Complex Number Formula

Table of Contents

Complex Number

A combination of real and imaginary numbers in the form of z = x + iy, where x, y ∈ R is called a complex number.

Real and Imaginary Parts of a Complex Number

Let z = x + iy be a complex number

then ‘x’ is called the real part of z; Re (Z) and
‘y’ is called the imaginary part of z and it may be denoted as Im (Z)

Iota

Mathematician Euler introduced the symbol i (read as iota) for √-1 with the property i² + 1 = 0 i.e. i² = -1. He also called this symbol the imaginary unit.

i = √-1
i² = -1
i³ = – i
i⁴= 1

Algebra of Complex Numbers

Addition of Complex Numbers
Subtraction of Complex Numbers
Multiplication of Complex Numbers
Division of Complex Numbers

Let z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂ be any two complex numbers, then their sum will be defined as

(1) Addition of Complex Numbers

z₁ + z₂= (x₁ + iy₁) + (x₂ + iy₂)

or, z₁ + z₂= (x₁ + x₂) + i(y₁ + y₂)

Properties

Closure Property – The sum of two complex numbers is also a complex number.
Commutative Property – z₁ + z₂= z₂ + z₁
Associative Property – (z₁ + z₂)+ z₃ = z₁ + (z₂+ z₃)

(2) Subtraction of Complex Numbers

z₁ – z₂= (x₁ + iy₁) – (x₂ + iy₂)

or, z₁ + z₂= (x₁ – x₂) + i(y₁ – y₂)

Note: z₁ – z₂ follows the closure property, but this operation is neither commutative nor associative.

(3) Multiplication of Complex Numbers

z₁ z₂ = (x₁ + iy₁) × (x₂ + iy₂)

= (x₁x₂ – y₁ y₂) + i(x₁y₂ + x₂y₁)

Properties

Closure Property – The sum of two complex numbers is also a complex number.
Commutative Property – z₁ × z₂= z₂ × z₁
Associative Property – (z₁ × z₂)× z₃ = z₁ × (z₂× z₃)
Distributive Property – z₁ × (z₂+ z₃) = z₁ × z₂+ z₁ × z₃

(4) Division of Complex Numbers

{z_1\over z_2} = {(x_1+iy_1)\over(x_2+iy_2)}

Note: division of two complex numbers ${z_1\over z_2}$ follows the closure property, but this operation is neither commutative nor associative.

Identities Related to Complex Numbers

For any complex number z₁, z₂, we have

(z₁ + z₂)² = z₁² + 2z₁z₂ + z₂²
(z₁ – z₂)² = z₁² – 2z₁z₂ + z₂²
(z₁ + z₂)³ = z₁³ + 3z₁²z₂ + 3z₁z₂²+ z₂³
(z₁ – z₂)³ = z₁³ – 3z₁²z₂ + 3z₁z₂²– z₂³
z₁² – z₂² = (z₁ + z₂)(z₁ – z₂)

Conjugate of a Complex Number

Let z = x + iy is a complex number, then the conjugate of z is denoted by $\overline{z}$

i.e. $\overline{z}$ = x – iy

Properties of Conjugate of Complex Number

For any complex number z, z₁, z₂, we have

$\overline {\overline{z} }$ =z
$\overline{z_1 + z_2} =\overline{z_1} + \overline{z_2}$
$\overline{z_1 - z_2} =\overline{z_1} - \overline{z_2}$
$\overline{z_1 . z_2} =\overline{z_1} . \overline{z_2}$
$(\overline{{z_1 \over z_2}}) = {\overline{z_1}\over \overline{z_2} }$

Reciprocal/Multiplicative Inverse of a Complex Number

Let z = x + iy is a non-zero complex number, then the multiplicative inverse

z^-1 = $1\over z$ = $1\over x+iy$

= ${1\over x+iy} × {x-iy\over x-iy}$

Modulus of a Complex Number

Let z = x + iy, then the modulus or magnitude of z is denoted by |z| and given by

|z| = \sqrt{x^2+y^2}

Properties of Modulus

|z| ≥ 0
|z| = | $\overline{z}$ |
z $\overline{z}$ = |z|²
|z₁| = |z₂|
$|{z_1\over z_2}|={|z_1|\over|z_2|}$
|z₁ ± z₂| ≤ |z₁| + |z₂| and |z₁ ± z₂| ≥ |z₁| – |z₂|
|zⁿ| = |z|ⁿ

Argand Plane and Argument

Argand Plane: Any complex number z = x + iy can be represented geometrically by a point (x, y) in a plane, called the Argand plane or Gaussian plane.

The length of the line segment OZ is the modulus of z,

i.e. |z| = length of OZ = $\sqrt{x^2+y^2}$

Argument

The angle made by the line joining point z to the origin, with the positive direction of real axis is called argument of that complex number. It is denoted by the symbol arg (z) or amp (z).

arg (z) = θ = $tan^{-1}({y\over x})$

Principal Value of Argument

The value of the argument which lies in the interval (-π, π} is called the principal value of the argument.

If x > 0 and y > 0, then arg (z) = θ
If x < 0 and y > 0, then arg (z) = π – θ
If x < 0 and y < 0, then arg (z) = – (π – θ)
If x > 0 and y < 0, then arg (z) = – θ

where, θ = $tan^{-1}({y\over x})$

Polar Form of a Complex Number

If z = x + iy is a complex number, then z can be written as z = r(cos θ + i sin θ), where θ = arg (z) and r = $\sqrt{x^2+y^2}$ this is called polar form.