# NCERT Solutions for Class 10 Maths Chapter 1 – Real Numbers

NCERT Solutions for Class 10 Maths Chapter 1 – Real Numbers begins with two very important properties of positive integers, namely Euclid’s division algorithm, which is related to the divisibility of integers. Here you will also learn that Fundamental Theorem of Arithmetic is related to multiplication of positive integers. Chapter 1 of NCERT Solutions for Class 10 further tells you about Euclid’s lemma, which is the basis of Euclid’s division algorithm. Next, you learn about some theorems for irrational numbers. Further, this chapter explores the situations when a decimal expansion of p/q is terminating, and when it is non-terminating, i.e. repeating or recurring.

All the questions asked in different exercises of chapter 1 of CBSE NCERT Class 10 Maths book are framed in order to provide you with enough practice of the concepts taught in this chapter. The questions asked in this chapter utilise knowledge of Euclid’s division algorithm, prime factorisation method, the fundamental theorem of the algorithm and terminating and non-terminating decimal expansions.

Our NCERT Solutions are carefully and meticulously written so that you can understand the topics efficiently. They are outlined in such a way that they make learning fun, interesting, engaging and interactive for you. These help you to build a strong conceptual foundation and also to perform well in your Class 10 board exams.

## NCERT Class 10 Maths Chapter 1: Important Topics

NCERT Solutions for Class 10 Maths Chapter 1 – Real Numbers includes all the important concepts of this chapter 1. They give you a comprehensive knowledge of the subject matter. The main points regarding topics covered in this chapter are as follows –

**Introduction**

This part of the CBSE NCERT Class 10 Maths Chapter 1 introduces you to the concepts of Euclid’s division algorithm related to the divisibility of integers and the Fundamental Theorem of Arithmetic for multiplication of positive integers.

**Euclid’s Division Lemma**

This part of the chapter teaches you about theorem known as Euclid’s Division Lemma according to which if there are two positive integers a and b, two other unique integers q and r exist such that they satisfy the following equation: a = bq + r, 0 ≤ r < b.

Further, you learn that Euclid’s division algorithm is based on this lemma. An algorithm is a sequence of steps which gives a procedure for solving a type of problem.

This chapter also explains Euclid’s division algorithm in detail according to which in order to obtain the HCF of two positive integers, for example, c and d, where c > d, three steps are needed. They are as follows.

Step 1: Applying Euclid’s division lemma, to c and d to find whole numbers, q and r such that c = dq + r, 0 ≤ r < d.

Step 2: In case r is equal to 0, d is the HCF of c and d. If r ≠ 0, division lemma can be applied to d and r.

Step 3: The process is to be continued until the remainder is zero. The divisor at this stage gives the required HCF.

**The Fundamental Theorem of Arithmetic**

This part of NCERT Class 10 Maths Chapter 1 explores whether natural numbers can be obtained by multiplying prime numbers. It further states that every composite number can be factored as a product of primes, and this is known as the Fundamental Theorem of Arithmetic, which is a study of integers. The formal version of this statement is ‘every composite number can be expressed as a product of primes, and this factorisation is unique and independent of the order in which the prime factors occur. You further come to know that this theorem has many applications, in a lot of fields apart from mathematics.

**Revisiting Irrational Numbers**

This part of chapter 1, Class 10 Science Real Numbers talks about irrational numbers where a number ‘s’ is called irrational if it cannot be written in the form, p/q where p and q are integers and q ≠ 0. This chapter further proves that √2, √3, √5 and √p (in general) where p is a prime number. It also teaches you the theorem: If p is a prime number and it divides a2, then a is divided by p, a being a positive integer.

**Rational Numbers and Their Decimal Expansions**

This part of the NCERT CBSE Class 10 Science Chapter 1 determines when exactly the decimal expansion of p/q is terminating and when it is non-terminating repeating (or recurring) considering a rational number, p/q where q ≠ 0.

This chapter further gives a theorem which says, if *x* is a rational number whose decimal expansion terminates, then *x* can be stated in the form, p/q given p and q are coprime numbers, and 2n5m is the prime factorisation of q, given, n, and m are non-negative integers. This part of the chapter further concludes that the decimal expansion of every rational number is either terminating or non-terminating repeating.

## CBSE NCERT Class 10 Maths Chapter 1 – Real Numbers Exercises

Our NCERT Solutions for Class 10 Maths Chapter 1 – Real Numbers consists of 4 exercises covering the important topics learnt here. The constituents of the exercises have been listed below.

**Exercise 1.1**

There are a total of 5 questions asked in the first exercise. Question 1 and 3 ask you to utilise Euclid’s division algorithm to find the answers to given direct questions and word problems, respectively. Question 2, 4 and 5 are based on Euclid’s division lemma. Our solutions provide step by step answers to these questions.

**Exercise 1.2**

This exercise contains 7 questions in total. Question 1 asks you to express the give numbers as a product of its prime factors. Question 2, 3 and 4 are based on finding LCM and HCF of the given pairs of integers. Question 5 and 6 ask you to use the prime factorisation method for solving the question. Question 7 is a word problem based on the concepts taught. NCERT solutions to these questions are given in a stepwise manner for easy understanding.

**Exercise 1.3**

This particular exercise contains only 3 questions, all of which are based on proving the given numbers or number expressions irrational in nature. To solve these questions, you need to assume certain conditions first and then reach a conclusion. Our solutions have used the best methods to solve these questions.

**Exercise 1.4**

This exercise also consists of 3 questions. Question 1 and 2 are based on terminating decimal expansion or a non-terminating repeating decimal expansion rational number.

Question 3 asks you to tell if the given numbers are rational or not based on the values of their decimal expansions. NCERT Solutions for Chapter 1 provides methodical answers to all these questions.

NCERT Solutions for Chapter 1 – Real Numbers are prepared using the best methods to solve these types of questions and help you understand the concepts very well. These are highly beneficial for revising and retaining the fundamentals of the chapter and ensure you score good marks in class 10th board exams.