Higher Engineering Mathematics, Sixth Edition

Complex numbers 221

=( 1. 732 +j 1. 000 )+( 3. 536 −j 3. 536 )

−(− 2. 000 +j 3. 464 )

= 7. 268 −j 6. 000 ,which lies in the fourth quadrant

=

√

[( 7. 268 )^2 +( 6. 000 )^2 ]∠tan−^1

(

− 6. 000

7. 268

)

= 9. 425 ∠− 39. 54 ◦

Now try the following exercise

Exercise 88 Further problemson polar

form

1. Determine the modulus and argument of
   (a) 2+j4(b)− 5 −j2(c)j( 2 −j).
   ⎡
   ⎢
   ⎣

(a) 4. 472 , 63. 43 ◦

(b)5. 385 ,− 158. 20 ◦

(c) 2. 236 , 63. 43 ◦

⎤

⎥

⎦

In Problems 2 and 3 express the given Cartesian

complex numbers in polar form, leaving answers

in surd form.

2. (a) 2+j3(b)−4(c)− 6 +j
   [
   (a)

√

13 ∠ 56. 31 ◦ (b)4∠ 180 ◦

(c)

√

37 ∠ 170. 54 ◦

]

3. (a)−j3(b)(− 2 +j)^3 (c)j^3 ( 1 −j)
   [
   (a)3∠− 90 ◦ (b)

√

125 ∠ 100. 30 ◦

(c)

√

2 ∠− 135 ◦

]

In Problems 4 and 5 convert the given polar com-

plex numbers into(a+jb)form giving answers

correct to 4 significant figures.

4. (a) 5∠ 30 ◦(b) 3∠ 60 ◦(c) 7∠ 45 ◦
   ⎡
   ⎢
   ⎣

(a) 4. 330 +j 2. 500

(b)1. 500 +j 2. 598

(c) 4. 950 +j 4. 950

⎤

⎥

⎦

5. (a) 6∠ 125 ◦(b) 4∠π(c) 3. 5 ∠− 120 ◦
   ⎡
   ⎢
   ⎣

(a)− 3. 441 +j 4. 915

(b)− 4. 000 +j 0

(c)− 1. 750 −j 3. 031

⎤

⎥

⎦

In Problems 6 to 8, evaluate in polar form.

6. (a) 3∠ 20 ◦× 15 ∠ 45 ◦
   (b) 2. 4 ∠ 65 ◦× 4. 4 ∠− 21 ◦
   [(a) 45∠ 65 ◦(b) 10. 56 ∠ 44 ◦]


7. (a) 6. 4 ∠ 27 ◦÷ 2 ∠− 15 ◦
   (b) 5∠ 30 ◦× 4 ∠ 80 ◦÷ 10 ∠− 40 ◦
   [(a) 3. 2 ∠ 42 ◦(b) 2∠ 150 ◦]


8. (a) 4∠

π

6

+ 3 ∠

π

8

(b) 2∠ 120 ◦+ 5. 2 ∠ 58 ◦− 1. 6 ∠− 40 ◦

[(a) 6. 986 ∠ 26. 79 ◦(b) 7. 190 ∠ 85. 77 ◦]

20.8 Applications of complex numbers

There are several applications of complex numbers

in science and engineering, in particular in electrical

alternating current theory and in mechanical vector

analysis.

The effect of multiplying a phasor by jis to rotate

it in a positive direction (i.e. anticlockwise) on an

Argand diagram through90◦without altering its length.

Similarly, multiplying a phasor by−jrotates the pha-

sor through− 90 ◦. These facts are used in a.c. the-

ory since certain quantities in the phasor diagrams

lie at 90◦to each other. For example, in the R−L

series circuit shown in Fig. 20.8(a),VL leads I by

90 ◦(i.e. I lagsVLby 90◦) and may be written as

jVL, the vertical axis being regarded as the imagi-

nary axis of an Argand diagram. ThusVR+jVL=V

and sinceVR=IR,V=IXL(whereXLis the induc-

tive reactance, 2πfLohms) andV=IZ(whereZis

the impedance) thenR+jXL=Z.

Phasor diagram Phasor diagram

VR VL

R

V

I

L

(a)

VR

V

I

VL



VR VC

R

V

I

C

(b)

VR

VC

V

I



Figure 20.8