Higher Engineering Mathematics

COMPLEX NUMBERS 251

E

−j 4

−j 3

−j

j

j 2

j 3

Imaginary

axis

(^034) 5 Real axis
R (5−j)
P (2+j3)
Q (3−j4)
1 2
(a)
−j 4
−j 3
−j 2
−j
j 2
j 3
0 1 2 3 Real axis
(b)
Q (3−j4)
− 3 − 2 − 1
j
j 4
j 5
j 7
Imaginary
axis
S (−1+j7)
P (2+j3)
Q'
j 6
−j 2
Figure 23.2
(a) Z 1 +Z 2 =(2+j4)+(3−j)
=(2+3)+j(4−1)= 5 +j 3
(b) Z 1 −Z 2 =(2+j4)−(3−j)
=(2−3)+j(4−(−1))=− 1 +j 5
(c) Z 2 −Z 1 =(3−j)−(2+j4)
=(3−2)+j(− 1 −4)= 1 −j 5
Each result is shown in the Argand diagram of
Fig. 23.3.
23 45Real axis
Imaginary
axis
− 1 1
−j
−j 2
−j 3
−j 4
−j 5 (1−j5)
(5+j3)
(− 1 +j5)
j 3
j 4
j 5
0
j
j 2
Figure 23.3
23.4 Multiplication and division of
complex numbers
(i)Multiplication of complex numbers is
achieved by assuming all quantities involved
are real and then usingj^2 =−1 to simplify.
Hence (a+jb)(c+jd)
=ac+a(jd)+(jb)c+(jb)(jd)
=ac+jad+jbc+j^2 bd
=(ac−bd)+j(ad+bc),
since j^2 =− 1
Thus (3+j2)(4−j5)
= 12 −j 15 +j 8 −j^210
=(12−(−10))+j(− 15 +8)
= 22 −j 7
(ii) Thecomplex conjugateof a complex num-
ber is obtained by changing the sign of the
imaginary part. Hence the complex conjugate
ofa+jbisa−jb. The product of a complex