Table of Contents

Toggle## 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 *i*^{2} + 1 = 0 i.e. *i*^{2} = -1. He also called this symbol the imaginary unit.

- i = √-1
- i
^{2}= -1 - i
^{3}= – i - i
^{4 }= 1

## Algebra of Complex Numbers

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

Let z_{1} = x_{1} + iy_{1} and z_{2} = x_{2} + iy_{2} be any two complex numbers, then their sum will be defined as

**(1) Addition of Complex Numbers**

z_{1} + z_{2 }= (x_{1} + iy_{1}) + (x_{2} + iy_{2})

or, z_{1} + z_{2 }= (x_{1} + x_{2}) + i(y_{1} + y_{2})

**Properties **

**Closure Property –**The sum of two complex numbers is also a complex number.**Commutative Property –**z_{1}+ z_{2 }= z_{2}+ z_{1}**Associative Property –**(z_{1}+ z_{2})+ z_{3}= z_{1}+ (z_{2 }+ z_{3})

**(2) Subtraction of Complex Numbers**

z_{1} – z_{2 }= (x_{1} + iy_{1}) – (x_{2} + iy_{2})

or, z_{1} + z_{2 }= (x_{1} – x_{2}) + i(y_{1} – y_{2})

*Note: z _{1} – z_{2} follows the closure property, but this operation is neither commutative nor associative.*

**(3) Multiplication of Complex Numbers**

z_{1} z_{2} = (x_{1} + iy_{1}) × (x_{2} + iy_{2})

= (x_{1}x_{2} – y_{1} y_{2}) + i(x_{1}y_{2} + x_{2}y_{1})

**Properties **

**Closure Property –**The sum of two complex numbers is also a complex number.**Commutative Property –**z_{1}× z_{2 }= z_{2}× z_{1}**Associative Property –**(z_{1}× z_{2})× z_{3}= z_{1}× (z_{2 }× z_{3})**Distributive Property –**z_{1}× (z_{2 }+ z_{3}) = z_{1}× z_{2 }+ z_{1}× z_{3}

**(4) Division of Complex Numbers**

*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 z_{1}, z_{2}, we have

- (z
_{1}+ z_{2})^{2}= z_{1}^{2}+ 2z_{1}z_{2}+ z_{2}^{2} - (z
_{1}– z_{2})^{2}= z_{1}^{2}– 2z_{1}z_{2}+ z_{2}^{2} - (z
_{1}+ z_{2})^{3}= z_{1}^{3}+ 3z_{1}^{2}z_{2}+ 3z_{1}z_{2}^{2}+ z_{2}^{3} - (z
_{1}– z_{2})^{3}= z_{1}^{3}– 3z_{1}^{2}z_{2}+ 3z_{1}z_{2}^{2}– z_{2}^{3} - z
_{1}^{2}– z_{2}^{2}= (z_{1}+ z_{2})(z_{1}– z_{2})

**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, z_{1}, z_{2}, 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} - |z
_{1}| = |z_{2}| - |{z_1\over z_2}|={|z_1|\over|z_2|}
- |z
_{1}± z_{2}| ≤ |z_{1}| + |z_{2}| and |z_{1}± z_{2}| ≥ |z_{1}| – |z_{2}| - |z
^{n}| = |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.

## Eulerian Form of a Complex Number

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

z = re^{iθ}

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

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

## Cube root of Unity

Cube root of unity are 1, ω, ω^{2}

Where, ω = -1+i \sqrt{3}\over 2

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

### Properties of Cube root of unity

- 1 + ω + ω
^{2}= 0 - ω
^{3}= 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