Complex Numbers for Rotating, Translating, and Scaling the PlaneJune 7th, 2009 Patrick Stein

A good friend of mine recently discovered some of the fun things you can do with complex numbers if you’re using them to represent points in the plane. Yesterday, I re-read a passage by Tony Smith about why one should be interested in Clifford algebras. Tony Smith’s passage included all of the fun one can have with the complex plane and extends it to three, four, five, and more dimensions. I thought, I should segue from the complex numbers in the plane to Clifford algebras to quaternions in 3-space to Clifford algebras again in a series of posts here.

What are Complex Numbers

Say you’re playing around with polynomials. You start playing with the equation $z^2 - 1 = 0$. WIth a little fiddling, you find this is equivalent to $z^2 = 1$. Then, you take the square root of both sides to find that $z = \pm \sqrt{1} = \pm 1$. We started with a polynomial equation in one variable in which the highest exponent was two and we found two answers.

Pounding your chest and sounding your barbaric yawp, you move on to $z^2 + 1 = 0$. This should be easy, right? With the same fiddling, we find $z^2 = -1$ and then $z = \pm \sqrt{-1}$.

Uh-oh. What do we do now? We can’t think of any number that when multiplied by itself gives us a negative number. If we start with zero, we end with zero. If we multiply a positive number by itself, we get a positive number. If we multiply a negative number by itself, we get a positive number. Again!