3. The Modulus and the Conjugate of a Complex Numbers

The Modulus and the Conjugate of a Complex Numbers

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.

The product of two complex conjugate numbers is real. = a - ib. , which is real. Thus, the modulus of any complex number is equal to the positive square root of the product of the complex number and its conjugate complex number.

Conjugate of a Complex Number

Conjugate of a complex number z = x + iy is denoted by

= x – iy. It is the reflection of the complex number about the real axis on Argand’s plane or the image of the complex number about the real axis on Argand’s plane. If we replace the ‘i’ with ‘- i’, we get conjugate of the complex number.

Geometrical representation of the complex number is shown in the figure given below:

Properties of conjugate of a complex number:

If z, z₁ and z₂ are complex number, then

Modulus of Complex Number

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| = r = √(x² + y² ).

Properties of Modulus of Complex Number:

(v) For any two complex numbers z₁ and z₂, we have |z₁ + z₂| ≤ |z₁| + |z₂|.

Proof

The distance between the two points z₁ and z₂ in complex plane is | z₁ - z₂ |

If we consider origin, z₁ and z₂ as vertices of a triangle, by the similar argument we have

Geometrical interpretation

Now consider the triangle shown in figure with vertices O, z₁  or z₂ , and z₁ + z₂. We know from geometry that the length of the side of the triangle corresponding to the vector  z₁ + z₂ cannot be greater than the sum of the lengths of the remaining two sides. This is the reason for calling the property as "Triangle Inequality".

It can be generalized by means of mathematical induction to finite number of terms:

 |z₁ + z₂ + z₃ + …. + z_n | ≤ |z₁| + |z₂| + |z₃| + … + |z_n| for n = 2,3,….

Note:

1. |z| > 0.

2. All the complex number with same modulus lie on the circle with centre origin and radius r = |z|.

Application:

Spectrum Analyzer

Those cool displays you see when music is playing? Yes, Complex Numbers are used to calculate them! Using something called "Fourier Transforms".

In fact many clever things can be done with sound using Complex Numbers, like filtering out sounds, hearing whispers in a crowd and so on.

It is part of a subject called "Signal Processing".

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

Solution: z = (1 + 2i)/(1 – 2i)

Rationalising given 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

⇒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 side we have

⇒ x² + 4² = 1 + x² – 2x

⇒ 2x = -15

or x = -15/2

Argand Plane and Polar Representation

The plane having a complex number assigned to each of its point is called the

complex plane or the Argand plane.

Argument of Complex Numbers 

The argument of a complex number is defined as the angle inclined from the real axis in the direction of the complex number represented on the complex plane. It is denoted by “θ” or “φ”. It is measured in the standard unit called “radians”.

The real and complex components of coordinates are found in terms of r and θ where r is the length of the vector, and θ is the angle made with the real axis.

The point P is uniquely determined by the ordered pair of real numbers (r, ), called the polar

coordinates of the point P.

Polar Form of Complex Numbers 

The argument of a complex number is defined as the angle inclined from the real axis in the direction of the complex number represented on the complex plane. It is denoted by “θ” or “φ”. It is measured in the standard unit called “radians”.

The real and complex components of coordinates are found in terms of r and θ where r is the length of the vector, and θ is the angle made with the real axis.

We have, x = r cos q, y = r sin q and therefore, z = r (cos q + i sin q).

The latter is said to be the polar form of the complex number.

q is called the argument (or amplitude) of z which is denoted by arg z. and

r = |z| =√(x² + y²) is the modulus of z

For any complex number z ¹ 0, there corresponds only one value of q in 0 £ q < 2p.

The value of q such that – p < q £ p, called principal argument of z and is denoted by arg z.

Properties of Argument of Complex Numbers

Suppose that z be a nonzero complex number and n be some integer, then

Let us assume, z₁ and z₂ be the two complex numbers, the following identities are: