1. 7.7 Complex Numbers

2. Imaginary Numbers Previously, when we encountered square roots of negative numbers in solving equations, we would say “no real solution” or “not a real number”. That’s still true. However, we will now introduce a new set of numbers. Imaginary numbers which includes the imaginary unit i.

3. Imaginary Numbers The imaginary unit, written i, is the number whose square is ‒ 1. That is,

4. Write using i notation. a. b. c. Examples

5. Multiply or divide as indicated. a. b. Example

6. A complex number is a number that can be written in the form a + bi where a and b are real numbers. a is a real number and bi would be an imaginary number. If b = 0, a + bi is a real number. If a = 0, a + bi is an imaginary number. Standard Form of Complex Numbers

7. Adding and Subtracting Complex Numbers

8. Add or subtract as indicated. a. b. (8 + 2i) – (4i) (4 + 6i) + (3 – 2i) Example

9. To multiply two complex numbers of the form a + bi, we multiply as though they were binomials. Then we use the relationship i 2 = – 1 to simplify. Multiplying Complex Numbers

10. Multiply: a. 8i · 7ib. -4i · 7 c. 3i ·3i ·3id. 2i ·3i ·i ·5i Example

11. Multiply. a. 5i(4 – 7i)b. 4i(3i + 5) Example

12. Multiply. (6 – 3i)(7 + 4i) Example

13. In the previous chapter, when trying to rationalize the denominator of a rational expression containing radicals, we used the conjugate of the denominator. Similarly, to divide complex numbers, we need to use the complex conjugate of the number we are dividing by. Complex Conjugate

14. The complex numbers a + bi and a – bi are called complex conjugates of each other. (a + bi)(a – bi) = a 2 + b 2 Complex Conjugate

15. Divide. Example

16. Divide. Example

17. Patterns of i

18. Find each power of i. a.b. c.d. Example