Representation of a Complex Number

A complex number is a number that can be expressed in the form of a + ib where a represents the real part, and b is the imaginary part; i is the imaginary unit which is defined as the square root of -1, or we can have i as the solution of x² = -1. In this article, you will learn the representation of a complex number.

A complex number, whose real part is zero, is said to be a purely imaginary number, and the points of these numbers lie on the vertical axis of the complex plane. Similarly, a complex number, whose imaginary part is zero, can be viewed as a real number, and its point lies on the horizontal axis of the complex plane.

Complex Number Geometrical Representation

Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. Thus, if given a complex number a+bi, it can be identified as a point P(a, b) in the complex plane. Complex numbers can also be represented in a polar form that associates each complex number with its distance from the origin as its magnitude and with a particular angle, and this is called the argument of the complex number.

Plotting a Complex Number

To plot a complex number a + ib, which can be thought of as a point P(a, b), we consider the s-plane similar to the x-y plane relating to the normal Cartesian system.

S-plane is also called as Argand plane, complex-plane or Argand diagram, named after Jean-Robert Argand. It plots the real and imaginary values a and b, respectively, of the point P on the horizontal axis, called the real axis and on the vertical axis, called the imaginary axis.

Consider a point (2, 3) in a complex plane. It can be defined as 2+3i in terms of a complex number plotted in the first quadrant, as both the real and imaginary terms are positive. Similarly, we can have -2 + 3i as (-2, 3) in the second quadrant, (-2, -3) as -2-3i in the third quadrant and (2, -3) as 2 – 3i in the fourth quadrant.

Polar Complex Plane

The above figure represents a line joining some points in the complex plane, whose length is r and makes an angle Ɵ with the real axis. From the line, draw a line projecting parallel to the real axis and meeting the imaginary axis and let it be b. Similarly, draw a line drawn from the plot touching the real axis parallel to the imaginary axis, and let it be a. Then, by Pythagoras’ theorem, we have

If the complex number has no imaginary part in it or b = 0, then r = |x|, i.e., the absolute value of the real number equals the absolute value of the complex number.

Consider the polar form z = r cos θ + r i sin θ

⇒ z = r (cos θ + i sin θ)

By using Euler’s formula, we have z = r e^iθ

⇒ z = r < θ

Also Read

Modulus and Conjugate of a Complex Number

Important Complex Numbers Properties

• For two complex numbers, z₁ and z_2, to be equal, the corresponding real and imaginary parts of both complex numbers are supposed to be equal.

Consider a complex number z₁ = a + ib and z₂ = c + id.

Then, if z₁ = z₂ ⇒ a = c and b = d

In general, Re(z₁) = Re(z₂) and Im(z₁) = Im(z₂) implies z₁ = z₂.

• Conjugate of a complex number

Consider a complex number z = a + ib. Then, its conjugate is expressed as z^* = a – ib. It is often called the reflection of z about the real axis.

Note: Conjugating a complex number twice results in the complex number itself, i.e., z** = z.

Consider z = a + ib. Then, z* = a – ib and z** = a + ib; thus, z** = z is justified.

Moreover, Re(z*) = Re(z) and |z*| = |z|.

Similarly, Im(z*) = -Im(z) and arg(z*) = -arg(z).

zz* = |z|² = |z*|²

Re(z) = (z + z*)/2

Im(z) = (z – z*)/2i

• Arithmetic operation on complex numbers

Consider z₁= a + ib and z₂ = c + id;

Addition: z₁ + z₂ = (a + c) + i(b + d)

Subtraction: z₁ – z₂ = (a – c) + i(b – d)

Multiplication: z₁ * z₂ = ac – bd + i(ad + bc).

Division: To work with complex division, we need to take the conjugate of the denominator and multiply both the numerator and denominator by the conjugate value.

• Complex number commutative property

z₁ + z₂ = z₂ + z₁

z₁ * z₂ = z₂ * z₁

|z₁ + z₂ | ≤ |z₁| + |z₂ |

• Construction as ordered pairs

(a, b) + (c, d) = (a + c, b + d)

(a, b) . (c, d) = (ac + bd, bc + ad)

Important Concepts Related to Complex Numbers

Complex Analysis

The study of functions of a complex variable is known as complex analysis. They have four-dimensional graphs and may be usefully illustrated by colour-coding a three-dimensional graph to suggest four dimensions.

Holomorphic Functions

A function f: C -> C is called holomorphic if it satisfies the Cauchy-Riemann equations.

Consider f(z) = u z + v z*, with complex coefficients u and v. This map is holomorphic if and only if b = 0.

Solved Problems

Problem 1: Represent the complex number z = 1+ i√3 in the polar form.

Solution:

Let r cos θ = 1 and r sin θ = √3.

By squaring and adding, we get

r² (cos²θ +sin²θ ) = 1+3

r² = 4

i.e. r = 2

Substitute r in r cos θ = 1

Therefore, cos θ = 1/2

Also, sin θ = √3/2

⇒ θ = π/3

Hence, the required polar form is z = 2 (cos  π/3 + i sin π/3).

Problem 2: Let a complex number be defined by z = 2 + 5i and another complex number a = 3 + xi and b = y + 2i. Given that z = a + b. Then, find the values of x and y.

Solution: z = a + b

(2 + 5i) = (3 + xi) + (y + 2i)

⇒ 2 = 3 + y and 5 = x + 2

⇒ Solving, we have x = 3 and y = -1.

Problem 3: Find the conjugate of z₁ if z₂ + z₃ = 0 and z₁ = 3 * z₃; given z₂ = 5 + 5i and z₃ = x + yi. Also, find the magnitude and argument of the conjugate of z₂.

Solution: z₂ + z₃ = 0 implies 5 + x = 0 and 5 + y = 0.

Solving, we have x = – 5 and y = -5.

Thus, z₃ = -5 – 5i.

Given z₁ = 3 * z₃

⇒ 3(-5 – 5i) = -15 -15i.

We need to find the complex conjugate of z₁.

Hence, z₁* = -15 + 15i.

Now, z₂ * = 5 – 5i.

|z₂ |= √(25 + 25) = 7.07

And θ = tan^-1 (y/x) = tan^-1 (-5/5) = tan^-1 (-1) = –π/4.

Problem 4:  

Solution:

Let 

Hence,

Problem 5:   

Solution:

Let 

Problem 6:

A) I quadrant

B) II quadrant

C) III quadrant

D) IV quadrant

Solution:

Problem 7:

Solution:

Let

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