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Real Numbers

    Introduction

    Irrational Numbers

    Real Numbers

    Representing Real Numbers on a Numberline Through Successive Magnification

    Operations on Real Numbers

    Surd

    Exercise 1.1

    Exercise 1.2

    Exercise 1.3

    Exercise 1.4

    What We Have Discussed

    Enhanced Curriculum Support

    Easy Level Worksheet

    Moderate Level Worksheet

    Hard Level Worksheet

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Real Numbers > Enhanced Curriculum Support

Enhanced Curriculum Support

This is a comprehensive educational resource designed to provide students with the tools and guidance necessary to excel. This support system is structured to cater to various aspects of learning, ensuring that students are well-prepared for academic challenges and practical applications of mathematical concepts. Some are the key benefits are mentioned below:

1.Comprehensive Learning: This holistic approach helps students gain a thorough understanding of the subject. Practical Application: The resources encourage students to apply mathematical concepts to real-life scenarios, enhancing their practical understanding and problem-solving skills.

2.Critical Thinking and Reasoning: Value-Based and HOTS questions promote critical thinking and reasoning abilities. These skills are crucial for students to tackle complex problems and make informed decisions.

3.Exam Preparedness: Sample Question Papers and Practice Questions provide ample practice for exams. They help students familiarize themselves with the exam format and types of questions, reducing exam anxiety.

4.Ethical and Moral Development: Value-Based Questions integrate ethical and moral lessons into the learning process, helping in the overall development of students' character and social responsibility. By incorporating these diverse elements, Enhanced Curriculum Support aims to provide a robust and well-rounded knowledge, preparing students for both academic success and real-world challenges.

Sample Question Paper

About the Section

Sec A

1. Explain whether the number 0 is a rational number or an irrational number. Justify your answer.

2. Determine whether the following statements are true or false:

(i) Every integer is a rational number.

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

3. Simplify and express in decimal form: 511.

4. Write the following in standard form: 0.0000045.

5. Write 73as a decimal.

6. Find the value of x if x = 5.

7. Simplify : 64 × 9.

8. Find the value of x if 1x = 2.

9. Find the value of 1634.

10. Write a rational number between 12 and 34.

11. Simplify ∛2 × ∛32.

12. The value of 20+3050 is: (A) 0 (B) 1 (C) 65 (D) 2

13. Number of perfect squares between 1 and 50. (A) 5 (B) 7 (C) 6 (D) 8

14. pq form of 3.8 is: (A) 389 (B) 359 (C) 349 (D) 329

15. 121 = ?: (A) 0.5 (B) 0.25 (C) -4 (D) 4

Sec B

1. The simplest rationalisation factor of 22−3 is:

(a) 2+3

(b) 22+3

(c) 2−3

(d) 22−3

2. Simplify and express in simplest form: 53+3123

3. Express 0.0027 in standard form.

4. Simplify the following: 49 + 16.

5. Prove that 6 is an irrational number.

6. Simplify 45+205.

7. Find the square root of 121 using prime factorization.

8. Express 1.252525..... in the form pq, where 'p' and 'q' are integers and q ≠ 0.

9. Is 930 a terminating decimal? Justify your answer.

10. Give an example of two irrational numbers whose quotient is a rational number.

11. Represent 45 and 54 on separate number lines.

12. By using suitable identity, find the value of 983.

Sec C

1. Verify whether 2 is a rational or irrational number. Use a proof by contradiction to justify your answer.

2. Prove that the sum of a rational number and an irrational number is always irrational.

3. Simplify the expression 12 + 27 - 3.

4. Solve for x if x+1 = 4.

5. Solve the following : x = 12.

6. Find the square of 7 + 2 and simplify.

7. Visualise 4.26̄ on the number line upto 4 decimal using successive magnification.

8. If 4−354+35 = a + b 5, then find the value of 'a' and 'b'.

Sec D

1. Show that 2 + 3 is irrational.

2. Prove that the sum of two irrational numbers can be rational in some cases.

3. Simplify the following: 12×7527.

4. Express 56 and 0.83\overline{3} as decimals and fractions, and compare them.

5. Prove that the square root of a non-perfect square is always irrational.

6. If x = 8 + 32, simplify the expression and find the value of x.

7. Express 0.6\overline{3} as a fraction and explain how it is derived.

8. Find the product of 45 × 5 and simplify.

9. Using prime factorization, find the square root of 324 and explain the steps.

10. Show that the sum of two irrational numbers, 2 + 5, is irrational.

Value-Based Questions

About the Section

Problem 1

A school committee is planning a field trip for 72 students. If each bus can accommodate 36 students, how many buses are required for the trip? Discuss the significance of division and rounding in logistical planning for group activities.

Problem 2

Maya wants to distribute 300 notebooks and 450 pens among students in her neighborhood. If each student receives the same number of notebooks and pens, what is the maximum number of students that can receive the stationery items without any left over? Illustrate how division helps in fair distribution of resources.

Problem 3

Akash and Neha are organizing a charity drive in their school to raise funds for underprivileged children. They collected a total of ₹15,000. They decide to distribute the funds among three charities: Charity A, Charity B, and Charity C, in the ratio of 2:3:5. How much money will each charity receive? Discuss the importance of understanding ratios and proportions in equitable distribution.

HOTS

About the Section

Q1

Raju observed that the sum of the digits of any multiple of 9 is always divisible by 9. Prove this statement using mathematical reasoning and provide examples.

Q2

Ananya claims that the square root of any positive integer is either an integer or an irrational number. Discuss with examples and counterexamples to validate her claim.

Practice Questions

About the Section

Questions

1. If a = 5+26 and b = 1a , then what will be the value of a2+b2?

2. Decimal representation of a rational number cannot be:

(a) terminating

(b) non-terminating

(c) non-terminating repeating

(d) non-terminating non-repeating

3. Which of the following is irrational?

(a) 49

(b) 123

(c) 7

(d) 81

4. Which of the following is irrational?

(a) 0.14

(b) 0.1416

(c) 0.1416

(d) 0.4014001400014........

5. A rational number between 2 and 3 is:

(a) 2+32

(b) 232

(c) 1.5

(d) 1.8

6. 23+3 is equal to:

(a) 26

(b) 6

(c) 33

(d) 46

7. If 2 = 1.4142 then 2−12+1 is equal to:

(a) 2.4142

(b) 5.8282

(c) 0.4142

(d) 0.1718

8. The product of 21321432112 equals:

(a) 2

(b) 2

(c) 2112

(d) 32112

9. Find which of the variables x, y, z and u represent rational numbers and which irrational numbers :

(a) x2=5

(b) y2=9

(c) z2=0.04

(d) u2=174

10. Show that 0.142857142857... = 17

11. Express 0.6 + 0.7 + 0.47 in the form pq, where p and q are integers and q ≠ 0.

12. Simplify : 7310+3 - 256+5 - 3215+32

13. Express 0.123 in the form pq , where p and q are integers and q ≠ 0.

14. Find the value of a in the following :

632−23 = 32 - a 3

15. Simplify: 5813+2713314

16. Are there two irrational numbers whose sum and product both are rationals? Justify.

Case-Based Questions

About the Section

Q 1

Rohan is participating in a math competition. The first round of the competition focuses on the number systems, and he is given the following scenario to solve:

Rohan’s school organizes a fundraiser event, and the total amount collected is ₹10,000. The collected amount is to be distributed equally among three different charity organizations: Organization A, Organization B, and Organization C. However, Rohan realizes that the distribution can also be represented using fractions and decimals, and he decides to explore the number system properties in the process.

Based on this scenario, answer the following questions:

1. How much money will each organization receive if the total amount collected is distributed equally among the three organizations? Represent your answer in both fraction and decimal form.

2. Verify your answer by summing the distributed amounts to ensure it equals the total amount collected. Show your calculations.

3. If the distributed amount to each organization is represented as a fraction, simplify the fraction to its lowest terms. Explain the steps taken to simplify the fraction.

4. Convert the fraction obtained in Question 3 to a decimal. Compare the decimal representation with your answer from Question 1 to check for consistency.

Sol 1

1. The total amount collected is ₹10,000, and it is to be divided equally among three organizations.

Amount per organization = TotalAmountNumberofOrganizations = 103

In fraction form: Amount per organization = 103

In decimal form: Amount per organization = 10,000 ÷ 3 = 3,333.33.

2. Total distributed = 3 × 3,333.33 = 9,999.99

There is a small rounding error due to the approximation of decimals. However, the distribution checks out logically.

3. The fraction representing the amount per organization is 103.

Since 3 is a prime number and does not divide 10,000 evenly, this fraction is already in its simplest form.

4. Convert 103 to a decimal by performing the division:

10,000 ÷ 3 = 3,333.33

This matches the decimal result from Question 1, confirming consistency.