What is a complex number?

Complex numbers are a set of ordered pairs (a,b) where a and b are real numbers, such that addition and multiplication are defined as above. Now "real" numbers are all those numbers which are positive, negative, or zero.

What is an imaginary number?

Imaginary numbers is a complex number whose squared value is a real number not greater than zero. Funny thing is they are not imaginary. They are a rotation.

Each time it rotates 90 degrees you multiply by i.

Imagine it as a clock which is helpful for simplifying expression involving "i":

You don't always have to rotate by 90 degrees it's possible to rotate by 45° too. It's also possible to use one complex number and one imaginary number. (a+bi) If you put it on a graph you would be at a 45° angle.

For example:
  • If you're going 3 units East, 4 units North = 3 + 4i
  • Rotate counter-clockwise by 45 degrees = multiply by 1 + i

(3+4i) * (1+i) = ?

Distribute and solve this, that would be the new coordinates.

3 + 4i + 3i + 4i2 = 3 – 4 + 7i= -1 +7i

Laws of Exponents Vs. Using the iClock.

When you have i^22 or 84 you don't have all the time in the world to get your iClock and go around the circle 22 or 84 times. That is where the laws of exponents comes in. It's an easier way to get the same answer in just a quicker time.

For example:

Instead of going around the clock 12 times you could, i^12 = (i^4)^3.