Number Systems

Notes of chapter: Number Systems are presented below. Indepth notes along with worksheets and NCERT Solutions for Class 9.      Locate √3 on the number line      Locate √3.5 on the number line geometrically                   Bar above the digits indicates the block of digits that is repeat of the digits.

Therefore, the decimal expansion of a rational number is either terminating or non – terminating recurring. Moreover, a number whose decimal expansion is terminating or non – terminating recurring is rational number. A Video showing real numbers and their decimal expansions-        Video below showing representation of 5.377 on number line-      (8) Methods to find out rational numbers between two rational numbers

(i) Eg: Find three rational numbers between 2 and 3.

First Method-

Add 2 and 3 and divide by 2.  Second Method –

We will change numbers in fractional numbers with denominators (3 + 1 = 4).

[To find out denominator, add 1 in number of rational numbers which we will have to find out.]   (9) Operations on real numbers

(i) The sum of two rational numbers is a rational number.

Eg:-

5 + 2.5 = 7.5  Rational number

√4 = 2

√36 = 6

Therefore, 2 + 6 = 8 Rational Number

(ii) The sum of a rational number and an irrational number is irrational number.

Ans –

= √2 + 3  Irrational number

(iii) The sum of two irrational numbers is irrational number.

Eg:-

Ans-

= 2 √2 + 5√3  Irrational number

Subtraction-

(i) The difference of two rational numbers is a rational number.

Eg:-

(a) Subtract 5 and 2.5

5 – 2.5 = 2.5  Rational number

(b) Subtract √36 – √4

√4  = 2

√36 = 6

Therefore,  6 – 2 = 4 Rational Number

(ii) The difference of a rational number and an irrational number is irrational number.

(b) Subtract √2 and  3

Ans –

= √2  – 3 Irrational number

(iii) The difference of two irrational numbers is irrational number.

Eg:-

Subtract 2√2  and 5√3

Ans-

2√2  – 5√3   Irrational number

Multiplication

(i) The product of a non – zero rational number with an irrational number is irrational.

Eg:- Multiply 4 by 2√5

Ans-

4 × 2√5

= 8 √5   Irrational number

(ii) The product of two irrational numbers may be rational or irrational.

Eg:- (a) Multiply √3 by 2√5

Ans-

√3 × 2√5

= 2√15   Irrational number

(b) Multiply √5 by √5

Ans-

√5 × √5

= 5  Rational number

(iii) The product of two rational numbers is rational.

Eg:- Multiply 2 by 3.5

Ans-

2 × 3.5 = 7 Rational number

Division

(i) The division of rational number with an irrational number is irrational. (ii) The division of two irrational numbers may be rational or irrational. (iii) The division of two rational numbers is rational.

Eg:-  Divide 10 by 2 (10)Methods to change decimal expansions in to fractional number or rational number-

(i) Method to change terminating decimal expansion in to fractional number or rational number-

Eg:-  Show that 2.1234 is a rational number. (ii) Method to change non – terminating recurring decimal expansion in to fractional number or rational number-  Helping Topics

NCERT Solutions Class 9

Worksheet Class 9

Exponents and powers

Steps to represent √9.3 on the number line

Steps to represent √7 on the number line

Steps to represent √5 on the number line

Is π rational number or irrational number?