PAGE 642 26 5cis0 9273 27 13cis5 1072

3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex Numbers 2

DEMOVIRE’S THEOREM Section 10. 4 Pre-Calculus AB Pre. AP/Dual, Revised © 2014 Viet. dang@humble.

REVIEW 3/5/2021 8: 16 AM 10. 3 - Complex Numbers and Polar Form •

EXAMPLE 1 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex

ABOUT DEMOIVRE A. Abraham De. Moivre (1667 -1754) used this theorem for analytical and

DEMOVIRE’S THEOREM A. One can raise exponents. One can repeat this process raising complex

STEPS A. B. C. D. Identify the r (moduli), the argument (θ), and the

EXAMPLE 2 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex

YOUR TURN 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex

EXAMPLE 3 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex

EXAMPLE 4 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex

EXAMPLE 5 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex

ROOTS OF COMPLEX NUMBERS A. B. C. D. Identify the r (moduli), the argument

Slides: 21

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PAGE 642 26) 5(cis(0. 9273)) 27) 13(cis(5. 1072)) 28) 4(cis(3. 9897)) 30) 33) 35) 37) 39) 41) (cis (π/2)) 43) 12(cis(2π/3)) 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex Numbers 1

3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex Numbers 2

REVIEW 3/5/2021 8: 16 AM 10. 3 - Complex Numbers and Polar Form • 4

EXAMPLE 1 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex Numbers 5

ABOUT DEMOIVRE A. Abraham De. Moivre (1667 -1754) used this theorem for analytical and probability B. He calculated his sleeping pattern and formed an arithmetic sequence C. Hence, he was famous for correctly predicting the day he would die 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex Numbers 6

DEMOVIRE’S THEOREM A. One can raise exponents. One can repeat this process raising complex numbers to powers B. If z = r(cos θ + i sin θ) is a complex number, then zn = rn (cos θ + i sin nθ) where n > 1 is a positive integer 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex Numbers 7

STEPS A. B. C. D. Identify the r (moduli), the argument (θ), and the exponent Apply the exponent to the r (moduli) Use the exponent to multiply the argument Simplify 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex Numbers 8

EXAMPLE 1 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex Numbers 9

EXAMPLE 1 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex Numbers 10

EXAMPLE 2 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex Numbers 11

YOUR TURN 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex Numbers 12

EXAMPLE 3 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex Numbers 13

EXAMPLE 3 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex Numbers 14

EXAMPLE 4 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex Numbers 15

EXAMPLE 5 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex Numbers 16

YOUR TURN 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex Numbers 17

ROOTS OF COMPLEX NUMBERS A. B. C. D. Identify the r (moduli), the argument (θ), and the exponent Apply the exponent to the r (moduli) Use the exponent to multiply the argument Simplify 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex Numbers 18

EXAMPLE 4 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex Numbers 19

EXAMPLE 5 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex Numbers 20

YOUR TURN 3/5/2021 8: 16 AM 10. 3 A - Multiply and Divide Complex Numbers 21