Saw a YouTube video by a guy named Michael Penn about why there is no 3 dimensional equivalent of the complex numbers. It’s going through an abridged version of the mathematical reasons and I was able to follow along until we got a point where he showed that it would have to commute ix with xi, which contradicts an initial required claim that ix does not commute with xi.
This is not satisfying to me intuitively. The thing that bothers me is that I can accept the argument that the definition is incoherent, but that doesn’t show me why we can’t get there using some different claim or procedure. Here’s what I came up with instead:
When we build the complex numbers out of the reals, rather than extending the reals by one dimension, what we are really doing is extending the reals by the size of the reals. So rather than:
reals + one dimension ⇒ complex
We have:
reals + reals ⇒ complex
Carrying this forward, extending the complex numbers to the next level up is applying the same procedure again, so rather than:
complex + one dimension ⇒ ?
We have:
complex + complex ⇒ quaternions
So if we do the thing we did to construct the complex numbers from the reals over again, what we get is the quaternions which have four dimensions rather than three.
I note that this intuition really answers the question of what you get when you extend the complex numbers, rather than the question of why you can’t have something like the complex numbers with three dimensions. For that, I think of the previous problem in reverse: In order to build a three dimensional number system using anything like the same procedure, we would need to have something extended by its own size to get us there:
something + something ⇒ threenions
Since the dimension is supposed to be three, that means we need a base number system of dimension1.5. That’s a fraction; we might be able to do this, since fractional dimensions are how a fractal is made. But the fraction for 1.5 is 3⁄2, which means we need a three dimensional number system in order to construct our three dimensional number system, which is circular and suggests there isn’t a way to construct it (at least not one from the reals or complex numbers).
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.
Saw a YouTube video by a guy named Michael Penn about why there is no 3 dimensional equivalent of the complex numbers. It’s going through an abridged version of the mathematical reasons and I was able to follow along until we got a point where he showed that it would have to commute ix with xi, which contradicts an initial required claim that ix does not commute with xi.
This is not satisfying to me intuitively. The thing that bothers me is that I can accept the argument that the definition is incoherent, but that doesn’t show me why we can’t get there using some different claim or procedure. Here’s what I came up with instead:
When we build the complex numbers out of the reals, rather than extending the reals by one dimension, what we are really doing is extending the reals by the size of the reals. So rather than:
reals + one dimension ⇒ complex
We have:
reals + reals ⇒ complex
Carrying this forward, extending the complex numbers to the next level up is applying the same procedure again, so rather than:
complex + one dimension ⇒ ?
We have:
complex + complex ⇒ quaternions
So if we do the thing we did to construct the complex numbers from the reals over again, what we get is the quaternions which have four dimensions rather than three.
I note that this intuition really answers the question of what you get when you extend the complex numbers, rather than the question of why you can’t have something like the complex numbers with three dimensions. For that, I think of the previous problem in reverse: In order to build a three dimensional number system using anything like the same procedure, we would need to have something extended by its own size to get us there:
something + something ⇒ threenions
Since the dimension is supposed to be three, that means we need a base number system of dimension1.5. That’s a fraction; we might be able to do this, since fractional dimensions are how a fractal is made. But the fraction for 1.5 is 3⁄2, which means we need a three dimensional number system in order to construct our three dimensional number system, which is circular and suggests there isn’t a way to construct it (at least not one from the reals or complex numbers).
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.