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# RD Sharma Class 10 Solutions Chapter 1 - Real Numbers (Ex 1.2) Exercise 1.2 Last updated date: 03rd Dec 2023
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## RD Sharma Class 10 Solutions Chapter 1

### Introduction to The Chapter

Chapter 1 Real Numbers of Class 10 form the foundation of the important Mathematical concepts that will be discussed in higher Classes. Real Numbers can be defined as numbers that exist, a quantity that can be expressed as an infinite decimal expansion. The two main components of Real Numbers are rational and irrational numbers. Rational numbers can be further Classified into integers, coal, natural numbers. These are the important topics that are already covered in the previous Classes.

Even if a student does not know these topics, practicing from RD Sharma will give you the much-needed boost and will help you score well in the 10th CBSE board examination. The innumerable number of questions given in the book is tailored for students so that they can practice and gain complete knowledge of the important topics that are necessary to learn and understand and not to get a good score in the examination. To understand Chapter 1 of Class 10 Real Numbers, two important properties of positive integers are needed : Euclid’s division algorithm and the fundamental theorem of arithmetic.

Euclid’s division algorithm can be defined as Euclid’s division algorithm, as the name suggests, has to do with the divisibility of integers. Stated simply, it says any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b. While the theorem of arithmetic means- every composite number can be expressed as a product of primes in a unique way—this important fact is the Fundamental Theorem of Arithmetic, it has a more deep and significant meaning in Mathematics which will be studied later.

### Important Topics Of Chapter 1

The important topics covered in this Chapter are-

• Euclid’s division lemma, - A lemma is a proven statement used for proving another statement.

• theorem 1-: Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.

• fundamental theorem of arithmetic,

• Theorem 2-Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.

• theorem 3- Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer.

• Theorem 4-  Root 2 is irrational

• re-visiting irrational numbers and their decimal expansion

• theorem 5- Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form,p/q where p and q are coprime, and the prime factorization of q is of the form 2n5m, where n, m are non-negative integers.

• Theorem 6 - Let x = p/q be a rational number, such that the prime factorization of q is of the form 2n5m, where n, m are non-negative integers. Then x has a decimal expansion that terminates.

• Theorem 7 - Let x = p/q, where p and q are coprimes, be a rational number, such that the prime factorization of q is not of the form 2n5m , where n, m are non-negative integers. Then, x has a decimal expansion which is non-terminating, repeating (recurring).

## FAQs on RD Sharma Class 10 Solutions Chapter 1 - Real Numbers (Ex 1.2) Exercise 1.2

1.  Why is the Chapter Real Numbers an important Chapter of Class 10?

Chapter 1 Real Numbers of Class 10 is an extremely important Chapter as it covers the basic concepts that form the foundation of Mathematics. It explains in-depth the various concepts like revising the number system, the basis of Euclid’s division lemma, forms of positive integers, questions based on Euclid’s lemma, and the fundamental theorem of arithmetic.

2.  Where can I find RD Sharma solutions of Rational Numbers?

RD Sharma solutions of Chapter 1 rational numbers of Class 10 can be found on Vedanta‘s website where the expert teachers have curated all the content and have made it available for free to all the students. Study material is available for free download in a PDF so that students can take it anywhere with them for revising and can work offline as well without the distraction of the internet.

3.  Define Real Numbers.

Real Numbers can be defined as numbers that exist, a quantity that can be expressed as an infinite decimal expansion. The two main components of Real Numbers are rational and irrational numbers. That can be further Classified. RD Sharma provides a comparative study of the various concepts so that students can use the theorems easily.

4.  Where can I find all the study material related to Real Numbers?

The important questions and revision notes are easily available on Vedantu’s website for free download. The links are easily accessible. After downloading the whole PDF students can focus on their studies without having to access the Internet. Students can revise and learn any time of the day and take them anywhere. Vedantu’s team has curated important questions and revision notes as well as previous year’s question papers. RD Sharma solutions are also available.

5.   Is RD Sharma enough to get a good percentage in board exams?

Rd Sharma is based on the CBSE curriculum set by the CBSE. It explains all the numerical problems in detail and gives reasonable solutions. HOTS are questions of higher aptitude that need more attention and knowledge of the subject. Practicing from RD Sharma is more than enough to get a good percentage in the board exams.