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Score Higher with Class 9 Number System Exercise 1.1 Solutions Practice
Chapter 1 of Class 9 Maths, "Number System," introduces students to the fundamental concepts of numbers, their classifications, and properties. Exercise 1.1 focuses on understanding different types of numbers, including natural, whole, integers, rational, and irrational numbers. Vedantu's NCERT Solutions for Class 9 Maths Chapter 1 - Number System are essential for exam success.
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Solutions of class 9 ex 1.1 prepared by maths experts help students understand important concepts, boost confidence, and improve exam scores. These comprehensive solutions are available for free on Vedantu, making subjects like Science, Maths, and English easier to study.
Topics Covered in Class 9 Maths Chapter 1 Exercise 1.1
Different Types of Numbers.
Representation of Numbers.
Operations on Numbers.
Properties of Numbers.
Classifying Numbers.
NCERT Solutions for Class 9 Maths Chapter 1 Number System
ShareShareExercise (1.1)
1. Is zero a rational number? Can you write it in the form $\dfrac{ {p}}{ {q}}$, where $ {p}$ and $ {q}$ are integers and $ {q}\ne {0}$? Describe it.
Ans: Remember that, according to the definition of rational number,
a rational number is a number that can be expressed in the form of $\dfrac{p}{q}$, where $p$ and $q$ are integers and $q\ne \text{0}$.
Now, notice that zero can be represented as $\dfrac{0}{1},\dfrac{0}{2},\dfrac{0}{3},\dfrac{0}{4},\dfrac{0}{5}.....$
Also, it can be expressed as $\dfrac{0}{-1},\dfrac{0}{-2},\dfrac{0}{-3},\dfrac{0}{-4}.....$
Therefore, it is concluded from here that $0$ can be expressed in the form of $\dfrac{p}{q}$, where $p$ and $q$ are integers.
Hence, zero must be a rational number.
2. Find any six rational numbers between $ {3}$ and $ {4}$.
Ans: It is known that there are infinitely many rational numbers between any two numbers. Since we need to find $6$ rational numbers between $3$ and $4$, so multiply and divide the numbers by $7$ (or by any number greater than $6$)
Then it gives,
$\begin{align} & 3=3\times \dfrac{7}{7}=\dfrac{21}{7} \\ & 4=4\times \dfrac{7}{7}=\dfrac{28}{7} \\ \end{align}$
Hence, $6$ rational numbers found between $3$ and $4$ are $\dfrac{22}{7},\dfrac{23}{7},\dfrac{24}{7},\dfrac{25}{7},\dfrac{26}{7},\dfrac{27}{7}$.
3. Find any five rational numbers between $\dfrac{ {3}}{ {5}}$ and $\dfrac{ {4}}{ {5}}$.
Ans: It is known that there are infinitely many rational numbers between any two numbers.
Since here we need to find five rational numbers between $\dfrac{3}{5}$ and $\dfrac{4}{5}$, so multiply and divide by $6$ (or by any number greater than $5$).
Then it gives,
$\dfrac{3}{5}=\dfrac{3}{5}\times \dfrac{6}{6}=\dfrac{18}{30}$,
$\dfrac{4}{5}=\dfrac{4}{5}\times \dfrac{6}{6}=\dfrac{24}{30}$.
Hence, $5$ rational numbers found between $\dfrac{3}{5}$ and $\dfrac{4}{5}$ are $\dfrac{19}{30},\dfrac{20}{30},\dfrac{21}{30},\dfrac{22}{30},\dfrac{23}{30}$.
4. State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number.
Ans: Write the whole numbers and natural numbers in a separate manner.
It is known that the whole number series is $0,1,2,3,4,5.....$. and
the natural number series is $1,2,3,4,5.....$.
Therefore, it is concluded that all the natural numbers lie in the whole number series as represented in the diagram given below.
Thus, it is concluded that every natural number is a whole number.
Hence, the given statement is true.
(ii) Every integer is a whole number.
Ans: Write the integers and whole numbers in a separate manner.
