S U M M A T I V E SSA S S E S S M E N T

CHAPTER

1Real Numbers

SYLLABUS Euclid’s division lemma, Fundamental Theorem of Arithmetic–statements after reviewing work done

earlier and after illustrating and motivating through examples, Proofs of results–irrationality of 2 , 3 , 5 , Decimal expansions of rational numbers in terms of terminating/non-terminating recurring decimals.

QUICK REVIEWÅ Algorithm : An algorithm means a series of well defined steps which gives a procedure for solving a type of

problem.

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

Euclid’s Division Lemma : For given positive integers a and b, there exist unique integers q and r, satisfying

a = bq + r, 0 ≤ r < b. The steps to find the HCF of two positive integers by Euclid’s division algorithm are given below :

(i) Let two integers are a and b such that a > b.

(ii) Take greater number a as dividend and the number b as divisor.

(iii) Now find whole numbers ‘q’ and ‘r’ as quotient and remainder respectively.

∴ a = bq + r, 0 ≤ r < b.

(iv) If r = 0, b is the HCF of a and b. If r ≠ 0, then take r as divisor and b as dividend.

(v) Repeat the step (iii), till the remainder is zero, the divisor thus obtained at this stage is the required HCF. I. We state Euclid’s division algorithm for positive integers only but it can be extended for all integers except zero,

b ≠ 0. II. When ‘a’ and ‘b’ are two positive integers such that a = bq + r, 0 ≤ r < b, then HCF (a, b) = HCF (b, r). Fundamental Theorem of Arithmetic : Every composite number can be expressed as a product of primes and this

decomposition is unique, apart from the order in which the prime factors occur. By using this theorem, we shall find the HCF and LCM of given numbers (two or more). This method is also called

prime factorization method. Prime Factorization Method to find HCF and LCM : (i) First find all the prime factors of given numbers. (ii) The product of least power of each common factor among all the prime factors is the required HCF and the

product of the greatest power of each prime factors in the number is the required LCM. If p and q are two positive integers, then : HCF (p, q) × LCM (p, q) = p × q. Let p be a prime number. If p is divided by a2, then p can also be divided a, where a is a positive integer.

UNIT-INumber Systems

2 ] OSWAAL CBSE CCE Question Bank (Term-1), Mathematics-X

Rational Number : The number of the form

pq

, where p and q are integers and q ≠ 0, is known as rational number.

Irrational Numbers : A number is called irrational if it cannot be written in the form

pq

, where p and q are integers

and q ≠ 0. For example 2 , 3 , 5 , π are irrational numbers.

Terminating and Recurring Decimals : If decimal expression of rational number pq

terminates, i.e., comes to an

end, then the decimal obtained from

pq

is called terminating decimal.

Non-terminating Repeating Decimals : The decimals obtained from

pq

repeats periodically, then it is called non-

terminating repeating recurring decimal.

Let x =

pq

be a rational number, such that the prime factorization of q is of the form 2n5m, where n and m are

non-negative integers. Then x has a decimal expansion which terminates after k places of decimals, where k is the largest of m and n.

Let x =

pq

be a rational number, such that the prime factorization of q is not of the form 2n5m, where n, m are

non-negative integers. Then x has a decimal expansion which is non-terminating repeating. The sum or difference of a rational and irrational number is irrational. The product and quotient of a non-zero rational and irrational number is irrational. The product of three numbers is not equal to the product of their HCF and LCM, i.e., HCF (p, q, r) × LCM (p, q, r) ≠ p × q × r, where p, q, r are positive integers. However, the following results hold good for three numbers p, q and r :

LCM (p, q, r) =

( , , ) ( , ) ( , ) ( , )

p q r HCF p q rHCF p q HCF q r HCF p r

× × ×× ×

HCF (p, q, r) =

( , , ) ( , ) ( , ) ( , )

p q r LCM p q rLCM p q LCM q r LCM p r

× × ×× ×

Euclid’s division lemma states that for any two positive integers a and b, we can find two whole numbers q and r, such that

a = bq + r; where 0 ≤ r <b Here a = Dividend, b = Divisor, q = Quotient, r = Remainder i.e., Dividend = (Divisor × Quotient) + Remainder

For example, 1 → Quotient Divisor 3) 5 – 3 2 → Remainder According to the above formula, 5 = (3 × 1) + 2 Application of Euclid’s Division Lemma to Compute HCF of Two Positive Integers Euclid’s Division Lemma / algorithm is a technique to compute the HCF of two positive integers a and b (a > b)

using the following steps : Step–1 : Applying Euclid’s Lemma to a and b to find the whole numbers q and r such that A = bq + r, 0 ≤ r<b. Step–2 : If r = 0, then b is the HCF of a and b. If r ≠ 0, then apply the Euclid’s Division Lemma to b and r. Step–3 : Continue the process till the remainder is zero, i.e., repeat the step 2 again and again until r = 0. Then the

divisor at this stage will be the required HCF. The Fundamental Theorem of Arithmetic Every composite number can be written as the product of powers of primes and this factorisation is unique,

apart from the order in which the prime factors occur. Fundamental theorem of arithmetic is also called a Unique Factorisation Theorem.