Conjugate of a Complex Number

Conjugate of a Complex Number

In this section, we study about conjugate of a complex number, its geometric representation, and properties with suitable examples. 

Definition 2.3

The conjugate of the complex number x + iy is defined as the complex number x − i y.

The complex conjugate of z is denoted by  . To get the conjugate of the complex number z , simply change i by −i in z. For instance 2 − 5i is the conjugate of 2 + 5i. The product of a complex number with its conjugate is a real number.

For instance, (i) (x+iy)( x+iy) = x² – (iy) = x²+y²

(ii) (1 + 3i)(1-3i) = (1)² – (3i)² = 1 + 9 =10.

Geometrically, the conjugate of z is obtained by reflecting z on the real axis.

1. Geometrical representation of conjugate of a complex number

Two complex numbers x+iy and x-iy are conjugates to each other. The conjugate is useful in division of complex numbers. The complex number can be replaced with a real number in the denominator by multiplying the numerator and denominator by the conjugate of the denominator. This process is similar to rationalising the denominator to remove surds.

2. Properties of Complex Conjugates

Let us verify some of the properties.

Property

For any two complex numbers z₁ and z₂, we have . 

Proof

Let z₁ = x₁ + iy₁ , z₂ = x₂ + iy₂ , and x₁ , x₂ , y₁ , and y₂ ∈ R

It can be generalized by means of mathematical induction to sums involving any finite number of terms 

Property

Proof

Let z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂.

Then, z₁z₂   = ( x₁ + iy₁ )( x₂ + iy₂ ) = ( x₁ x₂ - y₁ y₂ ) + i ( x₁ y₂ + x₂ y₁ ) .

Property

A complex number z is purely imaginary if and only if z = − 

Proof

 Let z = x i + y . Then by definition   = x âˆ’ iy

 Therefore, z = −

⇔ x + iy = − (x − iy)

 â‡” 2x = 0 ⇔  x = 0

 â‡” z is purely imaginary.

Similarly, we can verify the other properties of conjugate of complex numbers.

Example 2.3

Write 3+4i / 5-12i in the xi + y form, hence find its real and imaginary parts.

Solution

To find the real and imaginary parts of [3+4i] / [5-12i] , first it should be expressed in the rectangular form x + iy. To simplify the quotient of two complex numbers, multiply the numerator and denominator by the conjugate of the denominator to eliminate i in the denominator. 

This is in the x + iy form.

 Hence real part is – 33/169 and imaginary part is 56/169

Example 2.4

Simplify  into rectangular form 

Solution

Example 2.5

If  , find the complex number z in the rectangular form 

Solution

Example 2.6

If z₁ = 3 − 2i and z₂ = 6  + 4i , find z₁/z₂ in the rectangular form 

Solution

Using the given value for z₁ and z₂ the value of z₁/z₂ =  

Example 2.7

Find z^−1, if z = (2+3i) (1− i). 

Solution

We have z = (2 + 3i )(1- i ) = (2 + 3) + (3 - 2)i = 5 + i 

⇒  z^-1 = 1/ z = 1/[5 + i ].

Multiplying the numerator and denominator by the conjugate of the denominator, we get

Example 2.8

Solution