# RD Sharma Class 10 Chapter 1 Solutions (Real Numbers)

RD Sharma Solutions for Class 10 Maths Chapter 1-Real** **Numbers** **have been framed by expert Maths teachers. The solutions to the chapter are drafted as per the latest RD Sharma Class 10 Maths Solutions Syllabus. This chapter introduces you to real numbers. Here you will be learning about Euclid’s Division Lemma and Euclid’s Division Algorithm. This chapter also introduces you to the Fundamental Theorem of Arithmetic, prime factorisation, method of finding LCM, HCF, and application of HCF and LCM in real-world problems.

Chapter 1 of RD Sharma Class 10 Maths consists of 6 exercises with around 77 questions in it. Our RD Sharma Class 10 Maths Solutions Chapter 1-Real Numbers answer all questions in an easy understand language and in a step by step manner. They** **have been devised in a manner to help you build a strong conceptual foundation. The exercises cover all the major topics and the sub-topics of the chapter in a crisp and concise manner. They give you ample amount of practice to become thorough with the chapter and score high marks in your Class 10 Board exams.

Our RD Sharma for Class 10 Maths Solutions contain smart study techniques and innovative Maths shortcuts that help you learn and revise quickly and efficiently. They have been prepared in a manner to make learning more fun and interactive for the students.

## RD Sharma Class 10 Maths Chapter 1

RD Sharma Class 10 Maths Solutions Chapter 1-Real Numbers discusses all aspects related to Real Numbers in details. Some of the topics of the chapter have been discussed below.

**Introduction**

This part of RD Sharma Class 10 Maths Chapter 1 – Real Numbers, explains the meaning and attributes of the Real Numbers. It teaches you that real numbers constitute a union of irrational and rational numbers. It also tells you that all real numbers can be plotted on a number line.

**Euclid’s Division Lemma**

This section also teaches you about Euclid’s Division Lemma. It states that the given two integers a and b, there always exists a unique pair of integers which are named as q and r such that a=b×q+r and 0≤r<b.

**Euclid’s Division Algorithm**

This part of Class 10 Maths Real Numbers teaches you about Euclid’s algorithm theorem. It tells you how Euclid’s Division Algorithm is based on Euclid’s Division Lemma.

Here you will also come to know how to find the HCF of the two given numbers with Euclid’s Division Algorithm say p and g, where p>g. Now, if we are applying Euclid’s theorem to this, then p=g×q+r and 0≤r<g.

Suppose, if we consider r=0, with the HCF as g, now, if we apply Euclid’s division Lemma to g(the divisor) and r (the remainder). With this, we get another pair of remainder and quotient. This method is repeated until a remainder 1 is attained. The divisor that step is the HCF of the given set of numbers.

**The Fundamental Theorem of Arithmetic**

This section of RD Sharma Solutions for Class 10 Maths Chapter 1-Real Numbers, introduces you to prime factorisation and how it is the method of representing a natural number through the multiplication of prime numbers. For example, 36=2 x 2 x3 x 3 is the prime factorisation of 36.

It also explains the Fundamental Theorem of Arithmetic which states that the prime factorisation of the given number should be unique if the arrangement of prime factors is ignored.

For Example 36= 2 x 2 x 3 x 3 or 36 = 2 x 3 x 2 x 3.

In this section, you’ll also learn the method of finding LCM, method of finding HCF, the product of two numbers which is HCF X LCM of the two numbers and application of HCF and LCM in real-world problems.

**Revisiting Irrational Numbers**

This element of Class 10 Maths Real Numbers, explains about the irrational numbers. Basically, these are the numbers which cannot be expressed in the form of a fraction (p/q). Examples of Irrational numbers are √2,π, and e.

You’ll also learn about the number theory, proof of contradiction.

**Revisiting Rational Numbers and Their Decimal Expansions**

This element of RD Sharma Solutions for Class 10 Maths Chapter 1-Real Numbers, teaches you about the terminating and non-terminating decimals.

- When a number stops at a certain point, then it is called terminating decimal.

Example: 8.42, 7.58. 1.23, and so on. - When a decimal number doesn’t terminate after a certain point, then it is considered as non-terminating decimals.

Example: 3.4444444444444….., 5.6666666666666….

Further, you’ll also learn how to check if a given rational number is terminating or not.

### RD Sharma Solutions for Class 10th Maths Chapter 1 – Real Numbers Exercise Questions

There are a total of 6 exercises at the end of the Class 10 Maths Real Numbers. Below is a brief overview of the same.

**Exercise 1.1:**

- This exercise consists of 17 questions in total. Questions 1 to 9, 12 and 15 are theorem based questions from Euclid’s division algorithm.
- Question 4, 10, 11, 14 and 17 are asked from Euclid’s division lemma topic. Question 13 and 16 is a scenario-based question which is asked from Euclid’s division lemma and the fundamental Theorem of Arithmetic

**Exercise 1.2: **

- This exercise consists of 22 questions in total. In which, question 1, 3, 5, 6 and 22 are based on methods of finding HCF concepts.
- Question 2 and 16 are based on Euclid’s division algorithm. Question 4, 7, 8, 9, 10, 17, 18, 19, 20 and 21 are scenario-based questions. These questions are asked from the finding HCF topic. Question 11, 12, 13, 14, 15 and 16 are based on the fundamental theorem of arithmetic concept.

**Exercise 1.3: **

- This exercise consists of 5 questions in total. In this exercise, all the questions are based on the prime factorisation method. Where question 1 and 2 are expressing the given integers as a product of its prime.
- Question 3 and 5 are based on composite numbers. You’ll have to solve this used prime factorisation.

**Exercise 1.4:**

- This exercise consists of 16 questions in total. In which, question 1 to 7 asks you to find the LCM and HCF of the given pair of integers.
- Question 8, 9, 10 and 14 asks you to find the smallest number which could be divided by the given numbers. Question 11, 15 and 16 are scenario-based questions where these are based on real-world problems of HCF and LCM.
- Question 12 and 13 asks you to find the largest number, which is divisible by the given numbers.

**Exercise 1.5:**

- This exercise consists of 12 questions, and all the questions are based on rational and irrational numbers. All the questions are of the same pattern, and you’ll have to prove the questions based on the equation or the prime positive integers.

**Exercise 1.6:**

- This exercise consists of 5 questions in total. Question 1, 2 and 3 are based on rational numbers. In this, you’ll have to find the decimal expansions of the mentioned numbers.
- Question 4 is based on the prime factorisation method. Question 5 is based on the rational number.

### Benefits of RD Sharma Class 10 Maths Chapter 1

- Our experienced Maths Teachers have drafted these solutions in a simple language to make your board exam preparation more efficient, effective and quick.
- All topics and sub-topics from this chapter of the Class 10 Maths Syllabus have been comprehensively and sufficiently covered.
- They have been designed to make learning fun, interesting, stimulating and a stress-free process.