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, where n is 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 ∠
1
2 × 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
◦
= 2.415 cos 15.48
◦
+ j 2.415 sin15.48
◦
= 2.327 + j0.6446
and 2.415∠195.48
◦
= 2.415 cos 195.48
◦
+ j 2.415 sin195.48
◦
= −2.327 − j0.6446
Hence
[(5 + j 3)]
1
2 = 2.415∠15.48
◦ and
2.415∠195.48
◦ or
± (2.327 + j0.6446).
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.
1. (a) 1 + j (b) j
(a) ±(1.099 + j 0.455)
(b) ±(0.707 + j 0.707)
2. (a) 3 − j 4 (b) −1 − j 2
(a) ±(2 − j )
(b) ±(0.786 − j 1.272)
3. (a) 7∠60 ◦ (b) 12∠
3π
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.
4. (3 + j 4)
1
3
Moduli 1.710, arguments 17.71 ◦ ,
137.71 ◦ and 257.71 ◦
arguments, θ, are different. It is shown in Problem 3
that arguments are symmetrically spaced on an Argand
diagram and are (360/n) ◦ apart, where n is 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 ∠
1
2 × 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
◦
= 2.415 cos 15.48
◦
+ j 2.415 sin15.48
◦
= 2.327 + j0.6446
and 2.415∠195.48
◦
= 2.415 cos 195.48
◦
+ j 2.415 sin195.48
◦
= −2.327 − j0.6446
Hence
[(5 + j 3)]
1
2 = 2.415∠15.48
◦ and
2.415∠195.48
◦ or
± (2.327 + j0.6446).
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.
1. (a) 1 + j (b) j
(a) ±(1.099 + j 0.455)
(b) ±(0.707 + j 0.707)
2. (a) 3 − j 4 (b) −1 − j 2
(a) ±(2 − j )
(b) ±(0.786 − j 1.272)
3. (a) 7∠60 ◦ (b) 12∠
3π
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.
4. (3 + j 4)
1
3
Moduli 1.710, arguments 17.71 ◦ ,
137.71 ◦ and 257.71 ◦