These Class 9 Maths Ch 1 Ex 1.1 Solutions provide complete step-by-step answers for all four questions in the Number Systems chapter, helping CBSE students master the fundamental concepts of rational numbers for the 2025-26 academic session.
Chapter 1 of NCERT Class 9 Mathematics introduces students to the fascinating world of Number Systems, beginning with Exercise 1.1 that establishes the foundation for understanding how different types of numbers relate to each other. This exercise specifically focuses on rational numbers and their relationship with natural numbers, whole numbers, and integers—concepts that appear repeatedly throughout higher mathematics.
The Number Systems chapter carries significant weightage in CBSE board examinations, with Unit I (Number Systems) allocated 10 marks in the final assessment. Understanding Exercise 1.1 thoroughly is essential because it introduces the definitions and classifications that form the basis for more complex topics like irrational numbers and real numbers covered later in the chapter. Students who master these foundational concepts find subsequent exercises considerably easier to approach.
Whether you are preparing for your school examinations or building a strong mathematical foundation for competitive exams, these detailed solutions will clarify every concept tested in Exercise 1.1. If you wish to continue your mathematical journey, you might also find NCERT Solutions Class 10 Maths Chapter 1 Real Numbers helpful for understanding how these concepts evolve in higher classes.
Class 9 Maths Ch 1 Ex 1.1 Solutions Overview
Exercise 1.1 of Chapter 1 Number Systems contains four carefully designed questions that test students’ understanding of number classification and the concept of rational numbers. Each question builds upon the previous one, creating a logical progression that helps students develop a complete understanding of how numbers are organised in mathematics.
What is a Rational Number? A rational number is any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0. This definition is the cornerstone of Exercise 1.1 and must be thoroughly understood before attempting the questions.
The questions in this exercise require students to apply the definition of rational numbers to various scenarios. Question 1 asks whether every natural number is also a whole number, testing the understanding of subset relationships between number categories. Question 2 explores whether every integer is a whole number, which requires careful consideration of negative integers.
Question 3 is particularly important as it asks students to demonstrate that every rational number is not necessarily an integer by providing counterexamples. This question develops critical thinking skills and helps students understand that while all integers are rational numbers, the reverse is not always true. Question 4 rounds off the exercise by asking students to represent certain rational numbers on the number line, connecting abstract concepts with visual representation.
| Total Questions | 4 |
|---|---|
| With Solutions | 4 |
| Exercise Number | 1.1 |
| Updated For | 2025-26 Session |
Understanding these concepts is crucial not only for Class 9 but also for higher studies. Students preparing for Class 11 can explore Class 11 Maths NCERT Book PDF to see how number systems concepts are expanded at the senior secondary level.
Key Concepts Tested in Exercise 1.1
Each question in Class 9 Maths Ch 1 Ex 1.1 Solutions is designed to test specific mathematical concepts. Understanding what each question assesses helps students prepare more effectively and identify areas requiring additional practice.
Why This Matters: Knowing which concepts are tested in each question allows you to focus your revision strategically. If you struggle with a particular question, you know exactly which concept needs more attention.
Question 1 tests the relationship between natural numbers and whole numbers. Students must recognise that natural numbers (1, 2, 3, 4…) are counting numbers starting from 1, while whole numbers include all natural numbers plus zero (0, 1, 2, 3…). The concept being assessed is whether natural numbers form a subset of whole numbers—and the answer is yes, because every natural number is also a whole number.
Question 2 examines the relationship between integers and whole numbers. This is where many students make errors. Integers include all whole numbers as well as negative numbers (…-3, -2, -1, 0, 1, 2, 3…). The question tests whether students understand that not every integer is a whole number because negative integers like -1, -2, -3 are not included in the set of whole numbers.
Question 3 assesses understanding of the relationship between rational numbers and integers. Students must provide examples of rational numbers that are not integers, such as 1/2, 3/4, or 7/5. This question tests the ability to apply the definition of rational numbers (p/q form where q ≠ 0) and recognise that fractions with denominators other than 1 are rational but not integers.
Question 4 tests the practical skill of representing rational numbers on a number line. This visual representation is crucial for understanding the density property of rational numbers and prepares students for locating irrational numbers on the number line in later exercises.
| Resource | Access |
|---|---|
| NCERT Class 9 Mathematics Textbook | Download Book |
| NCERT Class 9 Science Solutions | View Solutions |
| RD Sharma Class 9 (Updated 2025-26) | View Solutions |
| NCERT Class 9 English (Beehive) | Download Book |
Step-by-Step Solutions for All Questions
These comprehensive Class 9 Maths Ch 1 Ex 1.1 Solutions provide detailed explanations following the NCERT methodology approved for CBSE 2025-26. Each solution includes the reasoning process that helps students understand not just the answer but the mathematical logic behind it.
Solution to Question 1: Natural Numbers and Whole Numbers
The question asks: Is every natural number a whole number?
