1. Section 7.8 Complex Numbers  The imaginary number i  Simplifying square roots of negative numbers  Complex Numbers, and their Form  The Arithmetic of Complex Numbers  Complex Conjugates  Division of Complex Numbers  Powers of i 7.81

2. Quadratic Equations with Non-Real Solutions Try to solve the equation : No real solutions – but perhaps we can extend our number system beyond real numbers … The Complex Number System also contains all Real Numbers: 7.82

3. Imaginary Numbers and the Complex Number System  The Number i i is the unique number for which  An Imaginary Number can only be written in form a + bi where a and b are real numbers, b≠0 3 + i 8i 0.3 – 2i 3 – πi etc  A Complex Number can be a Real Number or an Imaginary Number (both a and b can be 0) 7.83

4. Recall the Properties of Radicals These properties are used to write square roots with negative radicands in terms of i 7.84

5. Addition and Subtraction of Complex Numbers 7.85

6. Multiplying Complex Numbers -6 + 4i + 9i – 6i 2 = -6 + 4i + 9i + 6 = 13i 7.86

7. Warning …  When you have the square root of a negative number involved in any multiplication, always convert to i Form then multiply.  Without converting:  Correct way: 7.87

8. Complex Conjugates -2 + 3i the product is always a Real Number 7.88

9. Dividing Complex Numbers To divide complex numbers, we often have to rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator. The answer should be written in the form a +bi. 7.89

10. Examples Perform the operations on the given complex numbers. Write answers in the form a +bi. 7.810

11. Powers of i 7.811

12. Powers of i 7.812

13. Solving Quadratic Equations We have solved quadratic equations for rational number solutions using factoring and the principle of zero products. In future sections, we’ll learn some ways to find solutions that are irrational numbers or complex numbers. If a negative radicand results, we can put it in terms of i For example: 7.813

14. What Next? Quadratic Equations!  Present Section 8.1 Present Section 8.1 7.814