Complex Numbers Notes for IIT JEE, Download PDF!

Complex Number is an important topic from the JEE Main exam point of view. Every year 2-3 questions are asked. Further, the concept of complex numbers, iota, quadratic equation and other included topics are used very often in different topics of JEE Main syllabus. This short notes on Complex Number will help you in revising the topic before the JEE Main & IIT JEE Advanced Exam. You can also download Complex Number notes PDF at end of the post.

Complex Numbers

1. Introduction to complex numbers

A number of the form x+ iy where x and y belong to the set of real numbers and is called a complex number. Here i(iota) is defined as the Fundamental Imaginary Unit.

Iota is nothing but the root of the equation. The role of iota in a complex number is to keep the real part and the imaginary part separate. Here x is called the real part of a complex number z and denoted by Re(z) while the imaginary part y is denoted by Im(z).

Therefore z=x+ iy=Re(z)+i Im(z)

A complex is often defined as an ordered pair of real numbers x and y, and is denoted by (x, y).

A complex number x +iy may also be defined as a 2-dimensional vector in XY plane with the point of initiation as origin and point of termination as (x,y). In such a case the unit vector along the positive x-axis is 1 while that along the positive y-axis is i. When complex numbers are plotted as vectors on such a plane they must follow all the properties of the vector. Such a plane is called the Argand Plane.

a) Two complex numbers z₁ and z₂ are said to be equal if and only if their real and imaginary parts are equal separately.

b) Two unequal complex numbers do not possess order property i.e x₁+iy₁>x₂+iy₂or x₁+iy₁<x₂+iy₂does not make any sense.

c) A real number can be a complex number with imaginary part 0

d) A complex number z is said to be a purely real number if Im(z)=0 and it lies on the x-axis while it is said to be a purely imaginary number if Re(z)=0 and it lies on the y-axis.

2. Algebra of Complex Numbers

a) Addition: (x₁+iy₁) + (x₂+iy₂) = (x₁+x₂) + i(y₁+y₂)

b) Subtraction: (x₁+iy₁) - (x₂+iy₂) = (x₁+x₂) - i(y₁+y₂)

c) Multiplication:(x₁+iy₁)(x₂+iy₂) = (x₁x₂ - y₁y₂) + i(y₁x₂+x₁y₂)

d) Division: (x₁+iy₁)/(x₂+iy₂) = [(x₁+iy₁)(x₂- iy₂) / (x₂+iy₂)(x₂ - iy₂)] = [ (x₁x₂ + y₁y₂) + i(y₁x₂- x₁y₂)/ (x²₂+y²₂)]

e) Equality: If (x₁+iy₁)=(x₂+iy₂) then x₁=x₂ and y₁=y₂

f) i² = -1, i³= -i , i⁴= 1

3. Conjugate of a Complex Number

The mirror image of the complex number z=x+iy on the real axis i.e z’=x - iy is called the conjugate of the complex number z. Conversely, z is the conjugate of z’.

Properties of the Conjugate

a) (z')' = z

b) (z₁ + z₂)' = z₁' + z₂'

c) (z₁ - z₂)' = z₁' - z₂'

d) (z₁z₂)' = z₁' z₂'

e) (z₁ /z₂)' = z₁' / z₂'

f) z₁z₂' + z₁'z₂ = 2Re(z₁'z₂) = 2 Re (z₁z₂')

g) (zⁿ)' = (z')ⁿ

h) If z = f ( z₁) then z' = f ( z₁')

i) Re (z) = (z + z')/2

j) Im(z) = (z - z') / 2i

4. Modulus and Argument of a Complex Number

The distance r of the point P from the origin is called the modulus of the complex number z and is denoted by |z|

If θ is the angle made by the vector OP with the positive real axis then it is called the argument of the complex number and is denoted by arg(z)

where -π < θ ≤ π

Hence, x + iy = (rcos θ) + i (r sin θ) = r ( cos θ + i sin θ) = r e^iθ = |z|e^i.arg(z)

5. Representation of a Complex Number

Cartesian Representation

Any complex number z can be represented in terms of the Cartesian coordinates (x,y) as z=x+iy where x,y belong to the set of real numbers.

Polar Representation

Any complex number z can be represented by its modulus r and argument θ as z = r cosθ + i sinθ

Euler Representation

Any complex number z can be represented by its modulus r and argument θ as z = re^iθ

Properties of Modulus

6. Cube Roots of Unity

The complex roots are represented by w and w². It does not matter which of the complex number is represented by w. The other one will be its square. Thus 1, w and w² are the three roots of unity.

Properties of Cube Roots of Unity

1 + w + w² = 0
w³ = 1 but w≠1
w³ⁿ = 1, w³ⁿ⁺¹ = w, w^{3n + 2} = w² where n is an integer
w' = w² and (w²)'=w

Important Factorisation

x² + y² = (x + iy) ( x - iy)
x² + x + 1 = (x - w) (x - w²)
x² - x + 1 = (x + w) (x + w²)
x² + xy + y² = (x - yw) (x-yw²)
x² - xy + y² = (x + yw) (x + yw²)
x³ + y³ = (x + y) (x + yw) (x + yw²)
x³ - y³ = (x - y) (x - yw) (x - yw²)
x² + y² +z² - xy - yz - xz = (x + yw + zw²) ( x + yw² + zw)
x³ + y³ + z³ - 3xyz = (x + y + z) (x + yw + zw²) ( x + yw² + zw)

7. n^th Roots of Unity

The equation xⁿ = 1 has n roots and are called the n^th roots of unity.

xⁿ = 1 = cos 0 + i sin 0 = cos 2kπ + i sin 2kπ, where k is an integer.

From the theory of equations, we can say that the sum of n^th roots of unity is 0

Product of n^throots of unity = (-1)ⁿ⁺¹

These roots are located at the vertices of an n sided regular polygon inscribed in a unit circle having centre at origin and one vertex on positive real axis.

8. Rotation of Complex Numbers

If z₁, z₂, z₃ be the complex numbers of the vertices of a triangle ABC described in the counter-clockwise sense then,

9. Complex Number Geometry

Equation of circles

The equation of a circle with centre at origin and radius a is given by |z| = a

The equation of a circle with centre at and radius a is given by | z - z₀| = a

The general equation of a circle is given by zz' + az' + a'z + b = 0 where b is a purely real number.
The centre of this circle is at -a and its radius is (aa'-b)^1/2