Alrighty, time for some actual math. First, a little word about who I am. I graduated a little over a year ago from Northwestern University with a B.A. in math. I kind of half-assedly looked for jobs as an actuary, but ended up moving back home to Ohio and began working as a retail clerk, eventually getting promoted to management. Gradually, I realized that maybe the corporate world of insurance wasn’t for me, and I decided to go back and get my PhD. So, if all goes according to plan, I’ll be doing that in the fall of 2005. Until then, it’s just me, my old math books, and this blog.

So I whipped out my algebra book from winter and spring quarters my senior year and started just doing problems the other day. The first section deals primarily with complex numbers, and it got me to thinking. When we graph complex numbers, we visualize them in 2-dimensional space. A complex number ceases to really be the sum a+bi, but rather the vector (a,b). However, any homework problems in either high school or college dealing with the multiplication of complex numbers didn’t deal with graphing the products on the Cartesian plane. So I did a little messing around. The product of the two complex numbers a+bi and c+di is ac-bd+i(ad+bc). Converting this to our little system of using vectors in 2-space, we have (ac-bd,ad+bc). And what does this look like geometrically? It’s actually quite lovely. First, we’ll take a look at the length of the vector. sqrt((ac-bd)^2^2) ends up simplifying to sqrt((a^2+b^2)(c^2+d^2)), which happens to be exactly the product of the lengths of our two original vectors (the ones we multiplied together). Imagine that. Okay, how about the angle? This is a little bit trickier, as you need to use an obscure arctangent identity I can’t remember ever learning, but fortunately found at this site after a quick Google search. arctan x arctan y = arctan ((x+y)/(1-xy)). It turns out that the angle between our product and the x-axis is simply the sum of the angles between our original two vectors and the x-axis. Fantastic!

So what does all this mean? Throughout my college coursework, the only products of two vectors that ever had any application were the dot and cross products. I guess this “complex product” serves mostly to just say, “Hey look! Multiplication of complex numbers makes a lot of sense geometrically!”