Fractal Theory: Complex Numbers

After you get into more complicated math, you start coming across some equations where the answer is neither a real number nor an imaginary number, but the sum of both -- an answer that looks like this: 1 + 2i; or 7 + 56i. There's no way of simplifying these equations. You can't reduce them so that there's only one term. You can't derive a sum. You just have to write them down as 1 + 2i. These numbers, part real, part imaginary, are called complex numbers.

To add two complex numbers, say 7 + 4i and 3 + 9i, simply add the real and imaginary components separately (that is, 7 + 3 is 10 and 4i + 9i is 13i for a result of 10 + 13i).

What about multiplication? Do you have to learn brand new weird math to do that? Not really. Start with the easy part: 7 * 3 is 21. More complicated is 4i * 9i. Or is it? It's the same as saying 4 * 9 * i * i. 4 * 9 is 36, and you know (because we defined it as such) that i * i is -1. Therefore, 4i * 9i is -36. That leaves 7 * 9i, which is 63i, and 4i * 3, which is 12i. First, add up the real parts (21 + -36, which is -15). Then, add up the imaginary parts (63i + 12i, which is 75i). So, the answer is -15 + 75i. A bit cumbersome and error prone, but not too incomprehensible.

From all this the fractal is born.