A high school math problem and Euler's formula

I was on my high school's math team in the early '70s, and in one contest there was a problem that stumped us. Basically it was this:

Find two numbers, not 0 or 1, such that each is the square of the other.

If you remember that there is such a thing as a complex number, the problem isn't hard. Trouble was, we had all forgotten about such things in the moment.

But what happened afterward was very interesting to me. I noticed that the solution involved complex numbers, and a particular kind of complex number: the kind with a "length" or "magnitude" of 1. Somehow I noticed what happens when you multiply such complex numbers together.

It was so much fun learning all that, and I was thinking about it again for some reason recently. So I wrote a bunch of stuff down. Not in blogger; this requires LaTeX. Some topics mentioned:

• factoring
• quadratic formula
• addition formulas for sin and cos
• matrix multiplication (2x2)
• Maclaurin series (Taylor series centered at 0) This is the only part needing calculus.
• "isomorphic" which just means that this thing is like that thing and behaves in the same way. Probably I misuse the word somewhere.

It sure was fun to learn that stuff, and fun to think about it again and write some of it down.

I've uploaded my little paper here.