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CBSE NOTES ⇒ Class 11th ⇒ Mathematics ⇒ 5. Complex Number and Quadratic Equations

5. Complex Number and Quadratic Equations

Introduction


Introduction

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.

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 

Other Pages of this Chapter:

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Mathematics Class - 11th

NCERT Maths book for CBSE Students.

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Study Materials List:

CBSE NOTES ⇒ Class 11th ⇒ Mathematics
1. Sets
2. Relations and Functions
3. Trigonometric Functions
4. Principle of Mathematical Induction
5. Complex Number and Quadratic Equations
6. Linear Inequalities
7. Permutations and Combinations
8. Binomial Theorem
9. Sequences and Series
10. Straight Lines
11. Conic Sections
12. Introduction to Three Dimensional Geometry
13. Limits and Derivatives
14. Mathematical Reasoning
15. Statistics
16. Probability

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