Complex Numbers

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

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

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

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

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) =

Mini Quiz Answers (Adding and Subtracting) -4 - 42i….. 3 – 7 =-4 and -18i – 24i =-42 -7 - 8i…… 5 – 12 =-7 and 2i – 10i =-8i 13 + 78i….. 15 -2 = 13 and 66i + 12i = 78i 18 + 10i….. 8 + 10 = 18 and 21i + -11i = 10i 45i………… -13 + 13 = 0 and 47i + -2i 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