REAL NUMBERS 1.1 Introduction In Class IX, you began your exploration of the world of real numbers and encountered irrational numbers. We continue our discussion on real numbers in this chapter. We begin with two very important properties of positive integers in Sections 1.2 and 1.3, namely the Euclid’s division algorithm and the Fundamental Theorem of Arithmetic. Euclid’s division algorithm, as the name suggests, has to do with divisibility of integers. Stated simply, it says any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b. Many of you probably recognise this as the usual long division process. Although this result is quite easy to state and understand, it has many applications related to the divisibility properties of integers. We touch upon a few of them, and use it mainly to compute the HCF of two positive integers. The Fundamental Theorem of Arithmetic, on the other hand, has to do something with multiplication of positive integers. You already know that every composite number can be expressed as a product of primes in a unique way—this important fact is the Fundamental Theorem of Arithmetic. Again, while it is a result that is easy to state and understand, it has some very deep and significant applications in the field of mathematics. We use the Fundamental Theorem of Arithmetic for two main applications. First, we use it to prove the irrationality of many of the numbers you studied in Class IX, such as 2, 3 and 5 . Second, we apply this theorem to explore when exactly the decimal pexpansion of a rational number, say (q ≠ 0) , is terminating and when it is non-qterminating repeating. We do so by looking at the prime factorisation of the denominator q of qp .You will see that the prime factorisation of q will completely reveal the nature of the decimal expansion of qp . So let us begin our exploration. 1.2 Euclid’s Division Lemma Consider the following folk puzzle*. A trader was moving along a road selling eggs. An idler who didn’t have much work to do, started to get the trader into a wordy duel. This grew into a fight, he pulled the basket with eggs and dashed it on the floor. The eggs broke. The trader requested the Panchayat to ask the idler to pay for the broken eggs. The Panchayat asked the trader how many eggs were broken. He gave the following response: If counted in pairs, one will remain; If counted in threes, two will remain; If counted in fours, three will remain; If counted in fives, four will remain; If counted in sixes, five will remain; If counted in sevens, nothing will remain; My basket cannot accomodate more than 150 eggs. So, how many eggs were there? Let us try and solve the puzzle. Let the number of eggs be a. Then working backwards, we see that a is less than or equal to 150: If counted in sevens, nothing will remain, which translates to a = 7p + 0, for some natural number p. If counted in sixes, a = 6 q+ 5, for some natural number q. If counted in fives, four will remain. It translates to a = 5 w + 4, for some natural number w. If counted in fours, three will remain. It translates toa = 4s + 3, for some natural number s. If counted in threes, two will remain. It translates to a = 3t + 2, for some natural number t. If counted in pairs, one will remain. It translates to a = 2 u + 1, for some natural number u. That is, in each case, we have a and a positive integer b (in our example, b takes values 7, 6, 5, 4, 3 and 2, respectively) which dividesa and leaves a remainder r (in our case, r is 0, 5, 4, 3, 2 and 1, respectively), that is smaller than b. The * This is modified form of a puzzle given in ‘Numeracy Counts!’ by A. Rampal, and others. moment we write down such equations we are using Euclid’s division lemma, which is given in Theorem 1.1. Getting back to our puzzle, do you have any idea how you will solve it? Yes! You must look for the multiples of 7 which satisfy all the conditions. By trial and error (using the concept of LCM), you will find he had 119 eggs. In order to get a feel for what Euclid’s division lemma is, consider the following pairs of integers: 17, 6; 5, 12; 20, 4 Like we did in the example, we can write the following relations for each such pair: 17 = 6 × 2 + 5 (6 goes into 17 twice and leaves a remainder 5) 5 = 12 × 0 + 5 (This relation holds since 12 is larger than 5) 20 = 4 × 5 + 0 (Here 4 goes into 20 five-times and leaves no remainder) That is, for each pair of positive integers a and b, we have found whole numbers q and r, satisfying the relation: a = bq + r, 0 ≤ r < b Note that q or r can also be zero. Why don’t you now try finding integersq and r for the following pairs of positive integers a and b? (i) 10, 3; (ii) 4, 19; (iii) 81, 3 Did you notice that q and r are unique? These are the only integers satisfying the conditions a = bq + r, where 0 ≤ r < b. You may have also realised that this is nothing but a restatement of the long division process you have been doing all these years, and that the integers q and r are called the quotient and remainder, respectively. A formal statement of this result is as follows : Theorem 1.1 (Euclid’s Division Lemma) : Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b. This result was perhaps known for a long time, but was first recorded in Book VII of Euclid’s Elements. Euclid’s division algorithm is based on this lemma. An algorithm is a series of well defined steps which gives a procedure for solving a type of problem. The wordalgorithm comes from the name of the 9th century Persian mathematician al-Khwarizmi. In fact, even the word ‘algebra’ is derived from a book, he wrote, called Hisab al-jabr w’al-muqabala. A lemma is a proven statement used for Muhammad ibn Musa al-Khwarizmi proving another statement. Euclid’s division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers. Recall that the HCF of two positive integers a and b is the largest positive integer d that divides both a and b. Let us see how the algorithm works, through an example first. Suppose we need to find the HCF of the integers 455 and 42. We start with the larger integer, that is, 455. Then we use Euclid’s lemma to get 455 = 42 × 10 + 35 Now consider the divisor 42 and the remainder 35, and apply the division lemma to get 42= 35 × 1 + 7 Now consider the divisor 35 and the remainder 7, and apply the division lemma to get 35= 7 × 5 + 0 Notice that the remainder has become zero, and we cannot proceed any further. We claim that the HCF of 455 and 42 is the divisor at this stage, i.e., 7. You can easily verify this by listing all the factors of 455 and 42. Why does this method work? It works because of the following result. So, let us state Euclid’s division algorithm clearly. To obtain the HCF of two positive integers, say c and d, with c > d, follow the steps below: Step 1 : Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and r such that c = dq + r, 0 ≤ r < d. Step 2 : If r = 0, d is the HCF of c and d. If r ≠ 0, apply the division lemma to d and r. Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be the required HCF. This algorithm works because HCF (c, d) = HCF (d, r) where the symbol HCF (c, d) denotes the HCF of c and d, etc. Example 1 : Use Euclid’s algorithm to find the HCF of 4052 and 12576. Solution : Step 1 : Since 12576 > 4052, we apply the division lemma to 12576 and 4052, to get 12576 = 4052 × 3 + 420 Step 2 : Since the remainder 420≠ 0, we apply the division lemma to 4052 and 420, to get 4052 = 420 × 9 + 272 Step 3 : We consider the new divisor 420 and the new remainder 272, and apply the division lemma to get 420 = 272 × 1 + 148 We consider the new divisor 272 and the new remainder 148, and apply the division lemma to get 272 = 148 × 1 + 124 We consider the new divisor 148 and the new remainder 124, and apply the division lemma to get 148 = 124 × 1 + 24 We consider the new divisor 124 and the new remainder 24, and apply the division lemma to get 124 = 24 × 5 + 4 We consider the new divisor 24 and the new remainder 4, and apply the division lemma to get 24= 4 × 6 + 0 The remainder has now become zero, so our procedure stops. Since the divisor at this stage is 4, the HCF of 12576 and 4052 is 4. Notice that 4 = HCF (24, 4) = HCF (124, 24) = HCF (148, 124) = HCF (272, 148) = HCF (420, 272) = HCF (4052, 420) = HCF (12576, 4052). Euclid’s division algorithm is not only useful for calculating the HCF of very large numbers, but also because it is one of the earliest examples of an algorithm that a computer had been programmed to carry out. Remarks : 1. Euclid’s division lemma and algorithm are so closely interlinked that people often call former as the division algorithm also. 2. Although Euclid’s Division Algorithm is stated for only positive integers, it can be extended for all integers except zero, i.e., b ≠ 0. However, we shall not discuss this aspect here. Euclid’s division lemma/algorithm has several applications related to finding properties of numbers. We give some examples of these applications below: Example 2 : Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer. Solution : Let a be any positive integer and b = 2. Then, by Euclid’s algorithm, a = 2q + r, for some integer q ≥ 0, and r = 0 or r = 1, because 0 ≤ r < 2. So, a = 2q or 2q + 1. If a is of the form 2q, then a is an even integer. Also, a positive integer can be either even or odd. Therefore, any positive odd integer is of the form 2q + 1. Example 3 : Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer. Solution : Let us start with taking a, where a is a positive odd integer. We apply the division algorithm with a and b = 4. Since 0 ≤ r < 4, the possible remainders are 0, 1, 2 and 3. That is, a can be 4q, or 4q + 1, or 4q + 2, or 4q + 3, where q is the quotient. However, since a is odd, a cannot be 4q or 4q + 2 (since they are both divisible by 2). Therefore, any odd integer is of the form 4q + 1 or 4q + 3. Example 4 : A sweetseller has 420 kaju barfis and 130 badam barfis. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the number of that can be placed in each stack for this purpose? Solution : This can be done by trial and error. But to do it systematically, we find HCF (420, 130). Then this number will give the maximum number of barfis in each stack and the number of stacks will then be the least. The area of the tray that is used up will be the least. Now, let us use Euclid’s algorithm to find their HCF. We have : 420 = 130 × 3 + 30 130 = 30 × 4 + 10 30= 10 × 3 + 0 So, the HCF of 420 and 130 is 10. Therefore, the sweetseller can make stacks of 10 for both kinds of barfi. EXERCISE 1.1 1. Use Euclid’s division algorithm to find the HCF of : (i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255 2. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer. 3. An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march? 4. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m. [Hint : Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square each of these and show that they can be rewritten in the form 3m or 3m + 1.] 5. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8. 1.3 The Fundamental Theorem of Arithmetic In your earlier classes, you have seen that any natural number can be written as a product of its prime factors. For instance, 2 = 2, 4 = 2 × 2, 253 = 11 × 23, and so on. Now, let us try and look at natural numbers from the other direction. That is, can any natural number be obtained by multiplying prime numbers? Let us see. Take any collection of prime numbers, say 2, 3, 7, 11 and 23. If we multiply some or all of these numbers, allowing them to repeat as many times as we wish, we can produce a large collection of positive integers (In fact, infinitely many). Let us list a few : 7×11 × 23 = 1771 3 × 7 × 11 × 23 = 5313 2 × 3 × 7 × 11 × 23 = 10626 23 × 3 × 73 = 8232 22 × 3 × 7 × 11 × 23 = 21252 and so on. Now, let us suppose your collection of primes includes all the possible primes. What is your guess about the size of this collection? Does it contain only a finite number of integers, or infinitely many? Infact, there are infinitely many primes. So, if we combine all these primes in all possible ways, we will get an infinite collection of numbers, all the primes and all possible products of primes. The question is – can we produce all the composite numbers this way? What do you think? Do you think that there may be a composite number which is not the product of powers of primes? Before we answer this, let us factorise positive integers, that is, do the opposite of what we have done so far. We are going to use the factor tree with which you are all familiar. Let us take some large number, say, 32760, and factorise it as shown : So we have factorised 32760 as 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13 as a product of primes, i.e., 32760 = 23 × 32 × 5 × 7 × 13 as a product of powers of primes. Let us try another number, say, 123456789. This can be written as 32 × 3803 × 3607. Of course, you have to check that 3803 and 3607 are primes! (Try it out for several other natural numbers yourself.) This leads us to a conjecture that every composite number can be written as the product of powers of primes. In fact, this statement is true, and is called the Fundamental Theorem of Arithmetic because of its basic crucial importance to the study of integers. Let us now formally state this theorem. Theorem 1.2 (Fundamental Theorem of Arithmetic) : Every composite number can be expressed ( factorised ) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur. An equivalent version of Theorem 1.2 was probably first recorded as Proposition 14 of Book IX in Euclid’s Elements, before it came to be known as the Fundamental Theorem of Arithmetic. However, the first correct proof was given by Carl Friedrich Gauss in his Disquisitiones Arithmeticae. Carl Friedrich Gauss is often referred to as the ‘Prince of Mathematicians’ and is considered one of the three greatest mathematicians of all time, along with Archimedes and Newton. He has made fundamental contributions to both mathematics and science. The Fundamental Theorem of Arithmetic says that every composite number can be factorised as a product of primes. Actually it says more. It says that given any composite number it can be factorised as a product of prime numbers in a ‘unique’ way, except for the order in which the primes occur. That is, given any composite number there is one and only one way to write it as a product of primes, as long as we are not particular about the order in which the primes occur. So, for example, we regard 2 × 3 × 5 × 7 as the same as 3 × 5 × 7 × 2, or any other possible order in which these primes are written. This fact is also stated in the following form: The prime factorisation of a natural number is unique, except for the order of its factors. In general, given a composite number x, we factorise it as x = p1 p2 ... pn , where p, p,..., pare primes and written in ascending order, i.e., p ≤ p12n 12 ≤ . . . ≤ pn. If we combine the same primes, we will get powers of primes. For example, 32760 = 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13 = 23 × 32 × 5 × 7 × 13 Once we have decided that the order will be ascending, then the way the number is factorised, is unique. The Fundamental Theorem of Arithmetic has many applications, both within mathematics and in other fields. Let us look at some examples. Example 5 : Consider the numbers 4n , where n is a natural number. Check whether there is any value of n for which 4n ends with the digit zero. Solution : If the number 4n, for any n, were to end with the digit zero, then it would be divisible by 5. That is, the prime factorisation of 4n would contain the prime 5. This is not possible because 4n = (2)2n; so the only prime in the factorisation of 4n is 2. So, the uniqueness of the Fundamental Theorem of Arithmetic guarantees that there are no other primes in the factorisation of 4n. So, there is no natural number n for which 4n ends with the digit zero. You have already learnt how to find the HCF and LCM of two positive integers using the Fundamental Theorem of Arithmetic in earlier classes, without realising it! This method is also called the prime factorisation method. Let us recall this method through an example. Example 6 : Find the LCM and HCF of 6 and 20 by the prime factorisation method. Solution : We have : 6= 21 × 31and 20 = 2 × 2 × 5 = 22 × 51. You can find HCF(6, 20) = 2 and LCM(6, 20) = 2 × 2 × 3 × 5 = 60, as done in your earlier classes. Note that HCF(6, 20) = 21 = Product of the smallest power of each common prime factor in the numbers. LCM (6, 20) = 22 × 31 × 51 = Product of the greatest power of each prime factor, involved in the numbers. From the example above, you might have noticed that HCF(6, 20) × LCM(6, 20) = 6 × 20. In fact, we can verify that for any two positive integers a and b, HCF (a, b) × LCM (a, b) = a × b. We can use this result to find the LCM of two positive integers, if we have already found the HCF of the two positive integers. Example 7 : Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM. Solution : The prime factorisation of 96 and 404 gives : 96 = 25 × 3, 404 = 22 × 101 Therefore, the HCF of these two integers is 22 = 4. 96 × 404 96 × 404 Also, LCM (96, 404) = == 9696 HCF(96, 404) 4 Example 8 : Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method. Solution : We have : 6 = 2 × 3, 72=23 × 32, 120 = 23 × 3 × 5 Here, 21 and 31 are the smallest powers of the common factors 2 and 3, respectively. So, HCF (6, 72, 120) = 21 × 31 = 2 × 3 = 6 23, 32 and 51 are the greatest powers of the prime factors 2, 3 and 5 respectively involved in the three numbers. So, LCM (6, 72, 120) = 23 × 32 × 51 = 360 Remark : Notice, 6 × 72 × 120 ≠ HCF (6, 72, 120) × LCM (6, 72, 120). So, the product of three numbers is not equal to the product of their HCF and LCM. EXERCISE 1.2 1. Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429 2. Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54 3. Find the LCM and HCF of the following integers by applying the prime factorisation method. (i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25 4. Given that HCF (306, 657) = 9, find LCM (306, 657). 5. Check whether 6n can end with the digit 0 for any natural numbern. 6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers. 7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point? 1.4 Revisiting Irrational Numbers In Class IX, you were introduced to irrational numbers and many of their properties. You studied about their existence and how the rationals and the irrationals together made up the real numbers. You even studied how to locate irrationals on the number line. However, we did not prove that they were irrationals. In this section, we will prove that 2, 3, 5 and, in general, p is irrational, where p is a prime. One of the theorems, we use in our proof, is the Fundamental Theorem of Arithmetic. pRecall, a number ‘s’ is called irrational if it cannot be written in the form , qwhere p and q are integers and q ≠ 0. Some examples of irrational numbers, with which you are already familiar, are : 2 ,2, 3, 15, ,π− 0.10110111011110... , etc.3 Before we prove that 2 is irrational, we need the following theorem, whose proof is based on the Fundamental Theorem of Arithmetic. Theorem 1.3 : Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer. *Proof : Let the prime factorisation of a be as follows : a = pp . . . p, where p,p, . . ., p are primes, not necessarily distinct.12n12nTherefore, a2 = ( p1 p2 . . . pn)( p1 p2 . . . pn) = p12 p22 . . . p2 n. Now, we are given that p divides a2. Therefore, from the Fundamental Theorem of Arithmetic, it follows that p is one of the prime factors of a2. However, using the uniqueness part of the Fundamental Theorem of Arithmetic, we realise that the only prime factors of a2 are p1, p2, . . ., pn. So p is one of p1, p2, . . ., pn. Now, since a = pp . . . p, p divides a. 12n We are now ready to give a proof that 2 is irrational. The proof is based on a technique called ‘proof by contradiction’. (This technique is discussed in some detail in Appendix 1). Theorem 1.4 : is irrational.2Proof : Let us assume, to the contrary, that 2 is rational. rSo, we can find integers r and s (≠ 0) such that 2 = . s Suppose r and s have a common factor other than 1. Then, we divide by the common afactor to get 2 =, where a and b are coprime.b So, b 2 = a. Squaring on both sides and rearranging, we get 2b2 = a2. Therefore, 2 divides a2. Now, by Theorem 1.3, it follows that 2 divides a. So, we can write a = 2c for some integer c. * Not from the examination point of view. Substituting for a, we get 2b2 = 4c2, that is, b2 = 2c2. This means that 2 divides b2, and so 2 divides b (again using Theorem 1.3 with p = 2). Therefore, a and b have at least 2 as a common factor. But this contradicts the fact that a and b have no common factors other than 1. This contradiction has arisen because of our incorrect assumption that 2 is rational. So, we conclude that 2 is irrational. • Example 9 : Prove that 3 is irrational. Solution : Let us assume, to the contrary, that 3 is rational. aThat is, we can find integers a and b (≠ 0) such that 3 = ⋅ b Suppose a and b have a common factor other than 1, then we can divide by the common factor, and assume that a and b are coprime. So, b 3 a=⋅ Squaring on both sides, and rearranging, we get 3b2 = a2. Therefore, a2 is divisible by 3, and by Theorem 1.3, it follows that a is also divisible by 3. So, we can write a = 3c for some integer c. Substituting for a, we get 3b2 = 9c2, that is, b2 = 3c2. This means that b2 is divisible by 3, and so b is also divisible by 3 (using Theorem 1.3 with p = 3). Therefore, a and b have at least 3 as a common factor. But this contradicts the fact that a and b are coprime. This contradiction has arisen because of our incorrect assumption that 3 is rational. So, we conclude that 3 is irrational. In Class IX, we mentioned that : • the sum or difference of a rational and an irrational number is irrational and • the product and quotient of a non-zero rational and irrational number is irrational. We prove some particular cases here. Example 10 : Show that 5– 3 is irrational. Solution : Let us assume, to the contrary, that 5– 3 is rational. aThat is, we can find coprime a and b (b ≠ 0) such that 5 −3 =⋅ b Therefore, 5 a b 5b −aRearranging this equation, we get 35 – a ⋅== bb Since a and b are integers, we get 5– a is rational, and so 3 is rational. bBut this contradicts the fact that 3 is irrational. This contradiction has arisen because of our incorrect assumption that 5 – 3 is rational. So, we conclude that 5 −3 is irrational. Example 11 : Show that 32is irrational. Solution : Let us assume, to the contrary, that 32is rational. aThat is, we can find coprime a and b (b ≠ 0) such that 32 =⋅ b aRearranging, we get 2 =⋅ 3b a Since 3, a and b are integers, is rational, and so 2 is rational.3bBut this contradicts the fact that 2 is irrational. So, we conclude that 32is irrational. EXERCISE 1.3 1. Prove that is irrational.52. Prove that 32 5is irrational.3. Prove that the following are irrationals : +1(i) (ii) 75 (iii) 6 +2 2 1.5 Revisiting Rational Numbers and Their Decimal Expansions In Class IX, you studied that rational numbers have either a terminating decimal expansion or a non-terminating repeating decimal expansion. In this section, we are pgoing to consider a rational number, say (q ≠0) , and explore exactly when the qpdecimal expansion of is terminating and when it is non-terminating repeatingq(or recurring). We do so by considering several examples. Let us consider the following rational numbers : (i) 0.375 (ii) 0.104 (iii) 0.0875 (iv) 23.3408. 375 375 104 104 Now (i) 0.375 == 3 (ii) 0.104 == 31000 10 1000 10 875 875 233408 233408 (iii) 0.0875 == (iv) 23.3408 == 10000 104 10000 10 4 As one would expect, they can all be expressed as rational numbers whose denominators are powers of 10. Let us try and cancel the common factors between the numerator and denominator and see what we get : 375 3 ×53 3 104 13 ×23 13(i) 0.375 == = (ii) 0.104 == = 3 333 3 33310 2 ×52 102 ×55 875 7 233408 22 ××521 7(iii) 0.0875 == (iv) 23.3408 == 44 4410 2 ×5 105 Do you see any pattern? It appears that, we have converted a real numberpwhose decimal expansion terminates into a rational number of the form , where pqand q are coprime, and the prime factorisation of the denominator (that is, q) has only powers of 2, or powers of 5, or both. We should expect the denominator to look like this, since powers of 10 can only have powers of 2 and 5 as factors. Even though, we have worked only with a few examples, you can see that any real number which has a decimal expansion that terminates can be expressed as a rational number whose denominator is a power of 10. Also the only prime factors of 10 are 2 and 5. So, cancelling out the common factors between the numerator and the pdenominator, we find that this real number is a rational number of the form , where qthe prime factorisation of q is of the form 2n5m, and n, m are some non-negative integers. Let us write our result formally: Theorem 1.5 : Let x be a rational number whose decimal expansion terminates. pThen x can be expressed in the form , where p and q are coprime, and the qprime factorisation of q is of the form 2n5m, where n, m are non-negative integers. You are probably wondering what happens the other way round in Theorem 1.5. p ,That is, if we have a rational number of the form and the prime factorisation of qqpis of the form 2n5m, where n, m are non negative integers, then does have a qterminating decimal expansion? Let us see if there is some obvious reason why this is true. You will surely agree athat any rational number of the form , where b is a power of 10, will have a terminatingbdecimal expansion. So it seems to make sense to convert a rational number of the a ,form p , where q is of the form 2n5m, to an equivalent rational number of the formqb where b is a power of 10. Let us go back to our examples above and work backwards. 33 3 ×53 375 (i) == ==0.375 333 382 2 ×5 10 13 1313 ×23 104 (ii) == ==0.104 333 3125 5 2 ×5 10 777 ×53 875 (iii) == ==0.0875 4 44480 2 ×52 ×5 10 14588 22 ××521 26 7 521 233408 7 ××(iv) = = ==23.3408 4 444625 5 2 ×5 10 So, these examples show us how we can convert a rational number of the form pa ,, where q is of the form 2n5m, to an equivalent rational number of the formqb where b is a power of 10. Therefore, the decimal expansion of such a rational number terminates. Let us write down our result formally. Theorem 1.6 : Let x = p be a rational number, such that the prime factorisation qof q is of the form 2n5m, where n, m are non-negative integers. Then x has a decimal expansion which terminates. We are now ready to move on to the rational numbers whose decimal expansions are non-terminating and recurring. Once again, let us look at an example to see what is going on. We refer to Example 5, Chapter 1, from your Class IX 1textbook, namely, . Here, remainders are 3, 2, 6, 4, 5, 1, 3, 7 2, 6, 4, 5, 1, . . . and divisor is 7. Notice that the denominator here, i.e., 7 is clearly not of the form 2n5m. Therefore, from Theorems 1.5 and 1.6, we 1 know that will not have a terminating decimal expansion.7Hence, 0 will not show up as a remainder (Why?), and the remainders will start repeating after a certain stage. So, we will have a block of digits, namely, 142857, repeating in the 1 quotient of .7 1 What we have seen, in the case of , is true for any rational number not covered7 by Theorems 1.5 and 1.6. For such numbers we have : pTheorem 1.7 : Let x = be a rational number, such that the prime factorisation qof 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 (recurring). From the discussion above, we can conclude that the decimal expansion of every rational number is either terminating or non-terminating repeating. EXERCISE 1.4 1. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion: 13 17 6415 (i) (ii) (iii) (iv)3125 8 455 1600 29 23 1296 (v) (vi) 32 (vii) 27 5 (viii)343 25 257 15 35 77(ix) (x)50 210 2. Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions. 3. The following real numbers have decimal expansions as given below. In each case, pdecide whether they are rational or not. If they are rational, and of the form , what can qyou say about the prime factors of q? (i) 43.123456789 (ii) 0.120120012000120000. . . (iii) 43.123456789 1.6 Summary In this chapter, you have studied the following points: 1. Euclid’s division lemma : Given positive integers a andb, there exist whole numbers q and r satisfying a = bq + r, 0 ≤ r < b. 2. Euclid’s division algorithm : This is based on Euclid’s division lemma. According to this, the HCF of any two positive integers a and b, with a > b, is obtained as follows: Step 1 : Apply the division lemma to find q and r where a = bq + r, 0 ≤ r < b. Step 2 : Ifr = 0, the HCF is b. If r ≠ 0, apply Euclid’s lemma to b and r. Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be HCF (a, b). Also, HCF(a, b) = HCF(b, r). 3. The Fundamental Theorem of Arithmetic : Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur. 4. If p is a prime and p divides a2, then p divides a, where a is a positive integer. 5. To prove that 3 are irrationals. 6. Let x be a rational number whose decimal expansion terminates. Then we can express x in the form qp , where p and q are coprime, and the prime factorisation of q is of the form 2n5m, where n, m are non-negative integers. p7. Let x = be a rational number, such that the prime factorisation of q is of the form 2n5m,qwhere n, m are non-negative integers. Then x has a decimal expansion which terminates. p8. Let x = be a rational number, such that the prime factorisation of q is not of the form q2n 5m, where n, m are non-negative integers. Then x has a decimal expansion which is non-terminating repeating (recurring).

