De Moivre’s theorem 227
arguments,θ, are different. It is shown in Problem 3
that arguments are symmetrically spaced on an Argand
diagram and are( 360 /n)◦apart, wherenis the number
of the roots required. Thus if one of the solutions to the
cube root of a complex number is, say, 5∠ 20 ◦,the other
two roots are symmetrically spaced( 360 / 3 )◦,i.e. 120◦
from this root and the three roots are 5∠ 20 ◦,5∠ 140 ◦
and 5∠ 260 ◦.
Problem 4. Find the roots of [( 5 +j 3 )]1(^2) in
rectangular form, correct to 4 significant figures.
( 5 +j 3 )=
√
34 ∠ 30. 96 ◦
Applying De Moivre’s theorem:
( 5 +j 3 )
1
(^2) =
√
34
1
(^2) ∠^12 × 30. 96 ◦
= 2. 415 ∠ 15. 48 ◦or 2. 415 ∠ 15 ◦ 29 ′
The second root may be obtained as shown above, i.e.
having the same modulus but displaced( 360 / 2 )◦from
the first root.
Thus,( 5 +j 3 )
1
(^2) = 2. 415 ∠( 15. 48 ◦+ 180 ◦)
= 2. 415 ∠ 195. 48 ◦
In rectangular form:
2. 415 ∠ 15. 48 ◦= 2 .415cos15. 48 ◦
+j 2 .415sin15. 48 ◦
= 2. 327 +j 0. 6446
and 2. 415 ∠ 195. 48 ◦= 2 .415cos195. 48 ◦
+j 2 .415sin195. 48 ◦
=− 2. 327 −j 0. 6446
Hence [( 5 +j 3 )]
1
(^2) = 2. 415 ∠ 15. 48 ◦and
2. 415 ∠ 195. 48 ◦or
±( 2. 327 +j 0 .6446).
Problem 5. Express the roots of
(− 14 +j 3 )
− 2
(^5) in polar form.
(− 14 +j 3 )=
√
205 ∠ 167. 905 ◦
(− 14 +j 3 )
− 2
(^5) =
√
205
− 2
(^5) ∠
[(
−
2
5
)
× 167. 905 ◦
]
= 0. 3449 ∠− 67. 164 ◦
or 0. 3449 ∠− 67 ◦ 10 ′
There are five roots to this complex number,
(
x
− 2
(^5) =^1
x
2
5
1
√ (^5) x 2
)
The roots are symmetrically displaced from one
another ( 360 / 5 )◦,i.e.72◦ apart round an Argand
diagram.
Thus the required roots are 0. 3449 ∠− 67 ◦ 10 ′,
0. 3449 ∠ 4 ◦ 50 ′, 0. 3449 ∠ 76 ◦ 50 ′, 0. 3449 ∠ 148 ◦ 50 ′and
0. 3449 ∠ 220 ◦ 50 ′.
Now try the following exercise
Exercise 91 Further problems on the
roots of complex numbers
In Problems 1 to 3 determine the two square roots
of the given complex numbers in Cartesian form
and show the results on an Argand diagram.
- (a) 1+j(b)j
[
(a)±( 1. 099 +j 0. 455 )
(b)±( 0. 707 +j 0. 707 )
]- (a) 3−j4(b)− 1 −j 2
[
(a)±( 2 −j)
(b)±( 0. 786 −j 1. 272 )
]- (a) 7∠ 60 ◦(b) 12∠
3 π
[^2
(a)±( 2. 291 +j 1. 323 )
(b)±(− 2. 449 +j 2. 449 )]In Problems 4 to 7, determine the moduli and
arguments of the complex roots.- ( 3 +j 4 )
1
3
[
Moduli 1. 710 ,arguments 17. 71 ◦,
137. 71 ◦and 257. 71 ◦]