1. Complex Numbers 10.7 1.Write imaginary numbers using i. 2.Perform arithmetic operations with complex numbers. 3.Raise i to powers.

3. Imaginary unit: Imaginary number: A number that can be expressed in the form bi, where b is a real number and i is the imaginary unit.

5. Complex number: A number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. Examples: real imaginary

6. Complex Numbers: a + bi b = 0: Real numbers a = 0: Imaginary numbers realimaginary

7. Add Complex Numbers Add the real parts – add the imaginary parts

8. Subtract Complex Numbers

9. Slide 10- 9 Copyright © 2011 Pearson Education, Inc. Simplify. (  4 + 7i) – (2 + i) a) 2 + 7i 2 b) 2 + 8i c)  6 + 6i d)  6 + 8i

10. Slide 10- 10 Copyright © 2011 Pearson Education, Inc. Simplify. (  4 + 7i) – (2 + i) a) 2 + 7i 2 b) 2 + 8i c)  6 + 6i d)  6 + 8i

11. Multiply Complex Numbers

12. standard a + bi form

13. Multiply Complex Numbers

14. Rewrite & Foil

15. Slide 10- 15 Copyright © 2011 Pearson Education, Inc. Multiply. (  4 + 7i)(2 + i) a)  15 + 10i b) 1 + 10i c)  15 + 18i d) 15 + 18i

16. Slide 10- 16 Copyright © 2011 Pearson Education, Inc. Multiply. (  4 + 7i)(2 + i) a)  15 + 10i b) 1 + 10i c)  15 + 18i d) 15 + 18i

17. Divide Complex Numbers

18. Binomial denominator conjugate Divide Complex Numbers standard a + bi form

19. Slide 10- 19 Copyright © 2011 Pearson Education, Inc. Write in standard form. a) b) c) d)

20. Slide 10- 20 Copyright © 2011 Pearson Education, Inc. Write in standard form. a) b) c) d)

21. Powers of i:

23. Simplify: