Title 1. NUMBER SYSTEMS.pdf ✅

Chapter Kinematics 12 Chapter Number 1 Systems REMEMBER Before beginning this chapter, you should be able to: • Review the types of numbers and understand representation of numbers on a number line • Study properties of numbers KEY IDEAS After completing this chapter, you would be able to: • Study Euclid’s division lemma • Learn fundamental arithmetic theorem • Prove theorems related to irrational numbers and rational numbers • Study proof of irrationality M01_TRISHNA_78975.indd 1 2/1/2018 3:59:07 PM
1.2 Chapter 1 INTRODUCTION Earlier we have learnt about number line, irrational numbers, how to represent irrational numbers on the number line, real numbers and their decimal representation, representing real numbers on the number line and operations on the number line. We shall now present a detailed study of Euclid’s division algorithm and fundamental theorem of arithmetic. Further we continue the revision of irrational numbers and the decimal expansion of rational numbers. We are very familiar with division rule. That is, dividend = (divisor × quotient) + remainder. Euclid’s division lemma is based on this rule. We use this result to obtain the HCF of two numbers. And also we know that, every composite number can be expressed as the product of primes in a unique way. This is the fundamental theorem of arithmetic. This result is used to prove the irrationality of a number. In the previous class we have studied about rational numbers. In this chapter, we shall explore when exactly the decimal expansion of a rational number say p q (q ≠ 0) is terminating and when it is non-terminating and repeating. EUCLID’S DIVISION LEMMA For any two positive integers, say x and y, there exist unique integers say q and r satisfying x = yq + r, where 0 ≤ r < y. Example: Consider the integers 9 and 19. 19 = 9 × 2 + 1 Example: Consider the integers 6 and 24. 24 = 6 × 4 + 0 Notes 1. Euclid’s division algorithm is used for finding the greatest common divisor of two numbers. 2. An algorithm is a process of solving particular problems. Example 1.1 Find the HCF of 250 and 30. Solution By using Euclid’s division lemma, we get, 250 = 30 × 8 + 10 Now consider, the divisor and remainder. Again by using Euclid’s division lemma, we get 30 = 10 × 3 + 0 Here, we notice that the remainder is zero and we cannot proceed further. The divisor at this stage is 10. The HCF of 250 and 30 is 10. It can be verified by listing out all the factors of 250 and 30. M01_TRISHNA_78975.indd 2 2/1/2018 3:59:07 PM
Number Systems 1.3 Note Euclid’s division algorithm is stated for only positive integers, it can be also extended for all negative integers. Euclid’s division algorithm has several applications. The following examples give the idea of the applications. Example 1.2 Show that every positive even integer is of the form 2n and every positive odd integer is of the form 2n + 1. Solution For any integer x and y = 2, x = 2n + r, where n ≥ 0. But 0 ≤ r < 2 ⇒ r = 0 or 1 When r = 0, x = 2n ⇒ x is a positive even integer When r = 1, x = 2n + 1 ⇒ x is a positive odd integer. Example 1.3 A trader has 612 Dettol soaps and 342 Pears soaps. He packs them in boxes and each box contains exactly one type of soap. If every box contains the same number of soaps, then find the number of soaps in each box such that the number of boxes is the least. Solution The required number is HCF of 612 and 342. This number gives the maximum number of soaps in each box and the number of boxes with them be the least. By using Euclid’s division algorithm, we have 612 = 342 × 1 + 270 342 = 270 × 1 + 72 270 = 72 × 3 + 54 72 = 54 × 1 + 18 54 = 18 × 3 + 0 Here we notice that the remainder is zero, and the divisor at this stage is 18. ∴ HCF of 612 and 342 is 18. So, the trader can pack 18 soaps per box. FUNDAMENTAL THEOREM OF ARITHMETIC Every composite number can be expressed as the product of prime factors uniquely. Note In general a = p1 p2 p3 ... pn, where p1, p2, p3, ..., pn are primes in ascending order. M01_TRISHNA_78975.indd 3 2/1/2018 3:59:07 PM
1.4 Chapter 1 Example 1.4 Write 1800 as product of prime factors. Solution 2 1800 2 900 2 450 3 225 3 75 5 25 5 ∴ 1800 = 23 × 32 × 52. Let us see the applications of fundamental theorem. Example 1.5 Check whether there is any value of x for which 6x ends with 5. Solution If 6x ends with 5, then 6x would contain the prime number 5. But, 6x = (2 × 3)x = 2x × 3x ⇒ The prime numbers in the factorization of 6x are 2 and 3 By uniqueness of fundamental theorem, there are no prime numbers other than 2 and 3 in 6x. ∴ 6x never ends with 5. Example 1.6 Show that 5 × 3 × 2 + 3 is a composite number. Solution 5 × 3 × 2 + 3 = 3(5 × 2 + 1) = 3(11) = 3 × 11 ∴ The given number is a composite number. Example 1.7 Find the HCF and LCM of 48 and 56 by prime factorization method. Solution 48 = 24 × 31 56 = 23 × 71 HCF = 23 (The product of common prime factors with lesser index) LCM = 24 × 31 × 71 (The product of common prime factors with greater index). M01_TRISHNA_78975.indd 4 2/1/2018 3:59:08 PM