It is known that integers are those rational numbers that can be expressed in the form of $\dfrac{p}{q}$, where $q=1$.
Now, the series of integers is like $0,\,\pm 1,\,\pm 2,\,\pm 3,\,\pm 4,\,...$.
But the whole numbers are $0,1,2,3,4,...$.
Therefore, it is seen that all the whole numbers lie within the integer numbers, but the negative integers are not included in the whole number series.
Thus, it can be concluded from here that every integer is not a whole number.
Hence, the given statement is false.
(iii) Every rational number is a whole number.
Ans: Write the rational numbers and whole numbers in a separate manner.
It is known that rational numbers are the numbers that can be expressed in the form $\dfrac{p}{q}$, where $q\ne 0$ and the whole numbers are represented as $0,\,1,\,2,\,3,\,4,\,5,...$
Now, notice that every whole number can be expressed in the form of $\dfrac{p}{q}$
as \[\dfrac{0}{1},\text{ }\dfrac{1}{1},\text{ }\dfrac{2}{1},\text{ }\dfrac{3}{1},\text{ }\dfrac{4}{1},\text{ }\dfrac{5}{1}\],…
Thus, every whole number is a rational number, but all the rational numbers are not whole numbers. For example,
$\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{5},...$ are not whole numbers.
Therefore, it is concluded from here that every rational number is not a whole number.
Hence, the given statement is false.
Conclusion
Class 9th Maths Exercise 1.1 provides a foundational understanding of the Number System. It introduces natural numbers, whole numbers, integers, rational numbers, and irrational numbers. The exercise emphasizes representing these numbers on the number line, which enhances visual learning and comprehension. By completing this exercise, students build a critical base for more advanced mathematical concepts, fostering logical reasoning and problem-solving skills essential for future studies. Students that practise these kinds of questions will gain confidence and perform well on tests.
Class 9 Maths Chapter 1: Exercises Breakdown
CBSE Class 9 Maths Chapter 1 Other Study Materials
Chapter-Specific NCERT Solutions for Class 9 Maths
Given below are the chapter-wise NCERT Solutions for Class 9 Maths. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.
Important Study Materials for CBSE Class 9 Maths
FAQs on NCERT Solutions for Class 9 Maths Chapter 1 Number System
1. How is zero a rational number?
Express zero in the p/q form, where q is not zero. For example, write 0 as 0/1, 0/5, or 0/-10. Since it can be written as a fraction with a non-zero denominator, zero is a rational number as per the definition in class 9 maths chapter 1 exercise 1.1.
2. How to download the Class 9 number system exercise 1.1 solutions PDF?
Click the “Download PDF” button on this page to get the complete solutions. Save the file to your device for easy offline access to all answers. Check that the downloaded file opens correctly. This makes revision convenient anytime.
3. What is the main difference between whole numbers and integers?
Remember that whole numbers are the set of positive integers including zero (0, 1, 2, ...), so they cannot be negative. Integers include all whole numbers plus their negative counterparts (..., -2, -1, 0, 1, 2, ...). Therefore, every whole number is an integer, but not every integer is a whole number.
4. How do I use these solutions to check my work for Exercise 1.1?
First, solve all the problems from the NCERT textbook on your own. Then, open the class 9 maths number system exercise 1.1 solutions and compare your final answers and method step-by-step. This helps identify mistakes and correct your understanding.
5. Is every integer a rational number?
Yes, confirm this by writing any integer 'n' in the p/q form. Any integer can be expressed as a fraction with a denominator of 1 (e.g., 5 = 5/1). Since this fits the definition of a rational number, all integers are rational numbers.
6. What is the method for finding five rational numbers between 3 and 4?
Instruction: To find 'n' rational numbers between two integers, convert them into equivalent fractions with a denominator of (n+1). This method creates a gap between the numerators to fit the required numbers.
Why it matters: This technique provides a structured way to find any quantity of rational numbers between two given numbers, a key skill for class 9 maths number system exercise 1.1.
