The basic imaginary number i is the number whose square equals -1. Since it's impossible to multiply two identical real numbers together to produce a negative number, some mathematician just pulled the idea of imaginary numbers out of thin air, called it i, and proceeded to describe their attributes based on what is known about real numbers.

If you picture the imaginary number line as being perpendicular to the real number line, then powers of i go in a circle -- something unique in mathematics, or at least in the part of it I've studied. How is this possible? Take a look at the chart below, remembering that i * i = -1 by definition:

• i^1 = i
• i^2 = i * i = -1
• i^3 = i * i * i = -1 * i = -i
• i^4 = i * i * i * i = -1 * -1 = 1
• i^5 = i * i * i * i * i = 1 * i = i
• i^6 = i * i * i * i * i * i = 1 * -1 = -1
You get the idea. It keeps looping, and raising i to a power of a multiple of 4 always yields a result of 1.

To represent an imaginary number other than i itself, treat i as any ordinary variable by sticking a numerical coefficient in front of it to represent how big an imaginary quantity you've got. For example, 3i is the square root of -9. Don't confuse this with something like 3 + i, though -- that's a complex number, not on the real or imaginary number lines but on the same plane as both.