1. Sec Math II Performing Operations with Complex Numbers
2.2

2. Vocabulary −1 = ?
What Number multiplied by itself equals – 1? 𝑥 2 = - 1 ( 1 ) ( 1 ) = 1 ( -1 ) ( - 1 ) = 1 THERE IS NO NUMBER ! Imaginary number “i” − 1 = i 𝑖 2 = ? 𝑖 2 = ( − 1 ) ² = - 1

3. Think of the “complex numbers” as the universe of numbers
Think of the “complex numbers” as the universe of numbers. Complex Numbers
Real Numbers
Imaginary Numbers

4. Complex Numbers i −1 12 2 e Real # ‘s Imaginary # ‘s Rational # ‘s
Whole #’s 0,1,2,3,……… −1
Integers .. -2,-1, 0, 1, 2,3 ……..
Rational #’s ( a/b) (1/2), (2/3) 5,(1/3)…. i e
…..
Irrational # ‘s
Real # ‘s
Imaginary # ‘s

5. What does it mean? i = −1
2 (x) = 2 x 3 (i)= 3 i ( 3i ) ( 4i ) = 3 ∙ i ∙ 4 ∙ i = 3 ∙ 4 ∙ i ∙ i = 12 i ² i² = ( −𝟏 ) ² = i² = 12 ( -1) = - 12

6. Simplify the Square Roots
−5 = −1 5 = 𝑖 5

7. Your turn
Simplify the following square roots
1. −27
−20
3. −25

8. Vocabulary
A Complex Number: either a real number, and imaginary number, or a combination of the 2.
a + bi i
Standard Form Imaginary Number
Complex conjugates: two complex numbers of the form
a + bi a – bi
2 + 3i – 3i

9. Adding and Subtracting Complex Numbers
Real numbers and complex numbers are not “like term.” ( 5 + 5i ) + ( 4 + 6i) 5 + 5i i Get rid of the parenthesis i + 6i Combine like terms i Treat it just like any other variable when you add them.

10. Your Turn:
Simplify: (add/subtract) 13. (6i) + (7i) 13a. ( 2 + 3i ) + ( 5i + 6) 13b. (7i) + ( 5 + 2i) 14. (4i) – (8i) 14a. (6i) – ( -2 – 7i) 14b. ( 6i + 4 ) – ( i)

11. Multiplying Complex Numbers
1 - ( 3i) ( 4i ) = 3 * i * 4 * i = 3 * 4 * i * i = 12i² = 12 ( - 1 ) = - 12
2 - ( 3 + 2i ) ( 1 – 3i ) = ?? FOIL
i + 2i – 6i²
3 – 7i – 6 ( -1)
3 – 7i + 6
9 – 7i

12. Your turn:
i ( 6i)
i ( 4 + 3i )
( 3 + 4i ) ( 8 + 9i)

13. Division of complex numbers

14. Your turn:
𝟓 𝟔𝒊 𝟐+𝟑𝒊 𝟔𝒊
𝟓 −𝒊 𝟐𝒊

15. Conjugates: Your turn Simplify ( multiply the conjugates)
( x – 2 ) ( x + 2 )
( 𝟑 ) ( 𝟑 )
What happens with you multiply the two conjugates together?

16. Your turn: Multiply the conjugates together. ( 3 + i ) ( 3 – i )

17. Same thing with complex numbers
𝟏 𝟐 −𝟓𝒊 imaginary number not allowed 𝟏 𝟐 −𝟓𝒊 * 𝟐+𝟓𝒊 𝟐+𝟓𝒊 = 𝟐+𝟓𝒊 𝟐² −𝟐𝟓𝒊² = 𝟐+𝟓𝒊 𝟒+𝟐𝟓 = 𝟐+𝟓𝒊 𝟐𝟗 ( 𝟐 𝟐𝟗 + 𝟓𝒊 𝟐𝟗 ) = 𝟐 𝟐𝟏 + 𝟓𝒊 𝟐𝟏

18. Your turn:
25. 𝟑 𝟒 −𝟓𝒊 26. 𝟐 𝟑+𝒊 27. 𝟑 𝟐 −𝒊

19. More problems ( 3 + 4i ) ( 5 – 2i ) 29. 30. 𝟒 𝒊
30. 𝟒 𝒊
31. Solve : y = 8 ( x + 3) ² + 16
1 −2𝑖 2+3𝑖

20. The End