1. Imaginary and Complex Numbers

2. The imaginary number i is the square root of -1: Example: Evaluate i 2 Imaginary Numbers

3. Evaluating a Negative Square Root “Factor” out a -1 Rewrite Simplify Calculate the value of the expression below:

4. Complex Number A number consisting of a real and imaginary part. Usually written in the following form (where a and b are real numbers): Example: Solve 0 = 2x 2 – 2x + 10 a =b =c = 1-2 10

5. Complex Conjugates For any complex number: The Complex Conjugate is: The sum and product of complex conjugates are always real numbers Example: Find the sum and product of 5 – 3i and its complex conjugate.

6. Generalizing Powers of i PatternExamples After 4, it starts to repeat itself! Evaluate the examples by first finding a pattern for raising i to a power: Divide the exponent by 4 and find the remainder. The remainder is where it falls in the pattern: Divide by 4 Remainder

7. Graphing in the Complex Plane Real Axis Imaginary Axis Plot: 6 – 4i 6 – 4 What is the distance from 0+0i? 6 4

8. Complex Absolute Value The distance from the complex number to 0 in the complex plane: Example: Evaluate a =b = -211