1. Unit 2 – Quadratic, Polynomial, and Radical Equations and Inequalities Chapter 5 – Quadratic Functions and Inequalities 5.4 – Complex Numbers

2. In this section we will learn how to: Find square roots and perform operations with pure imaginary numbers Perform operations with complex numbers

3. 5.4 – Complex Numbers Square root – the square root of a number n is a number with a square of n. Ex. 7 is a square root of 49 because 7 2 = 49 Since (-7) 2 = 49, -7 is also a square root

4. 5.4 – Complex Numbers Product and Quotient Properties of Square Roots For nonnegative real numbers a and b, √ab = √a  √b Ex. √3  2 = √3  √2 √a/b = √a / √b Ex. √1/4 = √1 / √4

5. 5.4 – Complex Numbers Simplified square root expressions DO NOT have radicals in the denominator. Any number remaining under the square root has no perfect square factor other than 1.

6. 5.4 – Complex Numbers Example 1 Simplify √18 √10/81

7. 5.4 – Complex Numbers Simplify the expression: 1. 2. 3.4.

8. 5.4 – Complex Numbers Simplify the expression: 5.6. 7.8.

9. 5.4 – Complex Numbers Simplify the expression: 9.10. 11.12.

10. 5.4 – Complex Numbers Homework: 5.4 (Part 1) Worksheet

11. 5.4 – Complex Numbers Example 2 Simplify √-9 IMAGINARY NUMBER!!

12. 5.4 – Complex Numbers Imaginary number – created so that square roots of negative numbers can be found Imaginary unit – i i = √-1 i 2 = –1 i 3 = – i i 4 = 1 Pure imaginary number – square roots of negative real numbers Ex. 3i, -5i, and i√2 For any positive real number b, √-b 2 = √b 2  √-1 or bi

13. 5.4 – Complex Numbers Example 3 Simplify √-28 √-32y 4

14. 5.4 – Complex Numbers Example 4 Simplify -3i  2i √-12  √-2 i 35

15. 5.4 – Complex Numbers You can solve some quadratic equations by using the square root property Square Root Property For any real number n, if x 2 = n, then x = ±√n

16. 5.4 – Complex Numbers Example 5 Solve 5y 2 + 20 = 0.

17. 5.4 – Complex Numbers HOMEWORK Page 264 #22 – 29 (all) #42 – 45 (all)

18. 5.4 – Complex Numbers Complex number – any number that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit. a is called the real part, and b is called the imaginary part Ex. 5 + 2i and 2 – 6i = 2 + (-6)i If b = 0, the complex number is a real number If b ≠ 0, the complex number is imaginary If a = 0, the complex number is a pure imaginary number

19. 5.4 – Complex Numbers Two complex numbers are equal if and only if (IFF) their real parts are equal AND their imaginary parts are equal. a + bi = c + di IFF a = c and b = d

20. 5.4 – Complex Numbers Example 6 Find the values of x and y that make the equation 2x + yi = -14 – 3i true.

21. 5.4 – Complex Numbers To add or subtract complex numbers, combine like terms. Combine the real parts Combine the imaginary parts

22. 5.4 – Complex Numbers Example 7 Simplify (3 + 5i) + (2 – 4i) (4 – 6i) – (3 – 7i)

23. 5.4 – Complex Numbers You can also multiply 2 complex numbers using the FOIL method

24. 5.4 – Complex Numbers Multiplying Imaginary and Complex Numbers: be sure to simplify (remember i 2 = -1) Ex: 1. (3i)(-2i)2. 4i (11 – 9i) 3. (8 + 5i)(3 + 10i)4. (-1 + 2i)(5 – 11i)

25. 5.4 – Complex Numbers Complex Conjugates: The real number part stays the same; the imaginary part changes signs Ex: (3 + 2i) and (3 – 2i) (-1 – 7i) and (-1 + 7i) (4 – i) and (4 + i)

26. 5.4 – Complex Numbers Multiply the following complex conjugates: 5. (5 – 2i)(5 + 2i) 6. (-4 + 2i)(-4 – 2i)

27. 5.4 – Complex Numbers HOMEWORK Page 264 #30 – 39 (all)

28. 5.4 – Complex Numbers Complex conjugates – two complex numbers of the form a + bi and a – bi. The product of complex conjugates is always a real number. You can use this to simplify the quotient of two complex numbers.

29. 5.4 – Complex Numbers Dividing Complex Numbers: Can’t have i in the denominator. Need to multiply the top and bottom by the complex conjugate! Ex: 1. 2.

30. 5.4 – Complex Numbers 3. 4.

31. 5.4 – Complex Numbers 5. 6.

32. 5.4 – Complex Numbers HOMEWORK Page 264 #40, 41, 46 – 49 (all) #54 – 65 (all)