CBSE Class 9 Maths Chapter 1: Number Systems NCERT Solutions
This chapter, Number Systems, is a foundational part of the Class 9 Mathematics curriculum for CBSE students. It delves into the nature of numbers, starting with the definition of rational numbers and exploring how to represent them. The solutions cover key concepts such as identifying whether zero is a rational number and how to express it in the standard p/q form. A significant focus is placed on finding a specified quantity of rational numbers between two given numbers, whether they are integers or fractions. The exercise also tests the understanding of different number sets by asking students to determine the truthfulness of statements relating natural numbers, integers, rational numbers, and whole numbers, requiring justification. These detailed solutions are designed to help students grasp the fundamental principles of number systems and prepare effectively for their examinations.
Quick info
| Board | CBSE |
|---|---|
| Class | Class 9 |
| Subject | Mathematics |
| Session | 2026 |
| Language | English |
| Type | NCERT Solutions |
| Chapter | Chapter 1: Number Systems |
Chapter summary
Chapter 1, Number Systems, for CBSE Class 9 Maths, focuses on understanding rational numbers. The NCERT Solutions provide step-by-step guidance on identifying rational numbers, expressing zero as a rational number, and finding multiple rational numbers between any two given rational numbers. It also includes exercises to differentiate between various number sets like natural numbers, integers, and whole numbers, reinforcing their definitions and properties.
Learning outcomes
- Understand the definition and properties of rational numbers.
- Represent rational numbers in the form p/q.
- Find a specified number of rational numbers between two given numbers.
- Differentiate between natural numbers, integers, whole numbers, and rational numbers.
- Justify statements about number set classifications.
Topics covered
Paper topics
- Rational Numbers
- Definition of Rational Numbers
- Zero as a Rational Number
- Expressing Numbers in p/q form
- Finding Rational Numbers Between Two Numbers
- Methods for Finding Rational Numbers (Denominator n+1, Averaging)
- Properties of Number Sets
- Natural Numbers
- Integers
- Whole Numbers
- Classification of Numbers
- True/False Statements on Number Systems
Important topics
- Finding rational numbers between two given numbers
- Definition and representation of rational numbers
- Distinguishing between number sets (Natural, Whole, Integer, Rational)
- Justifying number classification statements
PDF preview
Read page by page below. PDF is streamed from the official NCERT website — no download button on this page.
Questions and Solutions
Page | 1
Number Systems
EXERCISE 1.1
Q.1. Is zero a rational number? Can you write it in the form <math>\frac{p}{a}</math>, where p and q are integers and <math>q \neq 0</math>?
Sol. Yes, zero is a rational number. It can be written as <math>\frac{0}{1}</math>, <math>\frac{0}{2}</math>, etc., in the form <math>\frac{p}{q}</math>, where p and q are integers and <math>q \neq 0</math>. Ans.
Q.2. Find six rational numbers between 3 and 4.
Sol. To find six rational numbers between 3 and 4 denominator should be made equal to <math>6 + 1 = 7</math>.
Therefore,
<math>3 = \frac{3 \times 7}{7} = \frac{21}{7}</math> <math>4 = \frac{4 \times 7}{7} = \frac{28}{7}</math>
Six rational numbers between 3 and 4 can be found by varying the numerator between 21 and 28.
Or, the numbers are <math>\frac{22}{7}, \frac{23}{7}, \frac{24}{7}, \frac{25}{7}, \frac{26}{7}, \frac{27}{7}</math>. Ans.
Q.3. Find five rational numbers between <math>\frac{3}{5}</math> and <math>\frac{4}{5}</math>.
Sol. To find five rational numbers between <math>\frac{3}{5}</math> and <math>\frac{4}{5}</math>, we may add the given numbers and divide by 2, and repeat the process.
<math display="block">\frac{\frac{3}{5} + \frac{4}{5}}{2} = \frac{7}{5 \times 2} = \frac{7}{10} = x_1</math>
<math display="block">\frac{7}{10} + \frac{4}{5} = \frac{7+8}{10} = \frac{15}{10}</math>
Next rational number = <math>\frac{15}{10 \times 2} = \frac{15}{20} = \frac{3}{4} = x_2</math>
<math display="block">\frac{3}{4} + \frac{4}{5} = \frac{15 + 16}{20} = \frac{31}{20}</math>
Next rational number = <math>\frac{31}{20 \times 2} = \frac{31}{40} = x_3</math>
<math display="block">\frac{31}{40} + \frac{4}{5} = \frac{31 + 32}{40} = \frac{63}{40}</math>
Next rational number = <math>\frac{63}{40 \times 2} = \frac{63}{80} = x_4</math>
<math display="block">\frac{63}{80} + \frac{4}{5} = \frac{63 + 64}{80} = \frac{127}{80}</math>
Next rational number = <math>\frac{127}{80 \times 2} = \frac{127}{160} = x_5</math>
Page | 2
<math>x_1 = \frac{7}{10}, x_2 = \frac{3}{4}, x_3 = \frac{31}{40}, x_4 = \frac{63}{80}, x_5 = \frac{127}{160}.</math> Ans.
