Chapter-5. Complex Number and Quadratic Equations Mathematics class 11 in english Medium CBSE Notes

Introduction

First we know about the real number;

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