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

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 2ⁿ5^m, 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 2ⁿ5^m, 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 2ⁿ5^m , where n, m are non-negative integers. Then, x has a decimal expansion which is non-terminating, repeating (recurring).