Higher Engineering Mathematics, Sixth Edition

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.

1. 
   

   x^2 + 25 =0[±j5]

   
   


2. 
   

   x^2 − 2 x+ 2 =0[x= 1 ±j]

   
   


3. 
   

   x^2 − 4 x+ 5 =0[x= 2 ±j]

   
   


4. 
   

   x^2 − 6 x+ 10 =0[x= 3 ±j]

   
   


5. 
   

   2x^2 − 2 x+ 1 =0[x= 0. 5 ±j 0 .5]

   
   


6. 
   

   x^2 − 4 x+ 8 =0[x= 2 ±j2]

   
   


7. 
   

   25x^2 − 10 x+ 2 =0[x= 0. 2 ±j 0 .2]

   
   


8. 
   

   2x^2 + (^3) [x+ 4 = 0
   −
   3
   4
   ±j
   √
   23
   4
   or− 0. 750 ±j 1. 199
   ]

   
   


9. 
   

   4t^2 − 5 t+ 7 =[ 0
   5
   8

   
   

±j

√

87

8

or 0. 625 ±j 1. 166

]

10. 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)