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.