# 1

## Real Numbers

#### 1.1 Introduction

In Class IX, you began your exploration of the world of real numbers and encountered irrational numbers. We continue our discussion on real numbers in this chapter. We begin with two very important properties of positive integers in Sections 1.2 and 1.3, namely the Euclid’s division algorithm and the Fundamental Theorem of Arithmetic.

Euclid’s division algorithm, as the name suggests, has to do with divisibility of integers. Stated simply, it says any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b. Many of you probably recognise this as the usual long division process. Although this result is quite easy to state and understand, it has many applications related to the divisibility properties of integers. We touch upon a few of them, and use it mainly to compute the HCF of two positive integers.

The Fundamental Theorem of Arithmetic, on the other hand, has to do something with multiplication of positive integers. You already know that every composite number can be expressed as a product of primes in a unique way—this important fact is the Fundamental Theorem of Arithmetic. Again, while it is a result that is easy to state and understand, it has some very deep and significant applications in the field of mathematics. We use the Fundamental Theorem of Arithmetic for two main applications. First, we use it to prove the irrationality of many of the numbers you studied in Class IX, such as and . Second, we apply this theorem to explore when exactly the decimal expansion of a rational number, say , is terminating and when it is non-terminating repeating. We do so by looking at the prime factorisation of the denominator q of . You will see that the prime factorisation of q will completely reveal the nature of the decimal expansion of .

