Understanding Complex Numbers in Mathematics

1. Section 4.8 – Complex Numbers Students will be able to: To identify, graph, and perform operations with complex numbers To find complex number solutions of quadratic equations Lesson Vocabulary: imaginary unit imaginary number complex number pure imaginary number complex number plane abs val of comp # complex conjugates

2. Section 4.8 – Complex Numbers Essential Understanding: The complex numbers are based on a number whose square is -1. The imaginary unit “i” is the complex number whose square is -1. So i2 = -1, and i =

3. Section 4.8 – Complex Numbers

4. Section 4.8 – Complex Numbers Problem 1: How do you write by using the imaginary unit i?

5. Section 4.8 – Complex Numbers Problem 1b: How do you write by using the imaginary unit i?

6. Section 4.8 – Complex Numbers Problem 1c: How do you write by using the imaginary unit i?

7. Section 4.8 – Complex Numbers Problem 1e: Explain why ?

8. Section 4.8 – Complex Numbers An imaginary number is any number of the form “a + bi”, where a and b are real number and b cannot equal 0. Imaginary numbers and real numbers together make up the set of complex numbers.

9. Section 4.8 – Complex Numbers

10. Section 4.8 – Complex Numbers In the complex number plane, the point (a, b) represents the complex number a + bi. To graph a complex number, locate the real part on the horizontal axis and the imaginary part on the vertical axis.

11. Section 4.8 – Complex Numbers The absolute value of a complex number is its distance from the origin in the complex plane.

12. Section 4.8 – Complex Numbers Problem 2: What are the graph and absolute value of each number? a. -5 + 3i b. 6i

13. Section 4.8 – Complex Numbers Problem 2b: What are the graph and absolute value of each number? a. 5 – i b. 1 + 4i

14. Section 4.8 – Complex Numbers To add or subtract complex numbers, combine the real parts and the imaginary parts separately. If the sum of two complex numbers is 0, or 0 + 0i, then each number is the opposite, or additive inverse, of the other. The associative and commutative properties apply to complex numbers as well.

15. Section 4.8 – Complex Numbers Problem 3: What is each sum or difference? (4 – 3i) + (-4 + 3i) (5 – 3i) – (-2 + 4i) (7 – 2i) + (-3 + i)

16. Section 4.8 – Complex Numbers You multiply complex numbers a + bi and c + di as you would multiply binomials. Problem 4: What is each product? (3i)(-5 + 2i) (4 + 3i)(-1 – 2i) (-6 + i)(-6 – i)

17. Section 4.8 – Complex Numbers You multiply complex numbers a + bi and c + di as you would multiply binomials. Problem 4b: What is each product? (3i)(7i) (2 – 3i)(4 + 5i) (-4 + 5i)(-4 – 5i)

18. Section 4.8 – Complex Numbers Number pairs of the form a + bi and a – bi are complex conjugates. The product of complex conjugates is a real number. (a + bi)(a – bi) = You can use complex conjugates to simplify quotients of complex numbers.

19. Section 4.8 – Complex Numbers Problem 5: What is each quotient? a. b.

20. Section 4.8 – Complex Numbers Problem 5: What is each quotient? a. b.

21. Section 4.8 – Complex Numbers Problem 5: What is the quotient?

22. Section 4.8 – Complex Numbers Problem 6: What is the factored form of 2x2 + 32? What is the factored form of 5x2 + 20?

23. Section 4.8 – Complex Numbers Problem 7: What are the solutions of 2x2 – 3x +5 = 0? What are the solutions of 3x2 – x + 2 = 0?

24. Section 4.8 – Complex Numbers

25. Section 4.8 – Complex Numbers