Complex Numbers

• Imaginary unit• Complex 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

Powers of

-1

1How to solve:

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:=

326

• 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

Adding Complex Numbers

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 4 + i - 2 - 21 = 2 - i

Multiplying Complex Numbers

• Find the moon!

(a + bi) (c + di) (2 + i) (1 + i) = 1 + 3i

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

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) = 14 – 7i – 8i + 4i 2 + i (2-i) (2-i) 4 – 2i + 2i - i

= 14 – 7i – 8i – 4 = 10 – 15i = 2 – 3i 4 + 1 5

Mini Quiz(Adding and Subtracting)

1. (3-18i) - (7+24i) =2. (5+2i) - (12+10i) =3. (15+66i) - (2-12i) =4. (8+21i) + (10-11i) =5. (-13+47i) + (13-2i) =6. (51+52i) + (53+54i) =

Mini Quiz(Multiplying and Dividing)

1. (-3-4i)(5+2i) =2. (5+6i)(8-6i) = 3. (2-8i)(-9-3i) =4. (4+10i) / (2-i) =5. (6-2i) / (2+2i) =6. (24+6i) / (-4+4i) =

Mini Quiz Answers(Adding and Subtracting)

1. -4 - 42i….. 3 – 7 =-4 and -18i – 24i =-42 2. -7 - 8i…… 5 – 12 =-7 and 2i – 10i =-8i3. 13 + 78i….. 15 -2 = 13 and 66i + 12i = 78i4. 18 + 10i….. 8 + 10 = 18 and 21i + -11i = 10i5. 45i………… -13 + 13 = 0 and 47i + -2i6. 104+106i…. 51 + 53 = 104 and 52i + 54i =106i

Mini Quiz Answers(Multiplying)

1. -7-26i…-3(5) =-15, -3(2i) = -6i, -4i(5)=-20i, -4i(2i)= -8i²=8

2. 76+18i…5(8)=40, 5(-6i)=-30i, 6i(8)=48i, 6i(-6i)=-36i²=36

3. -42+66i…2(-9)=-18, 2(-3i)=-6i, -8i(-9)=72i, -8i(-3i)=24i²=-24

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), -96-96i-24i-24i² / 16+16i-16i-16i², -96-96i-24i+24 / 16+16, -72-120i / 32