214 Higher Engineering Mathematics
Problem 3. Evaluate
(a)j^3 (b)j^4 (c) j^23 (d)
− 4
j^9
(a) j^3 =j^2 ×j=(− 1 )×j=−j,sincej^2 =− 1
(b) j^4 =j^2 ×j^2 =(− 1 )×(− 1 )= 1
(c) j^23 =j×j^22 =j×(j^2 )^11 =j×(− 1 )^11
=j×(− 1 )=−j
(d) j^9 =j×j^8 =j×(j^2 )^4 =j×(− 1 )^4
=j× 1 =j
Hence
− 4
j^9
=
− 4
j
=
− 4
j
×
−j
−j
=
4 j
−j^2
=
4 j
−(− 1 )
= 4 jorj 4
Now try the following exercise
Exercise 85 Further problems on the
introduction to cartesian complexnumbers
InProblems1to9,solvethequadraticequations.
x^2 + 25 =0[±j5]
x^2 − 2 x+ 2 =0[x= 1 ±j]
x^2 − 4 x+ 5 =0[x= 2 ±j]
x^2 − 6 x+ 10 =0[x= 3 ±j]
2x^2 − 2 x+ 1 =0[x= 0. 5 ±j 0 .5]
x^2 − 4 x+ 8 =0[x= 2 ±j2]
25x^2 − 10 x+ 2 =0[x= 0. 2 ±j 0 .2]
2x^2 + (^3) [x+ 4 = 0
−
3
4
±j
√
23
4
or− 0. 750 ±j 1. 199
]
4t^2 − 5 t+ 7 =[ 0
5
8
±j
√
87
8
or 0. 625 ±j 1. 166
]
- Evaluate (a)j^8 (b)−
1
j^7
(c)
4
2 j^13
[(a) 1 (b)−j(c)−j2]
20.2 The Argand diagram
A complex number may be represented pictorially on
rectangularorcartesianaxes.Thehorizontal(orx)axisis
used to represent the real axis and the vertical (ory)axis
is used to represent the imaginary axis. Such a diagram
is called anArgand diagram. In Fig. 20.1, the pointA
represents the complex number( 3 +j 2 )and is obtained
byplottingtheco-ordinates( 3 ,j 2 )as in graphical work.
Figure20.1alsoshowstheArgandpointsB,CandDrep-
resenting the complex numbers(− 2 +j 4 ),(− 3 −j 5 )
and( 1 −j 3 )respectively.
2221 2
2 j
j
2 j 2
j 2
2 j 3
j 3
2 j 4
3 Real axis
Imaginary
axis
A
B
D
C
23 0 1
2 j 5
j 4
Figure 20.1
20.3 Addition and subtraction of
complex numbers
Two complex numbers are added/subtracted by adding/
subtracting separately the two real parts and the two
imaginary parts.
For example, ifZ 1 =a+jbandZ 2 =c+jd,
then Z 1 +Z 2 =(a+jb)+(c+jd)
=(a+c)+j(b+d)
and Z 1 −Z 2 =(a+jb)−(c+jd)
=(a−c)+j(b−d)