Introduction

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First we know about the real number;

π√

Real Number includes;

(i)    Whole numbers: like 0, 1, 2, 3, 4.................................. etc.

(ii) Rational numbers: like 4/5, 0/6, 0.333........, etc.

(iii) irrational numbers: like π, √3, √2 etc.

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We have learnt quadratic equation in previous class. The nature of quadratic equations is

D > 0,         {Real and unequal roots}

D = 0,         {Real and equal roots}

D < 0,         {No Real roots, i.e. Imaginary root}

Look the following example   

x² + 3x + 5 = 0

a = 1, b = 3, c= 5

D =

    = 3²– 4 ×1 × 5

    = 9 – 20

    = –11

D < 0,        {so equation has no real but imaginary roots}

Now we have to find the roots

 

Here both the value of x is an imaginary number, which is made by the composition of (i), symbol “i” is called iota. Such number is called complex number. 

 

1. Imaginary Number: A number whose square is negative is known as an imaginary number.

Ex: ,  , ,  etc.

2. Complex number: Any number which is of the form of x + iy, where x and y are real number and i =  is called a complex number.

Ex : 3 + i5, 2 – i3, 5 + i2 and 4 +i3 etc.

It is denoted by z i.e. z = x +iy, in which Re(z) = x and Im(z) = y

A complex number has two parts;

(I)   real part Re(z)                       {∈ R}

              real part : 2, 3, 5, and 4 or may be any real number.

(II)   imaginary part Im(z)            {Real number with i(iota)}

              imaginary part: i, i2, i3, i4, and i5 etc.

Every Real number is a complex number if x∈ R and y ∈ R; such as

z = 3 ⇒ 3 +i0,    x = 3, y =0

z = –3⇒ – 3 +i0,  x = –3, y =0

z = 7 ⇒ 7 +i0,  x = 7, y =0

3. See the following complex numbers

z = 3, z = i3, z = 4, z = i7

z = 3 and  z = 4 are purely Real

 z = i3 and  z = i7 are purely Imaginary 

4. Power of i (iota)

∈

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7. Algebra of Complex Numbers

(A)   Addition of Complex numbers

Let z₁ = a + ib and z₂ = c + id be any two complex numbers. Then, the sum z₁ + z₂ is defined as follows:

z₁ + z₂ = (a+ib)+(c+id) = (a + c) + i (b + d), which is again a complex number.

Ex: z₁ =2 + i3 and z₂ = 7 + i5

z₁+ z₂ = (2+7) +i(3+5)

           = 9+i8

Alternative Method:

z₁+ z₂ = 2+i3 +7+i5

          = 2+7+i3+i5

          = 9+i(3+5)

          = 9+i8

(B) Subtraction of Complex Numbers

Given any two complex numbers z₁ and z₂, the difference z₁ – z₂ is defined as follows:

z₁ – z₂ = (a+ib) – (c+id) = (a–c)+i(b–d)

For example: z₁= 4+i3,  z₂ =3 + i7

z₁ – z₂ =(4–3)+i(3–7)= 1–i4                  

Note: {4 and 3are like term and i3 and i7 are another like term}

Alternative Method:

z₁ – z₂ =(4+i3) – (3+i7)

            =4+i3 – 3 – i7

            =4– 3+i3 – i7

            =1+i(3 – 7)

            =1-i4

(C) Multiplication of Complex numbers:

z₁ × z₂ = (a +ib) (c+id)

           =a(c+id) +ib(c+id)

           = ac + iad + ibc +( –bd)

           = ac –bd +i(ad+bc)

For example: z­₁= 2+3i, z₂=5 +2i

z­₁× z₂ = (2+3i)( 5 +2i)

           = (2×5 – 3×2)+ i(2×2+3×5)

            = 10 – 6 + i(4+15)

            = 4 + 19i

Alternative Method:

z­₁× z₂ = (2+3i)( 5 +2i)

           =2( 5 +2i) +3i( 5 +2i)

           = 10 + 4i + 15i + 6i²

           = 10 + 4i + 15i + 6(–1)     

           = 10 – 6 + 19i

           = 4 + 19i

(D)Some Important identities:

1. (z₁ + z₂)² = z₁² + 2 z₁ z₂ + z₂²
2. (z₁ – z₂)² = z₁² – 2 z₁ z₂ + z₂²
3. (z₁ + z₂)³ = z₁³ + 3 z₁² z₂ + 3 z₁z₂² + z₂³
4. (z₁ – z₂)³ = z₁³ – 3 z₁² z₂ + 3 z₁z₂² – z₂³
5. z₁² – z₂² = (z₁ + z₂)( z₁ – z₂)

The Modulus and Congugate of a Complex Number

Let z = a + ib be a complex number. 

Then, the modulus of z, denoted by |z|, is defined to be the non-negative real number

 ​, i.e  

Conjugate: The conjugate is where you chaange the sign in the middle of two terms like

and the congugate of z, denoted as;

 

The conjugate is where you change the sign in the middle of two terms like this:

d,;d;a;'

Polar representation of a complex number: 

Argand plane: The plane having a complex number assigned to each of its point is called the complex plane or the Argand plane.

Polar form: Polar form of a complex number is another way to represent a complex number on argand plane. 

if z = x + iy is any complex number then 

In polar representation a complex number z is represented by two parameters r and θ. Where parameter r is the modulus of a complex number and θ is the angle with the positive direction of x - axis. 

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 using pythagoros theorem 

Here Z of modulus = r and θ is called the argument (or amplitude) of z which is denoted by arg z.

Principle arguments of z : The value of θ such that – π < θ ≤ π, called principal argument of z and is denoted by arg z.

The point (x,y) represent a normal cartesian coordinate. But in polar form this point is reoresented by a special coordinate system which is called polar coordinate having (r, θ). 

This coordinate (r, θ) represents the each location of a point of a complex number. 

So, We have there is a relation between polar cordinates and cartesian coordinates. 

We know; 

z = Re(z) + Img(z) 

Then x-axis represent Re(z) and y-axis represent Img(z). 

We consider Origin (0, 0) as pole. 

For any complex number z = x + iy is represented as r (cosθ + i sinθ) as any point on complex plane. This is called polar representation of a complex number. Where θ is angle between r and x-axis.