Class 10 Maths Chap 1 Real Numbers Ex 1.4 NCERT introduces students to the concept of revisiting rational and irrational numbers through the lens of decimal expansions. In this part of the chapter, you will learn how to use Euclid’s Division Lemma to prove the irrationality of numbers like √2, √3, and √5. These questions are theoretical and help strengthen your understanding of number properties—an important concept in competitive exams too. Our detailed and step-by-step NCERT solutions for Ex 1.4 will make it easier for you to grasp the logic behind each proof and write answers in the way CBSE exam
Class 10 Maths Chap 1 Real Numbers Ex 1.4 NCERT Full Chapter
Question 1: Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or non-terminating repeating decimal expansion:
(i) 13 / 3125
3125=553125 = 5^53125=55 → Only prime factor is 5
Terminating decimal
(ii) 17 / 8
8=238 = 2^38=23 → Only prime factor is 2
Terminating decimal
(iii) 64 / 455
455=5×7×13455 = 5 × 7 × 13455=5×7×13 → Contains prime factors other than 2 or 5
Non-terminating repeating decimal
(iv) 15 / 1600
1600=26×521600 = 2^6 × 5^21600=26×52 → Only 2 and 5
Terminating decimal
(v) 29 / 343
343=73343 = 7^3343=73 → Contains prime factor 7
Non-terminating repeating decimal
(vi) 23 / (2³ × 5²)
Only 2 and 5 in denominator
Terminating decimal
(vii) 129 / (2² × 5 × 7⁵)
Denominator contains 7
Non-terminating repeating decimal
(viii) 6 / 15
Simplify to 2/5 → Only 5 in denominator
Terminating decimal
(ix) 35 / 50
Simplify to 7/10 → 10=2×510 = 2 × 510=2×5
Terminating decimal
(x) 77 / 210
210=2×3×5×7210 = 2 × 3 × 5 × 7210=2×3×5×7 → Contains 3 and 7
Non-terminating repeating decimal
Ex 1.4 Class 10 Maths Question 2.
Write down the decimal expansions of those rational numbers in question 1, which have terminating decimal expansions.
Answer:
(i) 13 / 3125 = 0.00416
(ii) 17 / 8 = 2.125
(iv) 15 / 1600 = 0.009375
(vi) 23 / (2³ × 5²) = 23 / 200 = 0.115
(viii) 6 / 15 = 0.4
(ix) 35 / 50 = 0.7
Ex 1.4 Class 10 Maths Question 3.
The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational and of the form pq, what can you say about the prime factors of q?
(i) 43. 123456789
(ii) 0.120120012000120000…
(iii) 43. 123456789¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
(i) 43.123456789
Solution:
- This is a terminating decimal.
- So, it is a rational number.
- Since the decimal terminates, the denominator qqq (after simplification) has only prime factors 2 and/or 5.
Rational; Prime factors of qqq: only 2 or 5 (or both).
(ii) 0.120120012000120000…
- This is non-terminating and non-repeating (the pattern of 120, 1200, 12000… keeps changing).
- So, it is not a rational number.
Irrational number
(iii) 43.123456789̅ (bar over 123456789, i.e., repeating block)
Since the decimal repeats, it can be expressed as pq\frac{p}{q}qp, but the prime factorization of qqq may include primes other than 2 or 5.
Rational; Prime factors of qqq: may include primes other than 2 and 5
This is a non-terminating but repeating decimal.
So, it is a rational number.
New Syllabus – Class 10 Maths Chap 1 Real Numbers Ex 1.4 NCERT: Detailed Summary
1. Fundamental Theorem of Arithmetic
- Statement:
Every composite number can be written as a product of prime numbers,
and this factorisation is unique — the only difference may be the order of the prime factors. - Example:
60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
Another way: 3 × 2 × 5 × 2 — same factors, different order. - This theorem is the basis for many operations in number theory, such as finding HCF and LCM.
