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Definition, Geometrical representation, Properties, Proof, Solved Example Problems - Conjugate of a Complex Number | 12th Mathematics : UNIT 2 : Complex Numbers

Chapter: 12th Mathematics : UNIT 2 : Complex Numbers

Conjugate of a Complex Number

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

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 xi 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   = − iy

 Therefore, z = −

x + iy = − (xiy)

  2x = 0   x = 0

  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




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12th Mathematics : UNIT 2 : Complex Numbers : Conjugate of a Complex Number | Definition, Geometrical representation, Properties, Proof, Solved Example Problems

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12th Mathematics : UNIT 2 : Complex Numbers


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