So let us begin our exploration.

#### 1.2 Euclid’s Division Lemma

Consider the following folk puzzle*.

A trader was moving along a road selling eggs. An idler who didn’t have much work to do, started to get the trader into a wordy duel. This grew into a fight, he pulled the basket with eggs and dashed it on the floor. The eggs broke. The trader requested the Panchayat to ask the idler to pay for the broken eggs. The Panchayat asked the trader how many eggs were broken. He gave the following response:

If counted in pairs, one will remain;

If counted in threes, two will remain;

If counted in fours, three will remain;

If counted in fives, four will remain;

If counted in sixes, five will remain;

If counted in sevens, nothing will remain;

My basket cannot accomodate more than 150 eggs.

So, how many eggs were there? Let us try and solve the puzzle. Let the number of eggs be a. Then working backwards, we see that a is less than or equal to 150:

If counted in sevens, nothing will remain, which translates to a = 7p + 0, for some natural number p. If counted in sixes, a = 6q+5, for some natural number q.

If counted in fives, four will remain. It translates to a = 5w + 4, for some natural number w.

If counted in fours, three will remain. It translates to a = 4s + 3, for some natural number s.

If counted in threes, two will remain. It translates to a = 3t + 2, for some natural number t.

If counted in pairs, one will remain. It translates to a = 2u + 1, for some natural number u.

That is, in each case, we have a and a positive integer b (in our example,

b takes values 7, 6, 5, 4, 3 and 2, respectively) which divides a and leaves a remainder r (in our case, r is 0, 5, 4, 3, 2 and 1, respectively), that is smaller than b. The moment we write down such equations we are using Euclid’s division lemma, which is given in Theorem 1.1.

Getting back to our puzzle, do you have any idea how you will solve it? Yes! You must look for the multiples of 7 which satisfy all the conditions. By trial and error (using the concept of LCM), you will find he had 119 eggs.

In order to get a feel for what Euclid’s division lemma is, consider the following pairs of integers:

17, 6; 5, 12; 20, 4

Like we did in the example, we can write the following relations for each such pair:

17 = 6 × 2 + 5 (6 goes into 17 twice and leaves a remainder 5)

5 = 12 × 0 + 5 (This relation holds since 12 is larger than 5)

20 = 4 × 5 + 0 (Here 4 goes into 20 five-times and leaves no remainder)

That is, for each pair of positive integers a and b, we have found whole numbers q and r, satisfying the relation:

a = bq + r, 0 ≤ r < b

Note that q or r can also be zero.

Why don’t you now try finding integers q and r for the following pairs of positive integers a and b?

(i) 10, 3; (ii) 4, 19; (iii) 81, 3

Did you notice that q and r are unique? These are the only integers satisfying the conditions a = bq + r, where 0 ≤ r < b. You may have also realised that this is nothing but a restatement of the long division process you have been doing all these years, and that the integers q and r are called the quotient and remainder, respectively.

A formal statement of this result is as follows :

#### Theorem 1.1

(Euclid’s Division Lemma) : Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.

This result was perhaps known for a long time, but was first recorded in Book VII of Euclid’s Elements. Euclid’s division algorithm is based on this lemma.

#### Muhammad ibn Musa al-Khwarizmi

#### (C.E. 780 – 850)

An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.

The word algorithm comes from the name of the 9th century Persian mathematician

al-Khwarizmi. In fact, even the word ‘algebra’ is derived from a book, he wrote, called Hisab

al-jabr w’al-muqabala.

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

Euclid’s division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers. Recall that the HCF of two positive integers a and b is the largest positive integer d that divides both a and b.

Let us see how the algorithm works, through an example first. Suppose we need to find the HCF of the integers 455 and 42. We start with the larger integer, that is, 455. Then we use Euclid’s lemma to get

455 = 42 × 10 + 35

Now consider the divisor 42 and the remainder 35, and apply the division lemma to get

42 = 35 × 1 + 7

Now consider the divisor 35 and the remainder 7, and apply the division lemma to get

35 = 7 × 5 + 0

Notice that the remainder has become zero, and we cannot proceed any further. We claim that the HCF of 455 and 42 is the divisor at this stage, i.e., 7. You can easily verify this by listing all the factors of 455 and 42. Why does this method work? It works because of the following result.

So, let us state Euclid’s division algorithm clearly.

To obtain the HCF of two positive integers, say c and d, with c > d, follow the steps below:

#### Step 1 :

Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and r such that c = dq + r, 0 ≤ r < d.

#### Step 2 :

If r = 0, d is the HCF of c and d. If r ≠ 0, apply the division lemma to d and r.

#### Step 3 :

Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

This algorithm works because HCF (c, d) = HCF (d, r) where the symbol

HCF (c, d) denotes the HCF of c and d, etc.

#### Example 1 :

Use Euclid’s algorithm to find the HCF of 4052 and 12576.

#### Solution :

#### Step 1 :

Since 12576 > 4052, we apply the division lemma to 12576 and 4052, to get

12576 = 4052 × 3 + 420

#### Step 2 :

Since the remainder 420 ≠ 0, we apply the division lemma to 4052 and 420, to get

4052 = 420 × 9 + 272

#### Step 3 :

We consider the new divisor 420 and the new remainder 272, and apply the division lemma to get

420 = 272 × 1 + 148

We consider the new divisor 272 and the new remainder 148, and apply the division lemma to get

272 = 148 × 1 + 124

We consider the new divisor 148 and the new remainder 124, and apply the division lemma to get

148 = 124 × 1 + 24

We consider the new divisor 124 and the new remainder 24, and apply the division lemma to get

124 = 24 × 5 + 4

We consider the new divisor 24 and the new remainder 4, and apply the division lemma to get

24 = 4 × 6 + 0

The remainder has now become zero, so our procedure stops. Since the divisor at this stage is 4, the HCF of 12576 and 4052 is 4.