2. Prime Divisibility Theorem
- If a prime number
pdivides a², then p divides a. - This is used in proving irrationality of square roots of primes.
- Example:
If 2 divides a², then 2 must divide a.
3. Irrational Numbers
- Definition:
A number that cannot be expressed in the form p/q, where p and q are integers and q ≠ 0. - Examples of irrational numbers: √2, √3, √5, π, etc.
- Using proof by contradiction, it is shown that:
- √2 is irrational
- √3 is irrational
- √5 is irrational
- The idea is:
- Assume the number is rational (say √2 = a/b)
- Show that both a and b must be divisible by the same prime (like 2)
- This contradicts the assumption that a and b are coprime
⇒ Therefore, the number is irrational
4. Operations with Irrationals
- Sum or difference of a rational number and an irrational number is irrational
- Example: 5 − √3 is irrational
- Product or quotient of a non-zero rational and an irrational number is irrational
- Example: 7√5 is irrational
5. HCF and LCM using Prime Factorisation
- HCF: Product of smallest powers of common prime factors
- LCM: Product of highest powers of all prime factors involved
- Example:
- 6 = 2 × 3
- 20 = 2² × 5
- HCF(6, 20) = 2
- LCM(6, 20) = 2² × 3 × 5 = 60
- 6 × 20 = HCF × LCM → 120 = 2 × 60 ✔️
- Note: For 3 numbers, product of HCF and LCM is not always equal to the product of the three numbers.
The Class 10 Maths Chap 1 Real Numbers Ex 1.4 NCERT help students understand how to prove the irrationality of numbers like √2, √3, and √5 through logical reasoning and structured arguments. This exercise focuses on applying the proof by contradiction method, a fundamental concept in mathematics that forms the basis of many higher-level topics.
By practicing these solutions, students not only learn to differentiate between rational and irrational numbers but also gain confidence in constructing mathematical proofs step by step. These types of questions are frequently asked in CBSE board exams and are important for competitive exams like NTSE and various Olympiads.
What makes the Class 10 Maths Chap 1 Real Numbers Ex 1.4 NCERT particularly valuable is their clear explanation and easy-to-understand format. They guide students through each proof, helping them develop logical thinking and accuracy in their answers.
Make sure to review these concepts regularly and practice writing the proofs in your own words. If you’ve understood this exercise well, you’re ready to move on to Exercise 1.5, which deals with the Fundamental Theorem of Arithmetic and its applications.
Keep learning, keep practicing, and build a strong mathematical foundation with these well-structured NCERT solutions.
Conclusion – Class 10 Maths Chap 1 Real Numbers Ex 1.4 NCERT
The NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Ex 1.4 offer a deep understanding of one of the most important concepts in number theory: irrational numbers and their proofs. This exercise emphasizes how to prove the irrationality of certain non-perfect square roots like √2, √3, and √5 using the proof by contradiction technique. Such questions may look theoretical at first, but they play a crucial role in helping students develop mathematical reasoning and logical thinking.
Each solution in Exercise 1.4 guides students through clear and systematic steps, ensuring that the logic is easy to follow. These questions often appear in the CBSE board exams and help build a foundation for higher-level topics like quadratic equations, surds, and real analysis.
The NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Ex 1.4 not only improve your understanding but also train you to write mathematically correct and examiner-friendly answers. By mastering this exercise, you are enhancing your proof-writing skills and preparing for academic and competitive success.
Continue practicing regularly, and revisit these solutions whenever you revise the chapter. Once you’re confident with these concepts, you can move on to Exercise 1.5, where you will explore the Fundamental Theorem of Arithmetic and its real-life applications.
Visit NCERT Official Website: NCERT Class 10 Maths Book
This resource will provide you with the complete Class 10 Maths syllabus and ensure that you’re following the latest NCERT guidelines. Happy learning! Class 10 Maths Chap 1 Real Numbers Ex 1.4 NCERT