Basic Concepts :

A number in the form of a + ib, where a, b are real numbers and i = √-1 is called a complex number.

A complex number can also be defined as an ordered pair of real numbers a and b and may be written as (a, b), where the first number denotes the real part and the second number denotes the imaginary part.

If z = a + ib, then the real part of z is denoted by Re (z) and the imaginary part by Im (z).

A complex number is said to be purely real if Im(z) = 0, and is said to be purely imaginary if Re(z) = 0.

The complex number 0 = 0 + i0 is both purely real and purely imaginary.

Two complex numbers are said to be equal if and only if their real parts and imaginary parts are separately equal i.e.

a + ib = c + id implies a = c and b = d.

However, there is no order relation between complex numbers and the expressions of the type a + ib < (or >) c + id are meaningless.

Remark:

⋄ Clearly i^{2} = -1 , i^{3} = -i , i^{4} = 1

In general , i^{4n} = 1 , i^{4n+1} = i , i^{4n+2 }= -1 for an integer n.

__Geomertical Representation Of Complex Number__

A complex number z = x + iy, written as an ordered pair (x, y), can be represented by a point P whose Cartesian coordinates are (x, y) referred to axes OX and OY, usually called the real and the imaginary axes.

The plane of OX and OY is called the Argand diagram or the complex plane.

Since the origin O lies on both OX and OY, the corresponding complex number z = 0 is both purely real and purely imaginary.

__Modulus and Argument of a Complex Number__

We define modulus of the complex number z = x + iy to be the real number √(x^{2} + y^{2}) and denote it by |z|.

It may be noted that |z| ≥ 0 and |z| = 0 would imply that

z = 0.

If z = x + iy, then angle θ given by tan θ= y/x is said to be the argument or amplitude of the complex number z and is denoted by arg(z) or amp(z).

In case of x = 0 (where y ≠ 0),

arg(z) = + π/2 or − π/2 depending upon y > 0 or y < 0 and the complex number is called purely imaginary.

If y = 0 (where x ≠ 0), then arg(z) = 0 or π depending upon x > 0 or x < 0 and the complex number is called purely real.

The argument of the complex number 0 is not defined.

We can define the argument of a complex number also as any value of the θ which satisfies the system of equations

$ \displaystyle cos\theta = \frac{x}{\sqrt{x^2 + y^2 }} $

$ \displaystyle sin\theta = \frac{y}{\sqrt{x^2 + y^2 }} $

The argument of a complex number is not unique.

If θ is a argument of a complex number ,

then 2nπ + θ (n integer) is also argument of z for various values of n.

The value of θ satisfying the inequality − π < θ ≤ π is called the principal value of the argument.

From figure 1, we can see that OP = √(x^{2} + y^{2}) = |z| and If

θ = ∠POM ,

then tanθ = y/x.

In other words |z| is the length of OP i.e. the distance of point z from the origin and arg(z) is the angle which OP makes with the positive x-axis.

__Trigonometric (or Polar) form of a Complex Number__

Let OP = r , then x = r cos θ , and y = r sin θ

z = x + iy

= r cos θ + ir sin θ

= r ( cos θ + i sin θ ).

This is known as Trigonometric (or Polar) form of a Complex Number.

Here we should take the principal value of θ .

For general values of the argument

z = r [ cos (2nπ + θ) + i sin (2nπ + θ)]

(where n is an integer)

Students should note that sometimes cos θ + i sin θ is, in short, written as cis(θ).

Euler’s formula : cos θ + i sin θ = e^{iθ}

Remark:

⋄ Method of finding the principal value of the argument of a complex number z = x + iy

Step I: Find tan θ = |y/x| and this gives the value of θ in the first quadrant.

Step II: Find the quadrant in which z lies , with the help of sign of x and y co-ordinates.

Step III: Then argument of z will be θ , π − θ, θ − π , and − θ according as z lies in the first second , third or fourth quadrant

Illustration 1. For z = √3 − i , find the principal value arg(z).

Solution: Here x = √3 , y = − 1

⇒ $ \displaystyle tan\theta = |-\frac{1}{\sqrt{3 }}| $

⇒ θ = π/6

⇒ Principal value of arg z = − π/6

(Since z lies in the fourth quadrant)

__Unimodular Complex Number__

A complex number z for which |z| = 1 is said to be unimodular complex number.

Since |z| = 1 , z lies on a circle of radius 1 unit and centre (0, 0).

If |z| = 1

⇒ z = cos θ + i sin θ ,

⇒ 1/z = (cos θ + i sinθ)^{−1}

= cos θ − i sinθ

__Algebraic Operations with Complex Numbers:__

# Addition : (a + ib) + (c + id) = (a + c) + i (b + d)

# Subtraction : (a + ib) − (c + id) = (a − c) + i ( b − d)

# Multiplication : (a + ib) (c + id) = (ac − bd) + i ( ad + bc)

# Division :

(when at least one of c and d is non-zero)

$ \displaystyle \frac{a+ib}{c+id} = \frac{ac+bd}{c^2 +d^2} + i\frac{bc-ad}{c^2+d^2} $