Transcript:

Here's the thing about complex numbers: numbers that are A+iB or X+iY. These are two-dimensional numbers. You cannot sort them. You cannot put them in ascending or descending order. No such thing can be done because complex numbers cannot be ordered. Pretty interesting! They can be compared, but they can be compared only to know if they are equal or not. Like if you take the magnitude of a complex number, then you can compare the magnitude and say, "Okay, they are equal." But if they are not equal, you don't know which one is greater than which one. You cannot sort them—that's one thing.

The second thing about complex numbers is, you can have dimension two complex numbers like X+iY, but you cannot have dimension three complex numbers X+iY+jZ, for example. You cannot have a three-dimensional complex number. And the reason is very complex. Well, the reason is complex, but it essentially boils down to three-dimensional complex numbers cannot be guaranteed to work with division. It is not possible to guarantee that you can divide any two three-dimensional complex numbers, but it will work for four dimensions. When you extend it to four dimensions, it's guaranteed to work. Brilliant!