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z = 1 + 2j and z = 2 + j
2
1
Solution: Grouping the real and imaginary parts separately, we obtain:

z + z = + 3j
1
2
and

z – z = –1 + j
1 2

Preparatory Exercise

Pb. 6.1 Given the complex numbers z , z , and z corresponding to the ver-
3
1
2
tices P , P , and P of a parallelogram, find z corresponding to the fourth ver-
1
2
4
3
tex P . (Assume that P and P are opposite vertices of the parallelogram).
2
4
4
Verify your answer graphically for the case:
z =+ j, z = + j 2 , z = + j 3
4
2
1
1 2 3
6.2.2 Multiplication by a Real or Imaginary Number
If we multiply the complex number z = a + jb by a real number k, the resultant
complex number is given by:
z
k
k ×= × ( a + jb =) ka + jkb (6.5)
What happens when we multiply by j?
Let us, for a moment, return to Example 6.1. We note the following proper-
ties for the three points P , P , and P :
3
1
2
1. The three points are equally distant from the origin of the axis.
2. The point P is obtained from the point P by a π/2 counter-
1
2
clockwise rotation.
3. The point P is obtained from the point P through another π/2
3
2
counterclockwise rotation.
We also note, by examining the algebraic forms of z , z , z that:
1
2
3
z = jz and z = jz = j z = − z
2
2 1 3 2 1 1

© 2001 by CRC Press LLC