Multiplying Complex Numbers/DeMoivre’sTheorem

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Preliminaries and Objectives

Preliminaries• Arithmetic of Complex Numbers• Multiplying by the conjugate to rationalize the denominator• Converting vectors between rectangular form and polar

form

Objectives• Multiply and divide complex numbers in polar form• Raise a complex number to a power• Find the roots of a complex number

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Complex Numbers

To the real numbers, add a new number called i , with theproperty i2 = −1. In other words, i =

√−1

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Complex Numbers

We now get the set of complex numbers a + bi where a and bare real numbers. If z = a + bi is a complex number, we call athe real part and bi the imaginary part.

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Complex Numbers

Operations on Complex Numbers

Addition: (a + bi) + (c + di) = (a + c) + (b + d)i

Subtraction: (a + bi)− (c + di) = (a− c) + (b − d)i

Multiplication:(a+bi)(c +di) = ac +adi +bci +bdi2 = (ac−bd)+ (ad +bc)i

Division:(a + bi)(c + di)

= ???

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Complex Multilplication

(3 + 8i)(6− 5i) = 18− 15i + 48i − 40i2 =

(18 + 40) + (−15 + 48)i = 58 + 33i

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Polar Form of Complex Numbers

x + yi = r cos θ + r sin θi

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Polar Form of Complex Numbers

x + yi = r cos θ + r sin θi = r(cos θ + i sin θ) = r cis θ

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Example

−3− 4i = 5 cis 233.1◦

sincer =

√32 + 42 = 5

andtan θ =

−4−3

with θ in the third quadrant.

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Multiplication of Complex Numbers in Polar Form

[r1(cos θ1 + i sin θ1)][r2(cos θ2 + i sin θ2)] =

r1r2(cos θ1 + i sin θ1)(cos θ2 + i sin θ2) =

r1r2(cos θ1 cos θ2+cos θ1i sin θ2+i sin θ1 cos θ2+i sin θ1i sin θ2) =

r1r2[(cos θ1 cos θ2 − sin θ1 sin θ2)+i(sin θ1 cos θ2 + cos θ1 sin θ2)] =

r1r2[cos(θ1 + θ2) + i sin(θ1 + θ2)] = r1r2 cis (θ1 + θ2)

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Multiplication of Complex Numbers in Polar Form

[r1(cos θ1 + i sin θ1)][r2(cos θ2 + i sin θ2)] =

r1r2(cos θ1 + i sin θ1)(cos θ2 + i sin θ2) =

r1r2(cos θ1 cos θ2+cos θ1i sin θ2+i sin θ1 cos θ2+i sin θ1i sin θ2) =

r1r2[(cos θ1 cos θ2 − sin θ1 sin θ2)+i(sin θ1 cos θ2 + cos θ1 sin θ2)] =

r1r2[cos(θ1 + θ2) + i sin(θ1 + θ2)] = r1r2 cis (θ1 + θ2)

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Multiplication of Complex Numbers in Polar Form

[r1(cos θ1 + i sin θ1)][r2(cos θ2 + i sin θ2)] =

r1r2(cos θ1 + i sin θ1)(cos θ2 + i sin θ2) =

r1r2(cos θ1 cos θ2+cos θ1i sin θ2+i sin θ1 cos θ2+i sin θ1i sin θ2) =

r1r2[(cos θ1 cos θ2 − sin θ1 sin θ2)+i(sin θ1 cos θ2 + cos θ1 sin θ2)] =

r1r2[cos(θ1 + θ2) + i sin(θ1 + θ2)] = r1r2 cis (θ1 + θ2)

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Multiplication of Complex Numbers in Polar Form

[r1(cos θ1 + i sin θ1)][r2(cos θ2 + i sin θ2)] =

r1r2(cos θ1 + i sin θ1)(cos θ2 + i sin θ2) =

r1r2(cos θ1 cos θ2+cos θ1i sin θ2+i sin θ1 cos θ2+i sin θ1i sin θ2) =

r1r2[(cos θ1 cos θ2 − sin θ1 sin θ2)+i(sin θ1 cos θ2 + cos θ1 sin θ2)] =

r1r2[cos(θ1 + θ2) + i sin(θ1 + θ2)] = r1r2 cis (θ1 + θ2)

