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That is, multiplying by j is geometrically equivalent to a counterclockwise
rotation by an angle of π/2.

6.2.3 Multiplication of Two Complex Numbers
The multiplication of two complex numbers follows the same rules of algebra
2
for real numbers, but considers j = –1. This yields:
z = a + jb and z = a + jb
1 1 1 2 2 2
If: ⇒ zz = a a( − b b ) + j a b( + b a ) (6.6)
1 2 1 2 12 12 1 2

Preparatory Exercises
Solve the following problems analytically.
Pb. 6.2 Find zz z z, 2 , 2 for the following pairs:
12 1 2
a. z = j 3 ; z = − j
1
1 2
b. z =+ j; z = − j 3
46
2
1 2
c. z = 1 ( 24 j); z = 1 ( 15− j)
+
1 2
3 2
d. z = 1 ( 24− j); z = 1 ( 15+ j)
1 2
3 2
Pb. 6.3 Find the real quantities m and n in each of the following equations:
a. mj + n(1 + j) = 3 – 2j
b. m(2 + 3j) + n(1 – 4j) = 7 + 5j
(Hint: Two complex numbers are equal if separately the real and imaginary
parts are equal.)

Pb. 6.4 Write the answers in standard form: (i.e., a + jb)
2
a. (3 – 2j) – (3 + 2j) 2
b. (7 + 14j) 7
  1   2
c.   (2 + j )  2 + 2j   

d. j(1 + 7j) – 3j(4 + 2j)

Pb. 6.5 Show that for all complex numbers z , z , z , we have the following
1
2
3
properties:
z z = z z (commutativity property)
1 2
2 1
z (z + z ) = z z + z z (distributivity property)
1 3
1
3
1 2
2
© 2001 by CRC Press LLC