Euler was the first mathematician to introduce the symbol (iota) for i.e. a solution of with the property . He also called this symbol as the imaginary unit.

There are two basic ways to represent a complex number algebraically:

Cartesian form:

polar form: with

Notice that, in the polar form , we use the identity

.

Here we consider as the definition of the expression . Anyway, there is a name for this identity: Euler’s formula. One of the most amazing things in complex numbers is the following so–called addition formula:

Using Euler’s formula, we can rewrite it as

Multiplying out the right hand side and comparing the real and imaginary parts of both sides, we have arrived at

which are well-known (but by no means obvious) identities in trigonometry

Geometrically, is represented as a vector with as its coordinates in a Cartesian plane, called complex plane.

The magnitude of this vector is called the absolute value or the modulus of and is denoted by . It is equal to r given in its polar form:

The angle between this vector and the axis is given by .

When a complex number is represented in Cartesian form, say , the real number is called the real part of and the real number is called the imaginary part of . We write

,

Purely real and purely imaginary complex numbers: A complex number is purely real if its imaginary part is zero i.e. and purely imaginary if its real part is zero i.e. .

Set of complex numbers : The set of all complex numbers is denoted by i.e.

Since a real number can be written as . Therefore, every real number is a complex number. Hence, , where is the set of all real numbers.

Two complex numbers are equal if and only if both of their real parts and their imaginary parts are equal.

Thus For and , we have and

Using the Cartesian form, addition and multiplication of complex numbers are straightforward, as long as we keep in mind that :

The usual algebraic identities still hold for complex numbers, such as

and

The relation between the Cartesian form and the polar form is given by

,

If we write for the point in the plane with Cartesian coordinates , then is the length of the line segment and is the angle between the line and the axis. The angle in the polar form is called the argument of .

Every complex number has a “twin sister” , called the complex conjugate of . The twins and do not quite have exactly the same look. They are more like mirror images to each other. We put the pair and together in the Cartesian form and the polar form as follows:

It is clear from the above identities that the complex conjugate of is . Also, is a real number if and only if .

One of the most useful identities about complex numbers is the following:

The proof of this is simple: writing , we have

**Integral Powers of **

To find the value of in for , we may follow the following steps:

If , then write

**Imaginary Quantities**

The square root of a negative real number is called an imaginary quantity or an imaginary number.

Theorem: If are positive real numbers, then

Proof:

LHS RHS. Hence proved.

Note: For any two real numbers it true only when at least one of and is either positive or zero. In other words is not valid if and both are negative.

Note: For any positive real number , we have

**Equality of Complex Numbers**

Two complex numbers and are equal if , and i.e. and .

Thus, and

**Addition of Complex Numbers**

Let and be two complex numbers. Then their sum is defined as the complex number

It follows from this definition that the sum is a complex number such that and,

Properties of addition of complex numbers

i) Addition is Commutative: For any two complex numbers,

ii) Addition is Associative: For any three complex numbers ,

iii) Existence of Additive Identity: The complex number is the identity element for addition: i.e. for all .

iv) Existence of Additive Inverse: For any complex number , there exists such that . The complex number is called the additive inverse of .

**Subtraction of Complex Numbers**

Let and be two complex numbers. Then the subtraction of from is denoted by and is defined as the addition of and . Thus,

**Multiplication of Complex Numbers**

Let and be two complex numbers. Then the multiplication of with is denoted and is defined as the complex number

Properties of Multiplication:

i) Multiplication is commutative: For any two complex numbers and for all .

ii) Multiplication is associative: For any three complex numbers , then for all

iii) Existence of identity element for multiplication: The complex number is the identity element for multiplication i.e. for every complex number .

iv) Existence of multiplicative inverse: Corresponding to every non-zero complex number , there exists a complex number such that .

The multiplicative inverse of is denoted by or

v) Multiplication of complex numbers is distributive over addition of complex numbers : For any three complex numbers and .

**Division of Complex Numbers**

The division of a complex number by a non-zero complex number is defined as the multiplication of by the multiplicative inverse of and is denoted by .

Let and be two complex numbers.

Therefore

**Properties of Conjugate **

If and are complex numbers, then

i)

ii)

iii)

iv) is purely real

v) is purely imaginary

vi)

vii)

viii)

ix)

x)

**Modulus of Complex Number**

The modulus of a complex number is denoted and is defined as

Clearly for all .

Properties of modulus:

i)

ii)

iii)

iv)

v)

vi)

vii)

viii)

ix)

x) , where

**Reciprocal of a Complex Number**

Let be a non-zero complex number. Then,

Clearly , is equal to the multiplicative inverse of

Thus, the multiplicative inverse of a non-zero complex number z is same as its reciprocal and is given by

**Square Roots of Complex Numbers**

Let be a complex number such that where and are real numbers. Then,

On equating real and imaginary parts, we get

… … … … … i)

and … … … … … ii)

Now,

… … … … … iii)

Solving the equations (i) and (ii), we get

If is positive, then and are of the same sign

If is negative, then and are of different sign

**Representation of Complex Numbers**

A complex number can be represented in the following forms: (i) Geometrical form (ii) Vectorial form (iii) Trigonometrical form or, Polar form.

Geometrical Representation of Complex Numbers

The plane in which we represent a complex number geometrically is known as the complex plane or Argand plane or the Gaussian plane. The point P, plotted on the Argand plane, is called the Argand diagram.

Let us represent on Argand Plane

If a complex number is purely real, then its imaginary part is zero. Therefore, a purely real number is represented by a point on x-axis. A purely imaginary complex number is represented by a point on y-axis. That is why x-axis is known as the real axis and y-axis, as the imaginary axis.

The length of the line segment is called the modulus of and is denoted by .

The angle which makes with positive direction of x-axis in anticlockwise sense is called the argument or amplitude of and is denoted by arg(z) or amp(z).

Polar or Trigonometrical from of Complex Numbers

Let be a complex number represented by a point in the Argand plane. Then by the geometrical representation of , we obtain

and

In , we obtain

and

, where and

This form of is called a polar form of . If we use the general value of the argument of then the polar form of is given by

, where and is an integer.

Case 1: Polar form of when and :

In this case, we have .

So, the polar form of is

Case 2: Polar form of when and

In this case, we have .

So, the polar form of is

Case 3: Polar form of when and

ln this case, we have .

So, the polar form of is given by

Case 4: Polar form of when and

In this case, we have .

So, the polar form of is