1. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall

2. Chapter 8 Rational Exponents, Radicals, and Complex Numbers

3. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 8.7 Complex Numbers

4. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Previously, when we encountered square roots of negative numbers in solving equations, we would say “no real solution” or “not a real number”. Imaginary Unit The imaginary unit, written i, is the number whose square is – 1. That is,

5. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Write with i notation. a. b. c.

6. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Multiply or divide as indicated. a. b. c.

7. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Complex Numbers A complex number is a number that can be written in the form a + bi, where a and b are real numbers.

8. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Recall that the graph of a real number is a point on a real number line. In the same manner, the graph of a complex number is a point in the complex plane. The horizontal axis is the real axis. The vertical axis is the imaginary axis.

9. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Graph the complex numbers in the complex plane. a. 2 + 3ib.  5 + 4i c.  3id. 5 Solution a. To graph 2 + 3i, think of graphing (2, 3). b. (  5, 4) c. (0,  3) d. (5, 0)

10. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Sum or Difference of Complex Numbers If a + bi and c + di are complex numbers, then their sum is (a + bi) + (c + di) = (a + c) + (b + d)i Their difference is (a + bi) – (c + di) = (a – c) + (b – d)i

11. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Add or subtract the complex numbers. Write the sum or difference in the form a + bi. a. (4 + 6i) + (3 – 2i)b. (8 + 2i) – (4i) Solution a. (4 + 6i) + (3 – 2i) = b. (8 + 2i) – (4i) = (4 + 3) + (6 – 2)i = 7 + 4i (8 – 0) + (2 – 4)i = 8 – 2i

12. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall The technique for multiplying complex numbers varies depending on whether the numbers are written as single term (either the real or imaginary component is missing) or two terms.

13. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Multiply the complex numbers. Write the product in the form a + bi. a.  4i  6i b. 5i(8 – 4i)c. (6 – i)(2 + i) Solution a.  4i  6i b. c. = –24i 2 = –24(–1) =24 (6 – i)(2 + i) F O I L = 12 + 6i – 2i – i 2 = 12 + 4i – (1)(–1) = 12 + 4i + 1 = 13 + 4i 5i(8 – 4i) = 5i(8) – 5i(4i) = 40i – 20i 2 = 40i – 20(–1) = 40i + 20

14. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Complex Conjugates The complex numbers (a + bi) and (a – bi) are called complex conjugates of each other, and (a + bi)(a – bi) = a 2 + b 2.

15. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall The conjugate of a + bi is a – bi. The conjugate of a – bi is a + bi. The product of (a + bi) and (a – bi) is (a + bi)(a – bi) a 2 – abi + abi – b 2 i 2 a 2 – b 2 (–1) a 2 + b 2, which is a real number.

16. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Divide. Write in the form a + bi. Solution

17. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Divide. Write in the form a + bi. Solution

18. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall The powers recycle through each multiple of 4. Patterns of i

19. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Find the following powers of i. a.i 53 b. i  17 Solution a. b.