Notice that 4 = HCF (24, 4) = HCF (124, 24) = HCF (148, 124) =

HCF (272, 148) = HCF (420, 272) = HCF (4052, 420) = HCF (12576, 4052).

Euclid’s division algorithm is not only useful for calculating the HCF of very large numbers, but also because it is one of the earliest examples of an algorithm that a computer had been programmed to carry out.

#### Remarks :

1. Euclid’s division lemma and algorithm are so closely interlinked that people often call former as the division algorithm also.

2. Although Euclid’s DivisionAlgorithm is stated for only positive integers, it can be extended for all integers except zero, i.e., b ≠ 0. However, we shall not discuss this aspect here.

Euclid’s division lemma/algorithm has several applications related to finding properties of numbers. We give some examples of these applications below:

#### Example 2 :

Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer.

#### Solution :

Let a be any positive integer and b = 2. Then, by Euclid’s algorithm,

a = 2q + r, for some integer q ≥ 0, and r = 0 or r = 1, because 0 ≤ r < 2. So,

a = 2q or 2q + 1.

If a is of the form 2q, then a is an even integer. Also, a positive integer can be either even or odd. Therefore, any positive odd integer is of the form 2q + 1.

#### Example 3 :

Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.

#### Solution :

Let us start with taking a, where a is a positive odd integer. We apply the division algorithm with a and b = 4.

Since 0 ≤ r < 4, the possible remainders are 0, 1, 2 and 3.

That is, a can be 4q, or 4q + 1, or 4q + 2, or 4q + 3, where q is the quotient. However, since a is odd, a cannot be 4q or 4q + 2 (since they are both divisible by 2).

Therefore, any odd integer is of the form 4q + 1 or 4q + 3.

#### Example 4 :

A sweetseller has 420 kaju barfis and 130 badam barfis. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the number of that can be placed in each stack for this purpose?

#### Solution :

This can be done by trial and error. But to do it systematically, we find HCF (420, 130). Then this number will give the maximum number of barfis in each stack and the number of stacks will then be the least. The area of the tray that is used up will be the least.

Now, let us use Euclid’s algorithm to find their HCF. We have :

420 = 130 × 3 + 30

130 = 30 × 4 + 10

30 = 10 × 3 + 0

So, the HCF of 420 and 130 is 10.

Therefore, the sweetseller can make stacks of 10 for both kinds of barfi.

#### Exercise 1.1

1. Use Euclid’s division algorithm to find the HCF of :

(i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255

2. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.

3. An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

4. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

[Hint : Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square each of these and show that they can be rewritten in the form 3m or 3m + 1.]

5. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

#### 1.3 The Fundamental Theorem of Arithmetic

In your earlier classes, you have seen that any natural number can be written as a product of its prime factors. For instance, 2 = 2, 4 = 2 × 2, 253 = 11 × 23, and so on. Now, let us try and look at natural numbers from the other direction. That is, can any natural number be obtained by multiplying prime numbers? Let us see.

Take any collection of prime numbers, say 2, 3, 7, 11 and 23. If we multiply some or all of these numbers, allowing them to repeat as many times as we wish, we can produce a large collection of positive integers (In fact, infinitely many). Let us list a few :

7 × 11 × 23 = 1771 3 × 7 × 11 × 23 = 5313

2 × 3 × 7 × 11 × 23 = 10626 23 × 3 × 73 = 8232

22 × 3 × 7 × 11 × 23 = 21252

and so on.

Now, let us suppose your collection of primes includes all the possible primes. What is your guess about the size of this collection? Does it contain only a finite number of integers, or infinitely many? Infact, there are infinitely many primes. So, if we combine all these primes in all possible ways, we will get an infinite collection of numbers, all the primes and all possible products of primes. The question is – can we produce all the composite numbers this way? What do you think? Do you think that there may be a composite number which is not the product of powers of primes? Before we answer this, let us factorise positive integers, that is, do the opposite of what we have done so far.

We are going to use the factor tree with which you are all familiar. Let us take some large number, say, 32760, and factorise it as shown :

So we have factorised 32760 as 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13 as a product of primes, i.e., 32760 = 23 × 32 × 5 × 7 × 13 as a product of powers of primes. Let us try another number, say, 123456789. This can be written as 32 × 3803 × 3607. Of course, you have to check that 3803 and 3607 are primes! (Try it out for several other natural numbers yourself.) This leads us to a conjecture that every composite number can be written as the product of powers of primes. In fact, this statement is true, and is called the Fundamental Theorem of Arithmetic because of its basic crucial importance to the study of integers. Let us now formally state this theorem.

#### Theorem 1.2

(Fundamental Theorem of Arithmetic) : Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

#### Carl Friedrich Gauss

#### (1777 – 1855)

An equivalent version of Theorem 1.2 was probably first recorded as Proposition 14 of Book IX in Euclid’s Elements, before it came to be known as the Fundamental Theorem of Arithmetic. However, the first correct proof was given by Carl Friedrich Gauss in his Disquisitiones Arithmeticae.

Carl Friedrich Gauss is often referred to as the ‘Prince of Mathematicians’ and is considered one of the three greatest mathematicians of all time, along with Archimedes and Newton. He has made fundamental contributions to both mathematics and science.

