NCERT Class 11-Math՚S: Chapter – 5 Complex Numbers and Quadratic Equations Part 1 (For CBSE, ICSE, IAS, NET, NRA 2022)

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5.1 Overview

We know that the square of a real number is always non-negative e. g. and . Therefore, square root of is . What about the square root of a negative number? It is clear that a negative number cannot have a real square root. So we need to extend the system of real numbers to a system in which we can find out the square roots of negative numbers. Euler was the first mathematician to introduce the symbol i (iota) for positive square root of i.e.. , .

5.1. 1 Imaginary numbers

Square root of a negative number is called an imaginary number. , for example,

5.1. 2 Integral powers of i

To compute for , we divide by and write it in the form , where is quotient and is remainder

Hence

For example,

And

(i) If a and b are positive real numbers, then

(ii) if a and b are positive or at least one them is negative or zero.

However, if a and b, both are negative.

5.1. 3 Complex numbers

(a) A number which can be written in the form , where a, b are real numbers and is called a complex number.

(b) If is the complex number, then a and b are called real and imaginary parts, respectively, of the complex number and written as .

(c) Order relations “greater than” and “less than” are not defined for complex numbers.

(d) If the imaginary part of a complex number is zero, then the complex number is known as purely real number and if real part is zero, then it is called purely imaginary number, for example, is a purely real number because its imaginary part is zero and is a purely imaginary number because its real part is zero.

5.1. 4 Algebra of complex numbers

(a) Two complex numbers and are said to be equal if .

(b) Let and be two complex numbers then .

5.1. 5 Addition of complex numbers satisfies the following properties

1. As the sum of two complex numbers is again a complex number, the set of complex numbers is closed with respect to addition.

2. Addition of complex numbers is commutative, i.e.. ,

3. Addition of complex numbers is associative, i.e.. ,

4. For any complex number there exist , i.e.. , complex number such that z + 0 = 0 + z = z, known as identity element for addition.

5. For any complex number , there always exists a number such that and is known as the additive inverse of .

5.1. 6 Multiplication of complex numbers

Let and , be two complex numbers. Then

1. As the product of two complex numbers is a complex number, the set of complex numbers is closed with respect to multiplication.

2. Multiplication of complex numbers is commutative, i.e.. , =

3. Multiplication of complex numbers is associative, i.e.. , () . .

4. For any complex number , there exists a complex number , i.e.. , such that , known as identity element for multiplication.

5. For any non-zero complex number , there exists a complex number such that multiplicative inverse of

6. For any three complex numbers and ,

And

i.e.. , for complex numbers multiplication is distributive over addition.

5.1. 7 Let and . Then

5.1. 8 Conjugate of a complex number

Let be a complex number. Then a complex number obtained by changing the sign of imaginary part of the complex number is called the conjugate of z and it is denoted by , i.e.. , .

Note that additive inverse of is but conjugate of is .

We have:

1.

2.

3. , if z is purely real.

4. is purely imaginary

5.

6.

7.