NCERT Class 11-Math's: Chapter –5 Complex Numbers and Quadratic Equations Part 1

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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.