Higher Engineering Mathematics, Sixth Edition

Complex numbers 215

Thus, for example,

( 2 +j 3 )+( 3 −j 4 )= 2 +j 3 + 3 −j 4

= 5 −j 1

and ( 2 +j 3 )−( 3 −j 4 )= 2 +j 3 − 3 +j 4

=− 1 +j 7

The addition and subtraction of complex numbers may
be achieved graphically as shown in the Argand diagram
of Fig. 20.2.( 2 +j 3 )is represented by vectorOPand

2

2 j

j

2 j 2

j 2

2 j 3

j 3

2 j 4

345 Real axis

R (5 2 j)

Q (3 2 j4)

P (2 1 j3)

Imaginary

axis

0 1

(a)

(b)

2221 2

2 j

j

2 j 2

j 2

2 j 3

j 3

2 j 4

3 Real axis

Q (3 2 j4)

P (2 1 j3)

S ( 211 j7)

Imaginary

axis

Q 9

23 0 1

j 4

j 5

j 7

j 6

Figure 20.2

( 3 −j 4 )byvectorOQ.InFig.20.2(a)byvectoraddition

(i.e. the diagonal of the parallelogram)OP+OQ=OR.

Ris the point( 5 ,−j 1 ).

Hence( 2 +j 3 )+( 3 −j 4 )= 5 −j 1.

In Fig. 20.2(b), vectorOQis reversed (shown asOQ′)

since it is being subtracted. (NoteOQ= 3 −j4and

OQ′=−( 3 −j 4 )=− 3 +j 4 ).

OP−OQ=OP+OQ′=OSis found to be the Argand

point(− 1 ,j 7 ).

Hence ( 2 +j 3 )−( 3 −j 4 )=− 1 +j 7

Problem 4. GivenZ 1 = 2 +j4andZ 2 = 3 −j

determine (a)Z 1 +Z 2 ,(b)Z 1 −Z 2 ,(c)Z 2 −Z 1 and

show the results on an Argand diagram.

(a) Z 1 +Z 2 =( 2 +j 4 )+( 3 −j)

=( 2 + 3 )+j( 4 − 1 )= 5 +j 3

(b)Z 1 −Z 2 =( 2 +j 4 )−( 3 −j)

=( 2 − 3 )+j( 4 −(− 1 ))=− 1 +j 5

(c) Z 2 −Z 1 =( 3 −j)−( 2 +j 4 )

=( 3 − 2 )+j(− 1 − 4 )= 1 −j 5

Each result is shown in the Argand diagram of

Fig. 20.3.

21 2

2 j

j

2 j 2

j 2

2 j 3

j 3

2 j 4

2 j 5

3 Real axis

(1 2 j5)

(5 1 j3)

( 211 j5)

Imaginary

axis

0 1 45

j 4

j 5

Figure 20.3