Introduction

The number system is a way to represent and work with numbers. In Class 9, you’ll learn about different types of numbers, such as natural numbers, whole numbers, integers, and rational and irrational numbers. It also explains how these numbers are organized on the number line and how to perform operations with them.

This topic helps you understand concepts like decimals, fractions, and roots in simple steps, forming the foundation for advanced math.

Some Important Definition

• A number line is a straight, horizontal line used to represent numbers. It has a zero in the center, with positive numbers to the right and negative numbers to the left.

• A natural number (N) is any number used for counting and starts from 1, 2, 3, and so on. These numbers are positive and do not include fractions, decimals, or zero. Natural numbers are the basic numbers we use in daily life to count objects or perform simple calculations.
• A whole number (W) is any number that starts from 0 and includes 1, 2, 3, and so on. These numbers are always positive, do not include fractions or decimals, and are used for counting and simple calculations. Whole numbers include zero, unlike natural numbers.

EXERCISE 1.1

1. Is zero a rational number? Can you write it in the form p/q, where p and q are integers and q ≠ 0?

Sol. Yes, zero is a rational number because it can be expressed in the form 𝑝/𝑞, where 𝑝 and 𝑞 are integers, and 𝑞 ≠ 0. For example:

0 = 0/1  = 0/2 and so on. Similarly, If we take the denominator in negative (-) as:

0/(-1) = 0/(-2) and so on that always gives you the 0.

2. Find six rational numbers between 3 and 4.

Sol. Multiply both 3 and 4 by 7 (because we want 6 numbers, so take 6+1 = 7 ).

3 x (7/7) = 21/7 and 4 X (7/7) = 28/7

Now, the rational numbers between 21/7 and 28/7 are:

22/7, 23/7, 24/7, 25/7, 26/7, 27/7

These are the 6 rational numbers between 3 and 4.

3. Find five rational numbers between 3/5 and 4/5.

Sol. There are infinite rational numbers between 3/5 and 4/5.

To find out 5 rational numbers between 3/5 and 4/5, we will multiply both the numbers 3/5 and 4/5with 5+1=6 (or any number greater than 5)

i.e., (3/5) × (6/6) = 18/30 and, (4/5) × (6/6) = 24/30

The numbers in between18/30 and 24/30 will be rational and will fall between 3/5 and 4/5.

19/30, 20/30, 21/30, 22/30, 23/30

These are the 5 rational numbers between 3/5 and 4/5.

4. State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number.
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.

Sol. (i) Natural numbers (1,2,3,4,…) are a part of the set of whole numbers (0,1,2,3,4,…). Every natural number is included in the set of whole numbers.

(ii) Integers (−3,−2,−1,0,1,2,3,…) include negative numbers, which are not whole numbers. Whole numbers only consist of 0 and positive numbers (0,1,2,3,…).

(iii) Rational numbers (

pq​, where $p$ and $q$ are integers, and q≠0) include fractions and decimals, which are not whole numbers. For example, 12​ and 2.5 are rational numbers but not whole numbers.

Important Questions Based On Ex. 1.1
( Ex. 1.1 पर आधारित महत्वपूर्ण प्रश्न )

1. Find five rational numbers between 3/5 and 4/5.
   ( 3/5 और 4/5 के बीच पाँच परिमेय संख्याएँ लिखें। )
2. Are the following statements are true or false? Give reasons for your answers.
   ( क्या निम्न कथन सही या गलत हैं? अपने उत्तरों के लिए कारण दें। )
   1. Every whole number is naural number
      ( प्रत्येक पूर्णांक संख्या एक प्राकृतिक संख्या होती है। )
   2. Every integer is a rational number
      ( प्रत्येक पूर्णांक एक परिमेय संख्या होती है। )
   3. Every rational number is an integer
      ( प्रत्येक परिमेय संख्या एक पूर्णांक होती है। )
3. Find five rational numbers between 6 and 7.
   ( 6 और 7 के बीच पाँच परिमेय संख्याएँ लिखें। )
4. Find seven rational numbers between 4/5 and 6/5.
   ( 4/5 और 6/5 के बीच पाँच परिमेय संख्याएँ लिखें। )
5. Find five rational numbers between 6/13 and 7/13.
   ( 6/13 और 7/13 के बीच पाँच परिमेय संख्याएँ लिखें।  )