Multiplying Complex Numbers/DeMoivre`s Theorem - Math-UMN

Multiplying Complex Numbers/DeMoivre’s
Theorem
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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
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Multiplying Complex Numbers/DeMoivre’s Theorem
Complex Numbers
To the real numbers, add a new number
√ called i, with the
property i 2 = −1. In other words, i = −1
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Multiplying Complex Numbers/DeMoivre’s Theorem
Complex Numbers
We now get the set of complex numbers a + bi where a and b
are real numbers. If z = a + bi is a complex number, we call a
the real part and bi the imaginary part.
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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 + bdi 2 = (ac − bd) + (ad + bc)i
Division:
(a + bi)
= ???
(c + di)
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Multiplying Complex Numbers/DeMoivre’s Theorem
Complex Multilplication
(3 + 8i)(6 − 5i) = 18 − 15i + 48i − 40i 2 =
(18 + 40) + (−15 + 48)i = 58 + 33i
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Multiplying Complex Numbers/DeMoivre’s Theorem
Polar Form of Complex Numbers
x + yi = r cos θ + r sin θi
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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
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Multiplying Complex Numbers/DeMoivre’s Theorem
Example
−3 − 4i = 5 cis 233.1◦
since
r=
p
32 + 42 = 5
and
tan θ =
−4
−3
with θ in the third quadrant.
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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 )] =
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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 )] =
r1 r2 (cos θ1 + i sin θ1 )(cos θ2 + i sin θ2 ) =
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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 )] =
r1 r2 (cos θ1 + i sin θ1 )(cos θ2 + i sin θ2 ) =
r1 r2 (cos θ1 cos θ2 +cos θ1 i sin θ2 +i sin θ1 cos θ2 +i sin θ1 i sin 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 )] =
r1 r2 (cos θ1 + i sin θ1 )(cos θ2 + i sin θ2 ) =
r1 r2 (cos θ1 cos θ2 +cos θ1 i sin θ2 +i sin θ1 cos θ2 +i sin θ1 i sin θ2 ) =
r1 r2 [(cos θ1 cos θ2 − sin θ1 sin θ2 )
+i(sin θ1 cos θ2 + cos θ1 sin 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 )] =
r1 r2 (cos θ1 + i sin θ1 )(cos θ2 + i sin θ2 ) =
r1 r2 (cos θ1 cos θ2 +cos θ1 i sin θ2 +i sin θ1 cos θ2 +i sin θ1 i sin θ2 ) =
r1 r2 [(cos θ1 cos θ2 − sin θ1 sin θ2 )
+i(sin θ1 cos θ2 + cos θ1 sin θ2 )] =
r1 r2 [cos(θ1 + θ2 ) + i sin(θ1 + θ2 )] = r1 r2 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 ] = r1 r2 cis (θ1 + θ2 )
To multiply complex numbers in polar form, multiply the lengths
and add the angles.
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Multiplying Complex Numbers/DeMoivre’s Theorem
Division of Complex Numbers in Polar Form
3 − 4i
5 + 2i
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Multiplying Complex Numbers/DeMoivre’s Theorem
Division of Complex Numbers in Polar Form
3 − 4i
5 + 2i
(3 − 4i) (5 − 2i)
15 − 8 − 20i − 6i
7 − 26i
=
=
(5 + 2i) (5 − 2i)
25 + 4
29
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Multiplying Complex Numbers/DeMoivre’s Theorem
Division of Complex Numbers in Polar Form
3 − 4i
5 + 2i
(3 − 4i) (5 − 2i)
15 − 8 − 20i − 6i
7 − 26i
=
=
(5 + 2i) (5 − 2i)
25 + 4
29
7
26
−
i
29 29
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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 )
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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 ) (cos θ2 − i sin θ2 )
r2 (cos θ2 + i sin θ2 ) (cos θ2 − i sin 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 ) (cos θ2 − i sin θ2 )
r2 (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 )
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 ) (cos θ2 − i sin θ2 )
r2 (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
(cos(θ1 − θ2 ) + i sin(θ1 − θ2 ))
r2
University of Minnesota
Multiplying Complex Numbers/DeMoivre’s Theorem
Division of Complex Numbers in Polar Form
r1 cis θ1
r1
=
cis (θ1 − θ2 )
r2 cis θ2
r2
To divide complex numbers in polar form, divide the lengths
and subtract the angles.
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Multiplying Complex Numbers/DeMoivre’s Theorem
Powers of Complex Numbers in Polar Form
√
√ !4
2
2
+
i
2
2
University of Minnesota
Multiplying Complex Numbers/DeMoivre’s Theorem
Powers of Complex Numbers in Polar Form
√
√ !4
2
2
+
i
2
2
√
√ !
2
2
+
i
2
2
√
√ !
2
2
+
i
2
2
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√
√ !
2
2
+
i
2
2
√
√ !
2
2
+
i
2
2
Multiplying Complex Numbers/DeMoivre’s Theorem
Powers of Complex Numbers in Polar Form
√
√ !4
2
2
+
i
2
2
To raise a complex number to a power, raise the length to the
power, and multiply the angle by the power.
University of Minnesota
Multiplying Complex Numbers/DeMoivre’s Theorem
Powers of Complex Numbers in Polar Form
√
√ !4
2
2
+
i
2
2
= (1 cis 45◦ )4
University of Minnesota
Multiplying Complex Numbers/DeMoivre’s Theorem
Powers of Complex Numbers in Polar Form
√
√ !4
2
2
+
i
2
2
= (1 cis 45◦ )4
= 14 cis 180◦
University of Minnesota
Multiplying Complex Numbers/DeMoivre’s Theorem
Powers of Complex Numbers in Polar Form
√
√ !4
2
2
+
i
2
2
= (1 cis 45◦ )4
= 14 cis 180◦
= −1 + 0i = −1
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Multiplying Complex Numbers/DeMoivre’s Theorem
Roots of Complex Numbers in Polar Form
Find the three cube roots of −8i = 8 cis 270◦
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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.
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 answers
between 0◦ and 360◦ , we will want to begin with initial angles
for 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◦
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◦ }
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}
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Multiplying Complex Numbers/DeMoivre’s Theorem
Recap
• To multiply complex numbers in polar form, multiply the
lengths and add the angles
• To divide complex numbers in polar form, divide the
lengths and subtract the angles
• To raise complex number to a power, raise the length to the
power and multiply the angle by the power
• To find the roots of a complex number, take the root of the
length and divide the angle by the power
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Multiplying Complex Numbers/DeMoivre’s Theorem
Credits
Written by:
Mike Weimerskirch
Narration:
Mike Weimerskirch
Graphic Design: Robbie Hank
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Multiplying Complex Numbers/DeMoivre’s Theorem
Copyright Info
c The Regents of the University of Minnesota & Mike
Weimerskirch
For a license please contact http://z.umn.edu/otc
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Multiplying Complex Numbers/DeMoivre’s Theorem

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