**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__

__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__

__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**

** **

** **

**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__

__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**

⇒ ** **

⇒** θ = π/6**

⇒ ** Principal value of arg z = − π/6**

**(Since z lies in the fourth quadrant) **

__Unimodular Complex Number__

__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:__

__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)**