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Multiplication of Complex Numbers in Polar Form

[r1(cos θ1 + i sin θ1)][r2(cos θ2 + i sin θ2)] =

r1r2(cos θ1 + i sin θ1)(cos θ2 + i sin θ2) =

r1r2(cos θ1 cos θ2+cos θ1i sin θ2+i sin θ1 cos θ2+i sin θ1i sin θ2) =

r1r2[(cos θ1 cos θ2 − sin θ1 sin θ2)+i(sin θ1 cos θ2 + cos θ1 sin θ2)] =

r1r2[cos(θ1 + θ2) + i sin(θ1 + θ2)] = r1r2 cis (θ1 + θ2)

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Multiplication of Complex Numbers in Polar Form

[r1 cis θ1][r2 cis θ2] = r1r2 cis (θ1 + θ2)

To multiply complex numbers in polar form, multiply the lengthsand add the angles.

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Division of Complex Numbers in Polar Form

3− 4i5 + 2i

(3− 4i)(5 + 2i)

(5− 2i)(5− 2i)

=15− 8− 20i − 6i

25 + 4=

7− 26i29

729− 26

29i

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Division of Complex Numbers in Polar Form

3− 4i5 + 2i

(3− 4i)(5 + 2i)

(5− 2i)(5− 2i)

=15− 8− 20i − 6i

25 + 4=

7− 26i29

729− 26

29i

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Division of Complex Numbers in Polar Form

3− 4i5 + 2i

(3− 4i)(5 + 2i)

(5− 2i)(5− 2i)

=15− 8− 20i − 6i

25 + 4=

7− 26i29

729− 26

29i

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Division of Complex Numbers in Polar Form

r1(cos θ1 + i sin θ1)

r2(cos θ2 + i sin θ2)

=r1(cos θ1 + i sin θ1)

r2(cos θ2 + i sin θ2)

(cos θ2 − i sin θ2)

(cos θ2 − i sin θ2)

=r1[(cos θ1 cos θ2 + sin θ1 sin θ2) + i(sin θ1 cos θ2 − cos θ1 sin θ2)]

r2(cos2 θ2 + sin2 θ2)

=r1

r2(cos(θ1 − θ2) + i sin(θ1 − θ2))

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Division of Complex Numbers in Polar Form

r1(cos θ1 + i sin θ1)

r2(cos θ2 + i sin θ2)

=r1(cos θ1 + i sin θ1)

r2(cos θ2 + i sin θ2)

(cos θ2 − i sin θ2)

(cos θ2 − i sin θ2)

=r1[(cos θ1 cos θ2 + sin θ1 sin θ2) + i(sin θ1 cos θ2 − cos θ1 sin θ2)]

r2(cos2 θ2 + sin2 θ2)

=r1

r2(cos(θ1 − θ2) + i sin(θ1 − θ2))

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Division of Complex Numbers in Polar Form

r1(cos θ1 + i sin θ1)

r2(cos θ2 + i sin θ2)

=r1(cos θ1 + i sin θ1)

r2(cos θ2 + i sin θ2)

(cos θ2 − i sin θ2)

(cos θ2 − i sin θ2)

=r1[(cos θ1 cos θ2 + sin θ1 sin θ2) + i(sin θ1 cos θ2 − cos θ1 sin θ2)]

r2(cos2 θ2 + sin2 θ2)

=r1

r2(cos(θ1 − θ2) + i sin(θ1 − θ2))

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Division of Complex Numbers in Polar Form

r1(cos θ1 + i sin θ1)

r2(cos θ2 + i sin θ2)

=r1(cos θ1 + i sin θ1)

r2(cos θ2 + i sin θ2)

(cos θ2 − i sin θ2)

(cos θ2 − i sin θ2)

=r1[(cos θ1 cos θ2 + sin θ1 sin θ2) + i(sin θ1 cos θ2 − cos θ1 sin θ2)]

r2(cos2 θ2 + sin2 θ2)

=r1

r2(cos(θ1 − θ2) + i sin(θ1 − θ2))

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Division of Complex Numbers in Polar Form

r1 cis θ1

r2 cis θ2=

r1

r2cis (θ1 − θ2)

To divide complex numbers in polar form, divide the lengthsand subtract the angles.

