Complex numbers 217
(a) (1 + j ) 2 = (1 + j )(1 + j ) =1 + j + j + j 2
= 1 + j + j − 1 = j 2
(1 + j ) 4 = [(1 + j ) 2 ] 2 = ( j 2) 2 = j 2 4 =−4
Hence
2
(1 + j ) 4 =
2
−4
= −
1
2
(b)
1 + j 3
1 − j 2
=
1 + j 3
1 − j 2
×
1 + j 2
1 + j 2
=
1 + j 2 + j 3 + j 2 6
1 2 + 2 2
=
−5 + j 5
5
= −1 + j 1 = −1 + j
1 + j 3
1 − j 2
2
= (−1 + j )
2
= (−1 + j )(−1 + j )
= 1 − j − j + j
2
=− j 2
Hence j
1 + j 3
1 − j 2
2
= j (− j 2) =− j 2 2 =2,
since j 2 =−1
Now try the following exercise
Exercise 86 Further problems on
operations involving Cartesian complex
numbers
1. Evaluate
(a)
(3 + j 2) +(5 − j ) and
(b) (−2 + j 6) −(3 − j 2) and show the
results on an Argand diagram.
[(a) 8 + j (b) −5 + j 8]
2. Write down the complex conjugates of
(a) 3 + j 4, (b) 2 − j .
[(a) 3 − j 4 (b) 2+ j ]
3. If z = 2 + j and w = 3 − j evaluate
(a) z + w (b) w − z (c) 3z − 2w (d)
5z + 2w (e) j (2w − 3z) (f ) 2 j w − j z
[(a) 5 (b) 1 − j 2 (c) j 5 (d) 16 + j 3
(e) 5 (f ) 3 + j 4]
In Problems 4 to 8 evaluate in a + j b form
given Z 1 = 1 + j 2, Z 2 = 4 − j 3, Z 3 =−2 + j 3
and Z 4 =−5 − j .
4. (a) Z 1 + Z 2 − Z 3 (b) Z 2 − Z 1 + Z 4
[(a) 7 − j 4 (b) −2 − j 6]
5. (a) Z 1 Z 2 (b) Z 3 Z 4
[(a) 10 + j 5 (b) 13 − j 13]
6. (a) Z 1 Z 3 + Z 4 (b) Z 1 Z 2 Z 3
[(a) −13 − j 2 (b) −35 + j 20]
7. (a)
Z 1
Z 2
(b)
Z 1 + Z 3
Z 2 − Z 4
(a)
−2
25
+ j
11
25
(b)
−19
85
+ j
43
85
8. (a)
Z 1 Z 3
Z 1 + Z 3
(b) Z 2 +
Z 1
Z 4
+ Z 3
(a)
3
26
+ j
41
26
(b)
45
26
− j
9
26
9. Evaluate (a)
1 − j
1 + j
(b)
1
1 + j
(a) − j (b)
1
2
− j
1
2
10. Show that
−25
2
1 + j 2
3 + j 4
−
2 − j 5
− j
= 57 + j 24
20.5 Complex equations
If two complex numbers are equal, then their real parts
are equal and their imaginary parts are equal. Hence if
a + j b =c + j d, then a = c and b = d.
Problem 7. Solve the complex equations:
(a) 2(x + j y) =6 − j 3
(b) (1 + j 2)(−2 − j 3) =a + j b
(a) 2(x + j y) =6 − j 3 hence 2x + j 2y = 6 − j 3
Equating the real parts gives:
2x = 6, i.e. x = 3
Equating the imaginary parts gives:
2y = −3, i.e. y = −
3
2