ABSTRACT
The complex conjugate of (a+ jb) is (a− jb). For example, the conjugate of (3−j2) is (3+j2) The product of a complex number and its complex
conjugate is always a real number, and this is an important property used when dividing complex numbers. Thus
(a+ jb)(a− jb) = a2 − jab+ jab− j 2b2 = a2 − (−b2) = a2 + b2 (i.e. a real number)
5 and (3− j4)(3+ j4)= 32 + 42 = 25
The expression of one complex number divided by another, in the form a+ jb, is accomplished by multiplying the numerator and denominator by the complex conjugate of the denominator. This has the effect of making the denominator a real number. Hence, for example,
2+ j4 3− j4 =
2+ j4 3− j4 ×
3+ j4 3+ j4 =
6+ j8+ j12+ j 216 32 + 42
= 6+ j8+ j12− 16 25
= −10+ j20 25
= −10 25
+ j20 25
or
− 0.4 + j0.8
The elimination of the imaginary part of the denominator bymultiplying both the numerator anddenominator by the conjugate of the denominator is often termed ‘rationalizing’.
