Understanding Modulus and Conjugate of a Complex Number - Testbook

Conjugate of a Complex Number

The conjugate of a complex number z, where z = x + iy, is represented by

. It denotes the reflection or the image of the complex number about the real axis on the Argand plane. This is achieved by replacing the ‘i’ with ‘- i’, resulting in the conjugate of the complex number.

 

The geometrical representation of the complex number is illustrated in the figure below:

Properties of the Conjugate of a Complex Number

The conjugate of a complex number holds certain properties. Let's discuss these properties along with their proofs.

(1)

Proof: Let z ₁ = a + ib and z ₂ = c + id

Then,

= a ± c – i(b ± d)

= a – ib ± c ± id

= a – ib ± (c – id)

(2)

• If z lies in the 1st quadrant, then
  will lie in the 2nd quadrant.
• If x + iy = f(a + ib) then x – iy = f(a – ib)
• Further, g(x + iy) = f(a + ib) ⇒ g(x – iy) = f(a – ib).

Modulus of a Complex Number

The modulus of a complex number signifies the distance of the point on the Argand plane that represents the complex number z from the origin.

Let P denote the point that represents the complex number, z = x + iy.

Then, OP = |z| = √(x ² + y ² ).

Note: 1. |z| > 0. 2. All the complex numbers with the same modulus lie on the circle with the centre at the origin and radius r = |z|.

Properties of Modulus of Complex Number

The modulus of a complex number also has a few important properties. Let's delve into these properties and their proofs.

(i) |z ₁ z ₂ | = |z ₁ ||z ₂ |

Proof: let z ₁ = a + ib and z ₂ = c + id

Then, |z ₁ z ₂ | = |(a + ib)(c + id)|

⇒ |ac + iad + ibc + i ² bd|

⇒ |ac + iad + ibc – bd|

⇒ |ac – bd + i(ad + bc)|

⇒ (ac – bd) ² + (ad+bc) ²

⇒ (ac) ² + (bd) ² – 2abcd + (ad) ² + (bc) ² + 2abcd

⇒ a ² c ² + b ² d ² + a ² d ² + b ² c ²

⇒ a ² c ² + b ² c ² + b ² d ² + a ² d ²

⇒ (a ² + b ² )c ² + (b ² + a ² )d ²

⇒ (a ² + b ² ) (c ² + d ² )

⇒ |z ₁ ||z ₂ |.

(ii) |z ₁ / z ₂ | = (|z ₁ |) / (|z ₂ |).

Proof: |z ₁ /z ₂ | = |z ₁ . 1/z ₂ |

Applying the multiplicative property of modulus, we have

⇒ |z ₁ | |1/z ₂ |

⇒ |z ₁ | 1/(|z ₂ |)

⇒ (|z ₁ |) / (|z ₂ |).

Examples on Modulus and Conjugate of a Complex Number

Example 1: Find the conjugate of the complex number z = (3 + 4i)/(2 – 3i).

Solution: z = (3 + 4i)/(2 – 3i)

To rationalize the complex number, we multiply the numerator and denominator by the conjugate of the denominator.

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

⇒ z = (6 + 9i + 8i + 12i ² )/(4 + 6i - 6i - 9)

⇒ z = (6 - 12 + 17i)/(-5)

⇒ z = (-6 + 17i)/(-5)

Example 2: Find the modulus of the complex number z = (5 – 3i)/4i

Solution: z = (5 – 3i)/4i

⇒ z = (5)/4i – 3i/4i

⇒ z = 5/4i – 3/4

⇒ z = 5i/(4i ² ) – 3/4

⇒ z = (-5i/4) – 3/4

Example 3: If z + |z| = 3 + 7i, then find the value of |z|.

Solution: Let z = x + iy

⇒ z + |z| = 3 + 7i

⇒ x + iy + |x + iy| = 3 + 7i

⇒ x + iy + √(x ² + y ² ) = 3 + 7i

⇒ y = 7 and x + √(x ² + y ² ) = 3

⇒ x + √(x ² + 7 ² ) = 3

⇒ √(x ² + 49) = 3 - x

Squaring both sides, we get

⇒ x ² + 49 = 9 + x ² - 6x

⇒ 6x = 40

or x = 40/6 = 20/3