1. Complex Numbers

2. Imaginary unit Complex number a + b Imaginary number a + b
Where a and b are real numbers
Imaginary number a + b
Where b 0
Pure imaginary number
a + b ,if a = 0 and b 0

3. Powers of 1 -1 How to solve: 326
Divide the power of i by 4. The remainder is how many time you go around the clock starting at the top of the clock.
Examples: 1 = -1
326

4. Adding Complex Numbers
Combine the like terms
Add all the i’s together and all the non i’s together
(a + bi) + (c + di)
(5 + 3i) + (4 – 2i) = 9 + I

5. Subtracting Complex Numbers
Distribute the (–)
Combine the like terms
Add all the i’s together and all the non i’s together
(a + bi) – (c + di) (4 + i) – (2 – 2i)
a + bi - c - di i = 2 - i

6. Multiplying Complex Numbers
Find the moon!
(a + bi) (c + di) (2 + i) (1 + i) = 1 + 3i

7. Complex Conjugate (a + bi) and (a – bi)
They undo each other. The purpose is to get rid of the i.
You need this for dividing complex numbers

8. Dividing Complex Numbers
Find the complex conjugate of the denominator
Find the moon to multiply the top and bottom
Simplify, and use
Combine like terms
Write in standard for, a + bi
7 – 4i = (7-4I) (2-i) = – 7i – 8i + 4i
2 + i (2-i) (2-i) – 2i + 2i - i
= – 7i – 8i – 4 = 10 – 15i = 2 – 3i

9. Mini Quiz (Adding and Subtracting)
(3-18i) - (7+24i) =
(5+2i) - (12+10i) =
(15+66i) - (2-12i) =
(8+21i) + (10-11i) =
(-13+47i) + (13-2i) =
(51+52i) + (53+54i) =

10. Mini Quiz (Multiplying and Dividing)
(-3-4i)(5+2i) =
(5+6i)(8-6i) =
(2-8i)(-9-3i) =
(4+10i) / (2-i) =
(6-2i) / (2+2i) =
(24+6i) / (-4+4i) =

11. Mini Quiz Answers (Adding and Subtracting)
i….. 3 – 7 =-4 and -18i – 24i =-42
-7 - 8i…… 5 – 12 =-7 and 2i – 10i =-8i
i… = 13 and 66i + 12i = 78i
i… = 18 and 21i + -11i = 10i
45i………… = 0 and 47i + -2i
i… = 104 and 52i + 54i =106i

12. Mini Quiz Answers (Multiplying)
i…-3(5) =-15, -3(2i) = -6i, -4i(5)=-20i, -4i(2i)= -8i²= i…5(8)=40, 5(-6i)=-30i, 6i(8)=48i, 6i(-6i)=-36i²= i…2(-9)=-18, 2(-3i)=-6i, -8i(-9)=72i, -8i(-3i)=24i²=-24

13. Mini Quiz Answers (Dividing)
(-2/5)+(24/5)i… CC: (2+i), (4+10i)(2+i) / (2-i)(2+i), 8+20i+4i+10i², i²=-1, 8+20i+4i-10 / 4+2i-2i-i², 4+2i-2i+1, -2+24i / 5 1-2i… CC: 2-2i, (6-2i)(2-2i) / (2+2i)(2-2i), 12-12i-4i+4i² / 4-4i+4i-4i², 12-12i-4i-4 / 4-4i+4i+4, 8-16i / 8 (-9/4)-(15/4)i… CC: (-4-4i), (24+6i)(-4-4i) / (-4+4i)(-4-4i), i-24i-24i² / 16+16i-16i-16i², i-24i+24 / 16+16, i / 32