What are real numbers in Class 10 Maths?

Real numbers are all the numbers that can be represented on a number line. They include natural numbers, whole numbers, integers, rational numbers, and irrational numbers. In this chapter, students learn how different types of numbers are related and how they can be used in mathematical operations. Real numbers are important because they form the foundation of algebra, geometry, trigonometry, and higher mathematics. Understanding real numbers helps students solve problems involving divisibility, decimal expansions, and number properties. The chapter also introduces concepts that are frequently used in board examinations and competitive exams.

What is the Euclid Division Lemma and why is it important?

The Euclid Division Lemma states that for any two positive integers, dividend and divisor, there exist unique whole numbers quotient and remainder such that dividend equals divisor multiplied by quotient plus remainder. This concept is important because it forms the basis of the Euclidean Algorithm used to find the Highest Common Factor (HCF) of two numbers. The method simplifies calculations and helps students solve application-based questions. Understanding this lemma also improves logical thinking and provides a strong foundation for advanced number theory concepts studied in higher classes.

How do you find the HCF of two numbers using the Euclidean Algorithm?

The Euclidean Algorithm is a systematic method for finding the HCF of two numbers. First, divide the larger number by the smaller number. Then divide the divisor by the remainder obtained. Continue this process until the remainder becomes zero. The last non-zero remainder is the HCF of the given numbers. This method is faster and more reliable than listing factors, especially for large numbers. Questions based on this algorithm are commonly asked in examinations because they test understanding of number properties and application of mathematical procedures.

What is the relationship between HCF and LCM?

The relationship between HCF and LCM of two positive integers is given by the formula: HCF × LCM = Product of the two numbers. This formula helps students verify answers and solve problems efficiently. It is especially useful when either the HCF or LCM is known and the other value needs to be calculated. Understanding this relationship strengthens problem-solving skills and provides insight into the connection between factors and multiples. Many board examination questions are directly based on this concept, making it one of the most important topics in the chapter.

What are terminating and non-terminating decimals?

A terminating decimal is a decimal number that ends after a finite number of digits, such as 0.25 or 3.125. A non-terminating decimal continues indefinitely. If the digits repeat in a pattern, it is called a non-terminating recurring decimal. If there is no repeating pattern, it is a non-terminating non-recurring decimal. Rational numbers have either terminating or recurring decimal expansions, while irrational numbers have non-terminating non-recurring decimals. This concept helps students identify the nature of numbers and understand their representation in different mathematical forms.

How can we determine whether a rational number has a terminating decimal expansion?

To determine whether a rational number has a terminating decimal expansion, first express it in its simplest form. Then examine the prime factors of the denominator. If the denominator contains only the prime factors 2 and/or 5, the decimal expansion will terminate. If any other prime factor appears, the decimal expansion will be non-terminating recurring. This rule provides a quick and accurate way to classify rational numbers. Questions based on this concept are frequently included in school tests and board examinations because they test conceptual understanding.

Why are irrational numbers important in mathematics?

Irrational numbers are numbers that cannot be expressed as a ratio of two integers. Their decimal expansions are non-terminating and non-recurring. Examples include square roots of non-perfect squares and certain special constants. Irrational numbers are important because they help represent exact values in geometry, algebra, and trigonometry. Many measurements in real life involve irrational numbers. Understanding their properties allows students to appreciate the completeness of the real number system and prepares them for advanced mathematical concepts in higher classes.

NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers

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NCERT Solutions for Class 10 Maths Chapter 1: Real Numbers

If you are preparing for your Class 10 CBSE board exams and looking for reliable NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers, you have landed at the right place. Chapter 1 of Class 10 Maths is the foundation of number theory, and understanding it thoroughly will help you solve problems related to Euclid's Division Lemma, HCF, LCM, and the nature of decimal expansions. At Myclass24, we have designed these solutions to be simple, accurate, and step-by-step — exactly the way CBSE expects you to present answers in your board exam.

Real Numbers is a topic that carries direct marks in board exams, and mastering it with the right NCERT solutions can make a real difference in your score. Our solutions are updated as per the latest CBSE curriculum and cover all exercises — Exercise 1.1, Exercise 1.2, and Exercise 1.3 — along with important theorems and proofs. Students who go through these solutions regularly find that they gain not just the answers, but also a deep conceptual clarity that helps in MCQs, short-answer questions, and long-answer questions.

Download PDF – NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers

Students across India can access and download the complete PDF of NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers from Myclass24. The PDF is structured exercise-wise and includes solutions to every question from the NCERT textbook. It is mobile-friendly and can be saved to your phone or tablet for offline study — ideal for students who study during commutes or in areas with limited internet connectivity.

📄 Exercise-1.1

📄 Exercise-1.2

📄 Exercise-1.3

📄 Exercise-1.4

Chapter 1 Real Numbers – Key Concepts, Facts & Exercise-Wise Overview

NCERT Class 10 Maths Chapter 1 Real Numbers covers some of the most fundamental concepts of mathematics. It starts with Euclid's Division Lemma, which states that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. This lemma is the backbone of finding the HCF of two numbers using the division algorithm. The chapter then moves to the Fundamental Theorem of Arithmetic, which states that every composite number can be expressed as a product of primes in a unique way (ignoring the order). This theorem is used extensively to find the HCF and LCM of numbers. The chapter also covers irrational numbers, proving that numbers like √2, √3, and √5 are irrational, and explains the nature of decimal expansions — terminating vs. non-terminating repeating — based on the prime factors of the denominator.

Exercise	Topics Covered	No. of Questions	Difficulty
Exercise 1.1	Euclid's Division Lemma & Algorithm	4	Easy–Medium
Exercise 1.2	Fundamental Theorem of Arithmetic, HCF & LCM	7	Medium
Exercise 1.3	Irrational Numbers & Decimal Expansions	3	Medium–Hard

Concept	Key Fact
Euclid's Division Lemma	a = bq + r, 0 ≤ r < b
Fundamental Theorem of Arithmetic	Every composite number has a unique prime factorisation
HCF × LCM	= Product of the two numbers (for two numbers)
Terminating Decimal	Denominator has only 2 and/or 5 as prime factors
Non-Terminating Repeating	Denominator has prime factors other than 2 and 5
√2, √3, √5	All are irrational numbers (proof by contradiction)

One important fact students often overlook is that the HCF of two numbers can never be greater than the smaller of the two numbers, while the LCM can never be less than the larger. These properties are frequently tested in objective-type questions. In Exercise 1.1, students apply Euclid's Division Algorithm to real values, which also appears in the context of finding the HCF of large numbers efficiently. Chapter 1 also lays the groundwork for understanding why certain numbers like 1/7 or 1/3 give non-terminating repeating decimals — a concept that directly links to rational number theory covered in earlier classes.

Question Type	Marks	Frequency in CBSE Exams
Prove irrational (e.g., √2, 3+√5)	2–3 Marks	Very High
Find HCF using Euclid's Algorithm	2 Marks	High
HCF & LCM using Prime Factorisation	3 Marks	High
Decimal expansion – terminating or not	1–2 Marks	Medium