If real, irrational, rational, whole and complex numbers are the only numbers you’re familiar with...boy have I got news for you. Some special kinds of numbers, like quaternions and octonions, describe more than our three dimensions. Just let that sink in a little bit. Can you even picture the kind of math involved for a reality that exists in more than 3D? How would we even think about those shapes? Let’s take it back to like, middle school for a sec.
If you want to travel in one dimension, you add and subtract real numbers. Here you are, the dot, on your little line, and you want to move 7 units to the right? Great, that’s plus seven. Let’s take it into two dimensions. To make moves like this you can use what’s called an imaginary number These can be combined with real numbers to make what are called complex numbers. Real numbers and complex numbers are the first two of four number systems in pure mathematics that we call the normed division algebras over the real numbers.
We can think of complex numbers as two-dimensional numbers...because they help you move things in 2D. But something kinda funny happens when you use these division algebras to manipulate an object in 3-dimensions.