.
The video sounded too complicated. My own “proof” is imagining that we already have a number system with independent units 1, i, and j, and asking how much would be i×j. Plausible answers are 1, −1, i, -i, j, or -j, and each of them quickly results in a problem. For example, if i×j=1, then j is actually 1/i, which already exists as a complex number, so it is not an independent third dimension. But if i×j=i, then j=1. Etc.
To put j outside the plane defined by 1 and i, the result of i×j must be yet another dimension, let’s call it k… and we have reinvented quaternions.
I like this one better! It’s a more direct appeal to geometric intuition, which is the only area of math where I have any intuition at all.