# Complex Number Formula

## 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

2. Subtraction of Complex Numbers
3. Multiplication of Complex Numbers
4. Division of Complex Numbers

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

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

1. (z1 +  z2)2 = z12 + 2z1z2 + z22
2. (z1 –  z2)2 = z12 – 2z1z2 + z22
3. (z1 +  z2)3 = z13 + 3z12z2 + 3z1z22+ z23
4. (z1 –  z2)3 = z13 – 3z12z2 + 3z1z22– z23
5. 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.

1. If x > 0 and y > 0, then arg (z) = θ
2. If x < 0 and y > 0, then arg (z) = π – θ
3. If x < 0 and y < 0, then arg (z) = – (π – θ)
4. 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.

## Eulerian Form of a Complex Number

If z = x + iy is a complex number, then it can be written as

z = re

Where, θ = arg (z) and r = \sqrt{x^2+y^2}

This is the Eulerian form and eiθ = cos θ + isin θ and e-iθ = cos θ – isin θ.

## Cube root of Unity

Cube root of unity are 1, ω, ω2

Where, ω = -1+i \sqrt{3}\over 2

and -1-i \sqrt{3}\over 2

### Properties of Cube root of unity

1. 1 + ω + ω2 = 0
2. ω3 = 1

## Square Root of a Complex Number

If z = x + iy, then

\sqrt{z}=\sqrt{x+iy}

\pm \Bigg[{ \sqrt{|z|+x} \over 2}+i{ \sqrt{|z|-x} \over 2}\Bigg], for y > 0

\pm \Bigg[{ \sqrt{|z|+x} \over 2}-i{ \sqrt{|z|-x} \over 2}\Bigg], for y < 0

Subscribe
Notify of