Steps:
- Identify the number of rational numbers needed (here, n = 5).
- Calculate the new denominator: n + 1 = 5 + 1 = 6.
- Write the given integers as fractions with this denominator: 3 = 18/6 and 4 = 24/6.
- List the fractions between 18/6 and 24/6: 19/6, 20/6, 21/6, 22/6, 23/6.
Check: Ensure all the new numbers you found (like 19/6) are greater than the starting number (3) and less than the ending number (4).
These five fractions are the required rational numbers between 3 and 4.
7. How can NCERT Solutions for Class 9 Maths Chapter 1 be used for effective revision?
Instruction: Use the NCERT Solutions for Class 9 Maths Chapter 1 Exercise 1.1 to reinforce concepts and problem-solving methods before an exam. The solutions are prepared by experts at Vedantu to align with the latest CBSE guidelines.
Steps:
- Review the question and try to recall the solution method mentally without writing it down.
- Read the first two steps of the provided solution to verify your approach.
- If your method was correct, move to the next question. If not, study the full step-by-step explanation to understand the logic.
- Pay special attention to the true/false questions, as the solutions explain the reasoning behind each answer.
Tip: Instead of just reading, cover the solution and try to solve the problem again after a short break. This active recall technique greatly improves retention.
This focused approach helps you quickly revise the entire exercise.
8. How do you prove that zero is a rational number by writing it in the p/q form?
Instruction: To prove zero is a rational number, demonstrate that it can be written in the form p/q, where p and q are integers and the denominator q is not equal to zero. This is a fundamental concept from class 9 maths chapter 1 exercise 1.1.
Why it matters: Understanding this confirms that zero meets the primary condition of being a rational number, which is a common question in assessments. It clarifies the rules governing number systems.
Example:
- Let p = 0.
- Choose any non-zero integer for q, for instance, q = 1. The fraction is 0/1.
- Choose another non-zero integer, like q = -7. The fraction is 0/-7.
- In both cases, the value of the fraction is 0, and the denominator is not zero.
Check: The only condition for a number to be rational is that it can be expressed as p/q where q ≠ 0. Since 0 can be written as 0/1, 0/2, etc., it is a rational number.
9. What is the best way to use the Class 9 number system exercise 1.1 solutions PDF for offline study?
Instruction: Download the Free PDF of the solutions to create an offline resource for uninterrupted practice, especially when you don't have internet access. This ensures you can study for class 9 maths chapter 1 exercise 1.1 anytime, anywhere.
Steps:
- Click the 'Download PDF' link and save the file to a dedicated folder on your computer, tablet, or phone.
- Before a study session, open the PDF alongside your NCERT textbook.
- Attempt each exercise question in your notebook first, without looking at the solutions.
- After completing a set of questions, use the PDF to cross-verify your answers and methods.
- Highlight or make notes on questions where your method was incorrect for later review.
Tip: Using the PDF offline helps minimize distractions from online notifications, allowing for more focused study sessions and better concentration on the material.
10. How do you find rational numbers between two fractions with the same denominator, like 3/5 and 4/5?
Instruction: To find rational numbers between two fractions like 3/5 and 4/5, create equivalent fractions by multiplying the numerator and denominator of both by a suitable integer. This widens the numerical gap between them.
Steps:
- Decide how many rational numbers you need. Let's say you need 3 (n=3).
- Multiply the numerator and denominator by a number greater than n, like n+1 = 4.
- Convert the fractions: 3/5 becomes (3×4)/(5×4) = 12/20.
- Similarly, 4/5 becomes (4×4)/(5×4) = 16/20.
- Now, list the fractions between 12/20 and 16/20: 13/20, 14/20, 15/20.
Check: Verify that the new fractions (e.g., 13/20) are numerically greater than the original starting fraction (3/5 or 0.6) and less than the ending fraction (4/5 or 0.8).
This method provides a reliable way to solve such problems in the class 9 number system exercise 1.1 solutions cbse syllabus.
