Operations Involving Cartesian Complex Numbers

(a) Addition and subtraction

and 

Thus, 

and 

(b) Multiplication

But j²=–1, thus,

For example,

(c) Complex conjugate

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,

For example, 

and 

(d) Division

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. For example,

The elimination of the imaginary part of the denominator by multiplying both the numerator and denominator by the conjugate of the denominator is often termed rationalizing.

Example 7.1

In an electrical circuit the total impedance Z_T is given by:

Determine Z_T in (a+jb) form, correct to two decimal places, when Z₁=5 –j3, Z₂=4+j7 and Z₃=3.9 –j6.7.

Solution

Thus, 

=8.65 –j6.26, correct to two decimal places.

Example 7.2

Given Z₁=3+j4 and Z₂=2 –j5 determine in Cartesian form correct to three decimal places:

Solution