The real numbers can be used to represent a 1-dimensional space, the complex numbers can be used to represent a 2-dimensional space, but a 3-dimensional space cannot be represented by a single number system, though a 4-dimensional space can be represented by quaternions. I read that this is because an m-dimensional space can be represented by a single number system if and only if there exists a non-negative integer n such that 2^n = m. For instance, 1 = 2^0, 2 = 2^1, 4 = 2^2, etc.
There seems to be a contradiction though. In a 3D space, you can move east or west, north or south, and up or down. A 2D space is a plane where you can only move east or west, and north or south. A 1D space is a line where you can only move east or west. By logical extension, A 0D space is a dot where you can't move at all. And a 0D space could easily be represented by the set that only contains the number 0. This set is closed under addition(0 + 0 = 0), multiplication(0*0 = 0), and has all the field axioms. However, there is no non-negative integer n such that 2^n = 0. So what is the pattern? Is 0D just an exception to this rule?
Thanks in advance!