Chapter 21
De Moivre’s theorem
21.1 Introduction
From multiplication of complex numbers in polar form,
(r∠θ) × (r ∠θ) = r
2
∠2θ
Similarly, (r∠θ)× (r∠θ)× (r∠θ)= r 3 ∠3θ, and so on.
In general, De Moivre’s theorem states:
[r∠θ]
n
= r n ∠nθ
The theorem is true for all positive, negative and
fractional values of n. The theorem is used to determine
powers and roots of complex numbers.
21.2 Powers of complex numbers
For example [3∠20
◦ ]
4
= 3
4
∠(4 × 20
◦
) = 81∠80
◦ by
De Moivre’s theorem.
Problem 1. Determine, in polar form
(a) [2∠35 ◦ ] 5 (b) (−2 + j 3) 6 .
(a) [2∠35
◦ ]
5
= 2
5
∠(5 × 35
◦
),
from De Moivre’s theorem
= 32∠175 ◦
(b) (−2 + j 3)=
[(−2) 2 + (3) 2 ]∠ tan
−1 3
−2
=
√
13∠123.69
◦
, since −2 + j 3
lies in the second quadrant
(−2 + j 3)
6
= [
√
13∠123.69
◦ ]
6
= (
√
13)
6
∠(6 × 123.69
◦
),
by De Moivre’s theorem
= 2197∠742.14
◦
= 2197∠382.14
◦
(since 742.14
≡ 742.14
◦
− 360
◦
= 382.14
◦
)
= 2197∠22.14
◦
(since 382.14
◦
≡ 382.14
◦
− 360
◦
= 22.14
◦
)
or 2197∠22
◦ 8
Problem 2. Determine the value of (−7 + j 5) 4 ,
expressing the result in polar and rectangular forms.
(−7 + j 5) =
[(−7) 2 + 5 2 ]∠ tan
−1 5
−7
=
√
74∠144.46
◦
(Note, by considering the Argand diagram, −7 + j 5
must represent an angle in the second quadrant and not
in the fourth quadrant.)
Applying De Moivre’s theorem:
(−7 + j 5)
4
= [
√
74∠144.46
◦ ]
4
=
√
74 4 ∠4 ×144.46
◦
= 5476∠577.84
◦
= 5476∠217.84
◦
or 5476∠217
◦ 50
in polar form
Since r∠θ = r cos θ + jr sin θ,
5476∠217.84
◦
= 5476 cos217.84
◦
+ j 5476 sin217.84
◦
= −4325 − j 3359
i.e.
(−7 + j5)
4
= −4325 −j3359
in rectangular form