To answer this question, we must first recall the definitions. Natural numbers are the counting numbers starting from 1: {1, 2, 3, 4, 5, …}. These are the numbers we naturally use when counting objects. Whole numbers include all natural numbers plus zero: {0, 1, 2, 3, 4, 5, …}.
Important: The key difference between natural numbers and whole numbers is the inclusion of zero. Zero is a whole number but not a natural number according to the NCERT definition used in Indian curriculum.
Since the set of natural numbers {1, 2, 3, 4, …} is completely contained within the set of whole numbers {0, 1, 2, 3, 4, …}, we can conclusively state that yes, every natural number is a whole number. However, the reverse is not true—zero is a whole number that is not a natural number.
Solution to Question 2: Integers and Whole Numbers
The question asks: Is every integer a whole number?
To solve this, we examine the definitions carefully. Integers include all positive numbers, zero, and negative numbers: {…, -3, -2, -1, 0, 1, 2, 3, …}. Whole numbers include only zero and positive numbers: {0, 1, 2, 3, 4, …}.
The answer is no, every integer is not a whole number. While positive integers (1, 2, 3…) and zero are whole numbers, negative integers (-1, -2, -3…) are not whole numbers. For example, -5 is an integer but not a whole number. This demonstrates that integers include a broader category of numbers than whole numbers.
Solution to Question 3: Rational Numbers and Integers
The question asks: Is every rational number an integer?
A rational number is defined as any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0. An integer is a whole number or its negative counterpart.
The answer is no, every rational number is not an integer. Consider the rational number 1/2. It is written in the form p/q where p = 1 and q = 2 (both integers, and q ≠ 0), so it is definitely a rational number. However, 1/2 = 0.5, which is not an integer. Similarly, 3/4, 7/9, and -5/3 are all rational numbers that are not integers.
However, it is important to note that every integer is a rational number. Any integer n can be written as n/1, which satisfies the definition of a rational number. For comprehensive practice with polynomial concepts that build on these number system foundations, students can refer to NCERT Solutions Class 10 Maths Chapter 2 Polynomials.
Solution to Question 4: Representing Rational Numbers on Number Line
This question asks students to represent certain rational numbers on the number line. The process involves dividing the unit length into equal parts corresponding to the denominator and then marking the appropriate number of parts.
For example, to represent 3/4 on the number line:
Step 1: Divide the distance between 0 and 1 into 4 equal parts (since denominator is 4).
Step 2: Count 3 parts from 0 towards 1 (since numerator is 3).
Step 3: Mark this point as 3/4.
For negative rational numbers like -2/3:
Step 1: Move to the left of zero on the number line.
Step 2: Divide the distance between -1 and 0 into 3 equal parts.
Step 3: Count 2 parts from 0 towards -1 and mark as -2/3.
| Formula / Concept | Description |
|---|---|
| Rational Numbers | A number is called rational if it can be expressed in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). |
| Decimal Expansion of Rational Numbers | The decimal expansion of a rational number is either terminating (ends after a finite number of digits) or non-terminating recurring (a block of digits repeats indefinitely). |
| Irrational Numbers | A number is called irrational if it cannot be expressed in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). |
| Decimal Expansion of Irrational Numbers | The decimal expansion of an irrational number is non-terminating and non-recurring (the digits continue infinitely without forming a repeating pattern). Examples include \( \sqrt{2} \), \( \sqrt{3} \), \( \pi \), etc. |
| Finding Rational Numbers Between Two Rational Numbers (Mean Method) | If \( a \) and \( b \) are two rational numbers, then \( \frac{a+b}{2} \) is a rational number that lies between \( a \) and \( b \). This method can be applied repeatedly to find more rational numbers. |
| Finding Rational Numbers Between Two Rational Numbers (Equivalent Fraction Method) | To find rational numbers between two given rational numbers (say \( \frac{a}{b} \) and \( \frac{c}{d} \)), first make their denominators equal. If more numbers are needed, multiply both the numerator and denominator of each fraction by a suitable common factor (e.g., 10). |
| Converting Non-Terminating Recurring Decimals to \( \frac{p}{q} \) Form |
To convert a non-terminating recurring decimal (e.g., \( 0.\overline{x} \), \( 0.\overline{xy} \), \( 0.a\overline{b} \)) to \( \frac{p}{q} \) form, follow these steps:
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Understanding the Number Classification Hierarchy
A thorough understanding of how different number types relate to each other is essential for mastering the Class 9 Maths Ch 1 Ex 1.1 Solutions. The number system follows a hierarchical structure where each category is a subset of a larger category.
Number Hierarchy (from smallest to largest set): Natural Numbers ⊂ Whole Numbers ⊂ Integers ⊂ Rational Numbers ⊂ Real Numbers. The symbol ⊂ means “is a subset of” or “is contained within.”
Natural Numbers (N) form the most basic set of numbers used for counting. These are {1, 2, 3, 4, 5, …} and extend infinitely. They do not include zero, fractions, decimals, or negative numbers. Natural numbers are used in everyday counting—”