Real Numbers Class 10 Notes Maths Chapter 1

CHAPTER AT A GLANCE

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.

Note:
(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.

Note:
(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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