Class 11 || Maths || Ch. 05 Complex Numbers And Quadratic Equations

Class 11 || Maths || Ch. 05 COMPLEX NUMBERS AND QUADRATIC EQUATIONS

z₁

z^–1

Mathematics is the Queen of Sciences and Arithmetic is the Queen of Mathematics - GAUSS

5.1 Introduction

In earlier classes, we have studied linear equations in one and two variables and quadratic equations in one variable with real solution. 

We have seen that the equation x² + 1 = 0 has no real solution 

as 

x² + 1 = 0 

x² = –1 

x = √–1 

x² +1= 0 gives x = √–1 and square of every real number is non-negative. 

So, we need to extend the real number system to a larger system so that we can find the solution of the equation x²=-1. In fact, the main objective is to solve the equation 

ax² + bx + c = 0, 

where D = b² – 4ac < 0, which is not possible in the system of real numbers.

5.2 Complex Numbers

Let us denote √ –1 by the symbol i called 'iota'. Then, we have i² = –1. This means that ' i ' is a solution of the equation x² + 1 = 0.

A number of the form a + ib, where a and b are real numbers, is defined to be a complex number. 

For example, 

2 + i 3, (–1) + i √3, 4 + i (–1/11) are complex numbers.

For the complex number 

z = a + ib, a is called the real part, denoted by Re(z) and b is called the imaginary part denoted by Im(z) of the complex number z. 

For example, 

if z = 2 + i5, 

then 

Re(z)=2 and 

Im(z) = 5.

Two complex numbers z, a +ib and z,=c + id are equal if a=c and b = d.

5.3.3 Multiplication of two complex numbers 

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

z_1. z₂ = (ac – bd) + i(ad + bc)

For example, 

(3 + i5) (2 + i6) 

= (3x2 –5x6) + i(3x6 +5 × 2)

=–24 + i28

The multiplication of complex numbers possesses the following properties, which we state without proofs.

(i) The closure law 

The product of two complex numbers is a complex number, the product z₁, z₂ is a complex number for all complex numbers z, and

                 z_1. z₂ = z      

(ii) The commutative law 

For any two complex numbers z_1. z₂, and z_1. z₂

                 z_1. z₂ = z_2. z₁

(iii) The associative law 

For any three complex numbers 

        (z_1. z₂) z₃ = z₁ (z_2 . z₃)

(iv) The existence of multiplicative identity 

There exists the complex number 1 + i0 (denoted as 1), called the multiplicative identity such that z. 1 = z, for every complex number z.

(v) The existence of multiplicative inverse 

For every non-zero complex number z = a + ib or a + bi (a ≠ 0, b ≠ 0), we have the complex number 

a /(a² + b²) + b/(a² + b²) + (denoted by 1/z or z^–1, called the multiplicative inverse of z such that

z . 1/z = 1(the multiplicative identity).

(vi) The distributive law 

For any three complex numbers   z₁, z₂, z₃

(a) z₁ (z₂ + z₃) = z₁₁ . z₂ + z_1. z₃

(b) (z₁ + z₂) z₃ = z_1. z₃ + z_3. z₂

5.3.4 Division of two complex numbers 

Given any two complex numbers z, and z

where ₂0, the quotient is defined by

1

For example, let

z₁=6+31 and 2₂-2-1

Then

+i -(-1) 2 - (6 +31) × 2) - (+31) ( 2² + (-1)³ *' 2²³ +(-1)³ ) (6+31)× Z2 2-1/ =(