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