The modulus of a complex number gives the distance of the complex number from the origin in the Argand plane, whereas the conjugate of a complex number gives the reflection of the complex number about the real axis in the Argand plane. In this section, we will discuss the modulus and conjugate of a complex number, along with a few solved examples.
Conjugate of a Complex Number
The conjugate of a complex number z = x + iy is denoted by
The geometrical representation of the complex number is shown in the figure given below:
Properties of the Conjugate of a Complex Number
Below are some properties of the conjugate of complex numbers, along with their proof.
Proof: Let z1 = a + ib and z2 = c + id
= a ± c – i(b ± d)
= a – ib ± c ± id
= a – ib ± (c – id)
|Points to Remember:
Modulus of a Complex Number
The modulus of the complex number is the distance of the point on the Argand plane representing the complex number z from the origin.
Let P is the point that denotes the complex number, z = x + iy.
Then, OP = |z| = √(x2 + y2 ).
1. |z| > 0.
2. All the complex numbers with the same modulus lie on the circle with the centre origin and radius r = |z|.
Properties of Modulus of Complex Number
Below are a few important properties of the modulus of a complex number and their proofs.
(i) |z1 z2| = |z1||z2|
Proof: let z1= a + ib and z2 = c + id
Then, |z1 z2| = |(a + ib)(c + id)|
⇒ |ac + iad + ibc + i2bd|
⇒ |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
⇒ a2 c2 + b2 d2 + a2 d2+ b2 c2
⇒ a2 c2 + b2 c2 + b2 d2 + a2 d2
⇒ (a2 + b2)c2 + (b2 + a2)d2
⇒ (a2 + b2) (c2 + d2)
(ii) |z1 / z2 | = (|z1|) / (|z2|).
Proof: |z1/z2 | = |z1 . 1/z2 |
Using the multiplicative property of modulus, we have
⇒ |z1| |1/z2|
⇒ |z1| 1/(|z2|)
⇒ (|z1 |) / (|z2|).
|Some other important results:
|z1 + z2| ≤ |z1| + |z2|
|z1 + z2| ≥ |z1| – |z2|
|z1 – z2| ≥ |z1| – |z2|.
Examples on Modulus and Conjugate of a Complex Number
Example 1: Find the conjugate of the complex number z = (1 + 2i)/(1 – 2i).
Solution: z = (1 + 2i)/(1 – 2i)
Rationalising given the complex number, we have
⇒ z = ((1 + 2i)/(1 – 2i) )× (1 + 2i)/(1 + 2i)
⇒ z = (1 + 2i)2/(12 – (2i)2)
⇒ z = (1 + 4i2 + 4i)/(1 + 4)
⇒ z = (1 – 4 + 4i)/(1 + 4)
⇒ z = (-3 + 4i)/5
Example 2: Find the modulus of the complex number z = (3 – 2i)/2i
Solution: z = (3 – 2i)/2i
⇒ z = (3 )/2i – 2i/2i
⇒ z = 3/2i – 1
⇒ z = 3i/(2i2 ) – 1
⇒ z = (-3i/2) – 1
Example 3: If z + |z| = 1 + 4i, then find the value of |z|.
Solution: Let z = x + iy
⇒ z + |z| = 1 + 4i
⇒ x + iy + |x + iy| = 1 + 4i
⇒ x + iy + √(x2 + y2 ) = 1 + 4i
⇒ y = 4 and x + √(x2 + y2 ) = 1
⇒ x + √(x2 + 42 ) = 1
⇒ √(x2 + 42 ) = 1 – x
Squaring both sides, we have
⇒ x2 + 42 = 1 + x2 – 2x
⇒ 2x = -15
or x = -15/2
Example 5: Let (α, β) be real, and z be a complex number. If z2 + αz + β = 0 has distinct roots on the line Re z = 1, then find the necessary condition.
Given Re z = 1
Let z1 = x + iy and z2 = x – iy are the roots.
Since Re z = 1, x = 1
Let z2 + αz + β = 0 has roots (1 + iy) and (1 – iy)
z1 z2 = β
(1 + iy)(1 – iy) = β
1 + y2 = β
β > 1
Example 6: If ω(≠ 1) is a cube root of unity, and (1 + ω)7 = A + Bω, then find (A, B).
Example 7: A complex number z is said to be unimodular if |z| = 1. Suppose z1 and z2 are complex numbers such that
⇒ |z1| = 2 is a circle of radius 2 and centre at origin.
Frequently Asked Questions
What do you mean by a complex number?
A complex number is a number in the form of a+ib, where a and b are real numbers and i = √-1.
What do you mean by the conjugate of a complex number?
The conjugate of a complex number z = a + ib is given by z̄ = a – ib.
Give the equation for the modulus of a complex number.
The modulus of a complex number z is given by |z| = √(x2 + y2).
What is z+z̄, if z is purely imaginary?
If z is purely imaginary, then z+z̄ = 0.