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Powers of Complex Numbers in Polar Form

(√2

2+

√2

2i

)4

= (1 cis 45◦)4

= 14 cis 180◦

= −1 + 0i = −1

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Powers of Complex Numbers in Polar Form

(√2

2+

√2

2i

)4

(√2

2+

√2

2i

)(√2

2+

√2

2i

)(√2

2+

√2

2i

)(√2

2+

√2

2i

)

= (1 cis 45◦)4

= 14 cis 180◦

= −1 + 0i = −1

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Powers of Complex Numbers in Polar Form

(√2

2+

√2

2i

)4

To raise a complex number to a power, raise the length to thepower, and multiply the angle by the power.

= (1 cis 45◦)4

= 14 cis 180◦

= −1 + 0i = −1

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Powers of Complex Numbers in Polar Form

(√2

2+

√2

2i

)4

= (1 cis 45◦)4

= 14 cis 180◦

= −1 + 0i = −1

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Powers of Complex Numbers in Polar Form

(√2

2+

√2

2i

)4

= (1 cis 45◦)4

= 14 cis 180◦

= −1 + 0i = −1

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Powers of Complex Numbers in Polar Form

(√2

2+

√2

2i

)4

= (1 cis 45◦)4

= 14 cis 180◦

= −1 + 0i = −1

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Roots of Complex Numbers in Polar Form

Find the three cube roots of −8i = 8 cis 270◦

DeMoivre’s Theorem: To find the roots of a complex number,take the root of the length, and divide the angle by the root.

Note: Since you will be dividing by 3, to find all answersbetween 0◦ and 360◦, we will want to begin with initial anglesfor three full circles.

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Roots of Complex Numbers in Polar Form

Find the three cube roots of −8i = 8 cis 270◦

DeMoivre’s Theorem: To find the roots of a complex number,take the root of the length, and divide the angle by the root.

Note: Since you will be dividing by 3, to find all answersbetween 0◦ and 360◦, we will want to begin with initial anglesfor three full circles.

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Roots of Complex Numbers in Polar Form

Find the three cube roots of −8i = 8 cis 270◦

DeMoivre’s Theorem: To find the roots of a complex number,take the root of the length, and divide the angle by the root.

Note: Since you will be dividing by 3, to find all answersbetween 0◦ and 360◦, we will want to begin with initial anglesfor three full circles.

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Roots of Complex Numbers in Polar Form

Find the three cube roots of−8i = 8 cis 270◦ = 8 cis 630◦ = 8 cis 990◦

Solution: {2 cis 90◦,2 cis 210◦,2 cis 330◦}

= {2i ,−√

3− i ,√

3− i}

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Roots of Complex Numbers in Polar Form

Find the three cube roots of−8i = 8 cis 270◦ = 8 cis 630◦ = 8 cis 990◦

Solution: {2 cis 90◦,2 cis 210◦,2 cis 330◦}

= {2i ,−√

3− i ,√

3− i}

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Roots of Complex Numbers in Polar Form

Find the three cube roots of−8i = 8 cis 270◦ = 8 cis 630◦ = 8 cis 990◦

Solution: {2 cis 90◦,2 cis 210◦,2 cis 330◦}

= {2i ,−√

3− i ,√

3− i}

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Recap

• To multiply complex numbers in polar form, multiply thelengths and add the angles

• To divide complex numbers in polar form, divide thelengths and subtract the angles

• To raise complex number to a power, raise the length to thepower and multiply the angle by the power

• To find the roots of a complex number, take the root of thelength and divide the angle by the power

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Credits

Written by: Mike Weimerskirch

Narration: Mike Weimerskirch

Graphic Design: Robbie Hank

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem

Copyright Info

c© The Regents of the University of Minnesota & MikeWeimerskirchFor a license please contact http://z.umn.edu/otc

University of Minnesota Multiplying Complex Numbers/DeMoivre’s Theorem