Complex Conjugate

A complex number is a number represented in the form of (x + i y); where x & y are real numbers, and i = √(-1) is called iota (an imaginary unit). It is also known as imaginary numbers or quantities. A complex number can be purely real or purely imaginary depending upon the values of x & y.

Examples: 

(5 + 3 i), (0 + 7 i), (-4 + 9 i), (1 + 0 i), (0 – i), etc.

Now we will discuss two very important concepts of complex numbers which are Conjugate and Modulus.

Conjugate of a Complex Number

Conjugate of a complex number is another complex number whose real parts Re(z) are equal and imaginary parts Im(z) are equal in magnitude but opposite in sign. Sometimes conjugate of a complex number is also called a complex conjugate. Conjugate of a complex number z is represented by  and (z & ) together known as complex-conjugate pair because z and  are conjugate of each other. If z = (x + i y) is a complex number then conjugate of z is defined as   = (x – i y). From the relation between z and , we can say conjugate of a complex is obtained by replacing i with (-i). The geometrical meaning of conjugate of a complex number  is the reflection or mirror image of the complex number z about the real axis (X-axis) in the complex plane or argand plane.

Conjugate of a complex number in Argand Plane

Properties of Conjugate

If z, z₁, & z₂ are complex numbers then they will have the following conjugate properties.

1. Conjugate of a purely real complex number is the number itself (z = ) i.e. conjugate of (7 + 0 i) = (7 – 0 i) = 7
2. Conjugate of a purely imaginary complex number is negative of that number (z +  = 0) i.e. conjugate of (0 -7 i) = (0 + 7 i) = 7 i
3.  = z
4. z +  = 2 Re(z)
5. z –  = 2 i . Im(z)
6. z  = {Re(z)}² + {Im(z)}²
7.  = 
8.  = 
9.  =  
10. z = (z₁ / z₂) then  =  /  ; z₂ ≠ 0

Note: In order to find out the conjugate of a complex number that complex number must be in its standard form that is Z = (x + i y). If the complex number is not in its standard form then it has to be converted into its standard form before finding its complex conjugate.

Examples

Example 1: If z = (5 + 7 i) is a complex number then its conjugate is given by

Solution:

 = (5 – 7 i).

Example 2: If z = 1 / (4 + 3 i) is a complex number then its conjugate is given by

Solution:

First z has to be converted into its standard form by multiplying the numerator and

denominator with the conjugate of (4 + 3 i)

z = (1 / (4 + 3 i)) × ((4 – 3 i) / (4 – 3 i))

z = (4 – 3 i) / (16 + 9)

z = (4 / 25) – (3 / 25) i

The conjugate of z is  = (4 / 25) + (3 / 25) i

Example 3: If (a + i b) is a complex number which is the complex conjugate of (8 – 3 i)  then values of a and b are  given by

Solution:

Let z = a + i b

 = (8 + 3 i)

z = 

z = (8 + 3 i)

Two complex numbers are equal only when their corresponding real & imaginary parts are equal

Equating the real and imaginary parts of z & (8 + 3 i)

Re(z) = a = 8 

Im(z) = b = 3

Hence 8 & 3 are the respective values of a and b.

Modulus of a Complex Number

The modulus of a complex number z = (x + i y) is defined as |z| = √(x² + y²); where x = Re(z) and y = Im(z). |z| ≥ 0 for all complex numbers. We cannot compare two complex numbers z₁ and z₂ i.e. (z₁ > z₂) or (z₁ < z₂) has no significance but (|z₁| > |z₂|) or (|z₁| < |z₂|) has definite meaning because |z₁| & |z₂| are real numbers. The geometrical meaning of modulus of a complex number (x + i y) is the distance of the point P (x, y) from the origin O (0, 0) in the complex plane or argand plane i.e. |z| =  = √((x – 0)² + (y – 0)²) = √(x² + y²).

Modulus of a complex number on Argand plane

Properties of Modulus

If z, z₁, z₂ & z_n are complex numbers then they will have the following modulus properties.

1. If a complex number z has equal real and imaginary parts then its modulus is zero i.e. Re(z) = Im(z) ⇒ |z| = 0
2. Real and imaginary part of a complex number z always lies between -|z| and  |z|  i.e. -|z| ≤ Re(z) ≤ |z| and -|z| ≤ Im(z) ≤ |z|
3. |z| = || = |-z|
4.  = |z|²
5. |Z₁ .  Z₂ …….. Z_n| = |Z₁| . |Z₂| …….. |Z_n|
6. |Z₁ / Z₂| = |Z₁| / |Z₂| ; Z₂ ≠ 0
7. |Z₁ + Z₂|² = |Z₁|² + |Z₂|² + 2 Re()
8. |Z₁ – Z₂|² = |Z₁|² + |Z₂|² – 2 Re()
9. |Z₁ + Z₂|² + |Z₁ – Z₂|² = 2( |Z₁|² + |Z₂|²)
10. |b. Z₁ + a. Z₂|² + |a. Z₁ – b. Z₂|² = (a² + b²) ( |Z₁|² + |Z₂|²) ; here a & b are real numbers.

Note: To find out the modulus of a complex number that complex number must be in its standard form that is Z = (x + i y). If the complex number is not in its standard form then it has to be converted into its standard form before finding its modulus.

Examples

Example 1: If (8 + 3 i) is a complex number then its modulus is given by

Solution:

|8 + 3 i| = √(8² + 3²) 

|8 + 3 i| = √(64 + 9)

|8 + 3 i| = √(73)

Example 2: If (3 + 4 i) and (√3 + √6 i) are two complex numbers then the modulus of their product is given by the product of their individual modulus.

Solution:

|3 + 4 i| = √(3² + 4²)

|3 + 4 i| = √(9 + 16)

|3 + 4 i| = √(25) = 5

|√3 + √6 i| = √((√3)² + (√6)²)

|√3 + √6 i| = √(3 + 6) 

|√3 + √6 i| = √9 =3

|(3 + 4 i).(√3 + √6 i)| = 5 × 3

|(3 + 4 i).(√3 + √6 i)| = 15

Example 3: If z = (2 . i) / (3 – 4 i) is a complex number then its modulus is given by

Solution:

First z has to be converted into its standard form by multiplying the numerator and

denominator with the conjugate of (3 – 4 i)

z = ((2.i) / (3 – 4 i)) × ((3 + 4 i) / (3 + 4 i))

z = (8 + 6 i) / (9 + 16)

z = (8 + 6 i) / 25

z = (8 / 25) + (6 / 25) i

The modulus of z is |z| = √((8 / 25)² + (6 / 25)²)

|z| = √((64 + 36) /625)

|z| = √(100 /625)

|z| = 10 / 25

|z| = 2 / 5