Section 1-5:  Complex Numbers 
Recall from Algebra II that square roots of negative numbers are not real numbers.  The uses of these types of numbers has become quite common.  Therefore, we have the following basic definition of an imaginary unit!
The powers of i form a cycle.  They repeat the same pattern as above.  Thus, i5 = i, i6 = -1 etc.  To make it easy to find higher powers of i, simply divide the power by 4 and the remainder tells you which of the above to use.  Example:  i43  Divide 43 by 4 and get a remainder of 3 thus it is equivalent to i3 = -i!!

Study the sample problems below for the basic arithmetic with imaginary numbers!
The above examples illustrate that to work with imaginary numbers, separate the imaginary part from the real part and simplify the real part.

Complex numbers
Any number in the form a + bi is an imaginary number.  The a is the real part and b is the imaginary part.
Adding complex numbers is easy.  simply combine like terms!!
(5 + 3i) + (4 + 6i) = 9 + 9i
(3 + 2i) - (4 - 3i) = -1 + 5i (remember to switch signs when subtracting)

Multiplying is also fairly easy.  Since complex numbers are binomials, to multiply use the foil method.
(2 + 3i)(1 + 2i) = 2 + 4i + 3i + 6i2 = (2 - 6) + (4i + 3i) = -4 + 7i
remember i2 = -1

Recall that a + bi and a - bi are complex conjugates.  Their sum and product will always be a real number!!  You can use this property to rationalize the division of two complex numbers by multiplying top and bottom by the complex conjugate of the denominator!!  Look at the example below:

That's our simple review of the complex numbers.  This topic should be easily remembered from last year!
On to quadratics!