Imaginary Numbers 
 Imaginary numbers are not real numbers, but they are the product of reals number and i.
 You cannot take the square root of a negative number, because any number squared will be a positive.
 A positive times a positive is a positive.
 A negative times a negative is a positive.
 So, the square root of a positive is real, but the square root of a negative is imaginary.

Simplifying with i 
 Simplifying with i
 Rule: , because:
 is not a real number, so x is replaced by i (imaginary)
 Example: Simplify:
 Step 1: Factor 1 • 4 • 5 (1 = i, 4 is the perfect square, and 5 is the remaining factor)
 Step 2: Bring the i outside of the square root:
 Step 3: Take the square root of 4, which is 2, and bring it out of the square root:
 Step 4: The 5 is not a perfect square, so leave it inside the square root:
 To solve the problem above, follow these steps
 Steps for solving negatives inside square roots
 Step 1: Factor 20 into 1 • any perfect squares • remaining factors
 Step 2: The square root of 1 is i, so pull the i to the outside
 Step 3: Take the square root of any factors that are prefect squares (4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on…)
 Step 4: Leave remaining factors inside the square root

Powers of i 
 Rule: To find really large powers of i, divide by 4 and look at the chart for the remainder
 The remainder of any number divided by 4 will be 1, 2, or 3.
 Use the chart above to see the result of
 Remember: and 1 multiplied by any number is that number.
 Example:
 remainder 2
 because 4(3) = 12, and 12 + remainder 2 = 14
 What about –i raised to a power?

Complex Numbers 

 Adding & Subtracting with Complex Numbers

 Multiplication (FOIL) with Complex Numbers
 How to Multiply Complex Numbers
 Use FOIL to multiply.
 Substitute 1 for .
 Simplify

 FOIL with Imaginary Numbers

Complex Conjugates 
 The complex conjugate of a number a + bi is a – bi.
 Ex. The complex conjugate of 3 – 2i is 3 + 2i.

 Division with Imaginary Numbers Requires Using Complex Conjugates
 Multiply the numerator and denominator by the complex conjugate of the denominator

Quadratics with Imaginary Numbers 
 Quadratics with Imaginary Numbers by Completing the Square
 Step 1: Complete the Square
 Step 2: When you take the square root of both sides, factor out i for 1
 Step 3: Simplify

 Quadratics with Imaginary Numbers Using the Quadratic Equation
 Step 1: Set up the equation in standard form
 Step 2: List the a, b, and c and substitute them into the equation
 Step 3: Solve and simplify