(Note: Many answers are possible. There are of course infinitely many rational numbers between <math>\frac{3}{5}</math> and <math>\frac{4}{5}</math>.)
Q.4. State whether the following statements are true or false. Give reasons for your answers.
- Every natural number is a whole number.
- Every integer is a whole number.
- Every rational number is a whole number.
Sol.
- True, since the collection of whole numbers contains all the natural numbers and in addition zero.
- False. Negative integers are not whole numbers.
- False. Numbers such as <math>\frac{2}{3}</math>, <math>\frac{3}{4}</math>, <math>\frac{-3}{5}</math>, etc., are rational numbers but not whole numbers.
Common mistakes
- Incorrectly applying the method to find rational numbers between two given numbers.
- Confusing the definitions of different number sets (integers vs. whole numbers).
- Errors in fraction arithmetic when finding intermediate rational numbers.
- Failing to provide valid reasons for true/false statements about number classifications.
Revision tips
- Practice finding rational numbers between integers and fractions using both the 'n+1' denominator method and the averaging method.
- Clearly define each number set (natural, whole, integer, rational) before attempting classification questions.
- Ensure all steps are shown when calculating rational numbers between fractions to avoid arithmetic errors.
- Review the reasons provided for true/false statements to solidify understanding of number properties.
Practice MCQs
Q1. Which of the following is a rational number?
Explanation: Zero can be expressed as 0/1, fulfilling the definition of a rational number (p/q, where q ≠ 0). √2 and π are irrational, and √9 simplifies to 3, which is rational but 0 is explicitly asked.
Q2. How many rational numbers can be inserted between 3 and 4?
Explanation: There are infinitely many rational numbers between any two distinct rational numbers. The exercise shows methods to find a specific count, but the total number is infinite.
Q3. The number 3/5 can be written as:
Explanation: Dividing 3 by 5 gives 0.6. This is a terminating decimal, which is a characteristic of many rational numbers.
Q4. Which statement is FALSE?
Explanation: Not every rational number is an integer; for example, 1/2 is rational but not an integer. The other statements are true.
Q5. To find 6 rational numbers between 3 and 4, what should be the denominator of the equivalent fractions?
Explanation: To find 'n' rational numbers between two numbers, we convert them to fractions with a denominator of n+1. Here, n=6, so the denominator is 6+1=7.
Frequently asked questions
What is a rational number according to NCERT Class 9 Maths?
A rational number is a number that can be expressed in the form p/q, where p and q are integers and q is not equal to zero. Examples include 1/2, -3/4, and 5 (which can be written as 5/1).
Is zero a rational number?
Yes, zero is a rational number because it can be written in the form p/q as 0/1, 0/2, etc., where p=0 (an integer) and q is any non-zero integer.
How do you find rational numbers between two given numbers in Class 9 Maths?
You can find rational numbers between two given numbers by converting them to equivalent fractions with a larger common denominator (n+1, where n is the number of rational numbers to find) and then selecting numerators between the new numerators. Alternatively, you can repeatedly find the average of two numbers.
What is the difference between integers and whole numbers?
Whole numbers include zero and all positive integers (0, 1, 2, 3,...). Integers include all whole numbers as well as all negative integers (-3, -2, -1, 0, 1, 2, 3,...).
Are all rational numbers whole numbers?
No, not all rational numbers are whole numbers. For example, 1/2, -3/4, and 2.5 are rational numbers but are not whole numbers.
How can these NCERT solutions help in exam preparation?
These solutions provide clear, step-by-step explanations for each problem, helping students understand the methods and concepts. They cover various types of questions, including definitions, calculations, and justifications, which are crucial for mastering the Number Systems chapter for exams.
Content reviewed by the NCERT Help team. Editorial Team and update policy
NCERT Solutions PDF PDF on NCERT Help. URL unchanged for search indexing.