RD SHARMA Solutions for Class 10 Mathematics Chapter 1: Real Numbers

This chapter is a continuation of the Real Numbers' topic that the students were first introduced to in Class 9. Chapter 1- Real Numbers introduces two important properties of positive integers, namely 'Euclid's Division Algorithm' and 'The Fundamental Theorem of Arithmetic'.

It also explains Euclid's Division Lemma that states that: Given positive integers a and b (say), there exist unique integers, q and r satisfying a=bq+r, 0≤ r

R.D. Sharma for Class 10, Chapter 1 explains Euclid's division algorithm in detail, which, as the name suggests, has to do with the divisibility of integers. It states that any positive integer a can be divided by another positive integer b so that it leaves behind a remainder r that is smaller than b. This method is slightly similar to the Long Division Method.

Chapter 1-Real Numbers then sheds light on the Fundamental Theorem of Arithmetic used to multiply positive integers. It states that every composite number can be expressed as a product of primes in a unique way. It finds its use to prove the irrationality of numbers and explore when exactly the decimals of a rational number are terminating or non-terminating. We find the prime factors of the numbers in p/q and express them using the fundamental theorem of arithmetic. The expression of the numbers in prime factors helps simplify the decimal expansion and determine whether the given number would be irrational or rational by means of non-terminating and non-repeating decimals or terminating and repeating decimals, respectively

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