Real Numbers Class 10 Notes Maths Chapter 1


1. Euclid’s Division Lemma: For given any two positive integers a and b,
there exist unique integers q and r satisfying
a = bq + r, 0 :.::; r < 2.

2. Lemma : A lemma is a proven statement used for proving another statement.

3. Euclid’s Division Algorithm: Euclid’s division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers. To get HCF of two positive integers c and d, c > d following steps are to be followed:
(I) Apply Euclid’s division lemma to c and d to get whole numbers q and r such that c = dq + r, 0 :.::; r < d.
(ii) If r = 0, then d is HCF of c and d. If r ‘I’- 0, apply division lemma to d and r.
(iii) Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

(I) Euclid’s division lemma and algorithm are so closely interlinked that people often call former as the division algorithm also.
(ii) Euclid’s division algorithm is stated for only +ve integers but it can be extended for all integers except zero.

4. Fundamental Theorem of Arithmetic: 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.
The prime factorization of a natural number is unique, except for the order of its factors. In general, given a composite number x, we factorize it as
x = p1 p2 p3 ·······.Pn, where p1, p2, p3 , Pn are primes and written in
ascending order, i.e.,
p1 :-s; p2 :-s; p3:.:.:.; :.::; Pn· Ifwe combine the same primes, we will get powers
of primes.
For Example:

So, in each of the cases prime factors of l56 is 2 x 2 x 3 x 13
Hence, we can conclude that the prime factorisation of a number is unique.

(i) For two positive integers a, b
HCF (a, b) x LCM (a, b)=ax b
(ii) If p is a prime number and it divides a2, then p also, divides a where ‘a’
is the positive integer.

5. HCF & LCM of Three Numbers

6. Irrational Numbers : Number which is not a rational number whose decimal expansion is non-terminating and non-repeating

7. Rational Numbers and Their Decimal Expansion
(I) If denominator of a rational number is of the form 2n sm, where n, mare non-negative integers then x has decimal expansion which terminates.
(ii) If decimal expansion of rational number terminates then its denominator has prime factorisation of the form 2n sm, where n, mare non-negative integers.
(iii) If denominator of a rational number is not of the form 2n 5m, where n and m are non-negative integers then the rational number has decimal expansion which is non-terminating repeating
Thus we conclude that the decimal expansion of every rational number is either terminating or non-terminating repeating.

Regarding decimal expansion of rational number x =pq integers and q not = 0, we have, where p, q are co-prime
(I) x is a terminating decimal expansion if the prime factorization of q is of the form 2m 5n where m, n are non-negative integers.
(ii) If prime factorisation of q is not of the form 2m 5n then x is a non-terminating repeating decimal expansion

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