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**â‡’ Euclidâ€™s Division Lemma/Euclidâ€™s Division Algorithm:**

Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 â‰¤ r < b. This statement is nothing but a restatement of the long division process in which q is called the **quotient** and r is called the **remainder**.

â‡’ **Lemma** is a proven statement used for proving another statement.

â‡’ Euclidâ€™s Division Algorithm can be extended for all integers, except zero i.e., bâ‰ 0.**â‡’ HCF of two positive integers:**

HCF of two positive integers a and b is the largest integer (say d) that divides both a and b and is obtained by the following method:**Step 1.** Obtain two integers q and r, such that a = bq + r, 0 â‰¤ r < b.**Step 2.** If r = 0, then b is the required HCF.**Step 3.** If r â‰ 0, then again obtain two integers using Euclidâ€™s Division Lemma and continue till the remainder becomes zero. The divisor when remainder becomes zero, is the required HCF.**â‡’ The Fundamental Theorem of Arithmetic : **

Every composite number can be factorised as a product of primes and this factorisation is unique, apart from the order in which the prime factors occur.**â‡’ Irrational Number:**

A number is an irrational if and only if, its decimal representation is non-terminating and non-repeating (non-recurring).**Or**

A number which cannot be expressed in the form of p/q, q â‰ 0 and p, q âˆˆ I, will be an irrational number. The set of irrational numbers is generally denoted by S.

â‡’ The rational number p/q will have a terminating decimal representation only, if in standard form, the prime factorisation of q, the denominator is of the form 2^{n} 5^{m}, where n, m are some non-negative integers.

â‡’ The rational number p/q will have a non-terminating repeating (recurring) decimal representation, if in standard form, the prime factorisation of q, the denominator is not of the form 2^{n} 5^{m}, where n, m are some non-negative integers.

â‡’ The decimal expansion of every rational number is either terminating or non-terminating repeating.

â‡’ If p is a prime and p divides a^{2}, then p divides a, where a is a positive integer.

â‡’ For any two positive integers p and q, we have

HCF (p,q) x LCM [p, q] = p x q

â‡’ For any three positive integers a, b and c, we have

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