> The key idea of imaginary numbers is to consider multiplication -1 to be a rotation by half a turn rather than a reflection. That is a lot easier to do half o

> The key idea of imaginary numbers is to consider multiplication -1 to be a rotation by half a turn rather than a reflection. That is a lot easier to do half of! As soon as you have multiplication by -1 as a rotation by half a turn, it is obvious to identify i as rotation by a quarter turn. It gets even more interesting when you add additional degrees of freedom so that such rotations can happen via more than one path. For example, quaternions add two extra degrees of freedom, and this lets you have an infinite number of square roots of -1. Any imaginary unit quaternion (i.e. ip+jq+kr where p^2+q^2+r^2=1) is a square root of -1.