The Fundamental Theorem of Arithmetic says that every composite number can be factorised as a product of primes. Actually it says more. It says that given any composite number it can be factorised as a product of prime numbers in a ‘unique’ way, except for the order in which the primes occur. That is, given any composite number there is one and only one way to write it as a product of primes, as long as we are not particular about the order in which the primes occur. So, for example, we regard 2 × 3 × 5 × 7 as the same as 3 × 5 × 7 × 2, or any other possible order in which these primes are written. This fact is also stated in the following form:

The prime factorisation of a natural number is unique, except for the order of its factors.

In general, given a composite number x, we factorise it as x = p1p2 ... pn, where p1, p2,..., pn are primes and written in ascending order, i.e., p1 ≤ p2

≤ . . . ≤ pn. If we combine the same primes, we will get powers of primes. For example,

32760 = 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13 = 23 × 32 × 5 × 7 × 13

Once we have decided that the order will be ascending, then the way the number is factorised, is unique.

The Fundamental Theorem of Arithmetic has many applications, both within mathematics and in other fields. Let us look at some examples.

#### Example 5 :

Consider the numbers 4n, where n is a natural number. Check whether there is any value of n for which 4n ends with the digit zero.

#### Solution :

If the number 4n, for any n, were to end with the digit zero, then it would be divisible by 5. That is, the prime factorisation of 4n would contain the prime 5. This is not possible because 4n = (2)2n; so the only prime in the factorisation of 4n is 2. So, the uniqueness of the Fundamental Theorem of Arithmetic guarantees that there are no other primes in the factorisation of 4n. So, there is no natural number n for which 4n ends with the digit zero.

You have already learnt how to find the HCF and LCM of two positive integers using the Fundamental Theorem of Arithmetic in earlier classes, without realising it! This method is also called the prime factorisation method. Let us recall this method through an example.

#### Example 6 :

Find the LCM and HCF of 6 and 20 by the prime factorisation method.

#### Solution :

We have : 6 = 21 × 31 and 20 = 2 × 2 × 5 = 22 × 51.

You can find HCF(6, 20) = 2 and LCM(6, 20) = 2 × 2 × 3 × 5 = 60, as done in your earlier classes.

Note that HCF(6, 20) = 21 = Product of the smallest power of each common prime factor in the numbers.

LCM (6, 20) = 22 × 31 × 51 = Product of the greatest power of each prime factor, involved in the numbers.

From the example above, you might have noticed that HCF(6, 20) × LCM(6, 20) = 6 × 20. In fact, we can verify that for any two positive integers a and b,

HCF (a, b) × LCM (a, b) = a × b. We can use this result to find the LCM of two positive integers, if we have already found the HCF of the two positive integers.

#### Example 7 :

Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.

#### Solution :

The prime factorisation of 96 and 404 gives :

96 = 25 × 3, 404 = 22 × 101

Therefore, the HCF of these two integers is 22 = 4.

Also, LCM (96, 404)

=

#### Example 8 :

Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.

#### Solution :

We have :

6 = 2 × 3, 72 = 23 × 32, 120 = 23 × 3 × 5

Here, 21 and 31 are the smallest powers of the common factors 2 and 3, respectively.

So, HCF (6, 72, 120) = 21 × 31 = 2 × 3 = 6

23, 32 and 51 are the greatest powers of the prime factors 2, 3 and 5 respectively involved in the three numbers.

So, LCM (6, 72, 120) = 23 × 32 × 51 = 360

#### Remark :

Notice, 6 × 72 × 120 ≠ HCF (6, 72, 120) × LCM (6, 72, 120). So, the product of three numbers is not equal to the product of their HCF and LCM.

#### Exercise 1.2

1. Express each number as a product of its prime factors:

(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

2. Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.

(i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54

3. Find the LCM and HCF of the following integers by applying the prime factorisation method.

(i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25

4. Given that HCF (306, 657) = 9, find LCM (306, 657).

5. Check whether 6n can end with the digit 0 for any natural number n.

6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

#### 1.4 Revisiting Irrational Numbers

In Class IX, you were introduced to irrational numbers and many of their properties. You studied about their existence and how the rationals and the irrationals together made up the real numbers. You even studied how to locate irrationals on the number line. However, we did not prove that they were irrationals. In this section, we will prove that and, in general, is irrational, where p is a prime. One of the theorems, we use in our proof, is the Fundamental Theorem of Arithmetic.

Recall, a number ‘s’ is called irrational if it cannot be written in the form where p and q are integers and q ≠ 0. Some examples of irrational numbers, with which you are already familiar, are :

, etc.

Before we prove that is irrational, we need the following theorem, whose proof is based on the Fundamental Theorem of Arithmetic.

#### Theorem 1.3 :

Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer.

#### *Proof :

Let the prime factorisation of a be as follows :

a = p1p2 . . . pn, where p1,p2, . . ., pn are primes, not necessarily distinct.

Therefore, a2 = (p1p2 . . . pn)(p1p2 . . . pn) = p21p22 . . . p2n.

Now, we are given that p divides a2. Therefore, from the Fundamental Theorem of Arithmetic, it follows that p is one of the prime factors of a2. However, using the uniqueness part of the Fundamental Theorem of Arithmetic, we realise that the only prime factors of a2 are p1, p2, . . ., pn. So p is one of p1, p2, . . ., pn.

Now, since a = p1 p2 . . . pn, p divides a.

We are now ready to give a proof that is irrational.

The proof is based on a technique called ‘proof by contradiction’. (This technique is discussed in some detail in Appendix 1).

#### Theorem 1.4 :

is irrational.

#### Proof :

Let us assume, to the contrary, that is rational.

So, we can find integers r and s (≠ 0) such that = .

Suppose r and s have a common factor other than 1. Then, we divide by the common factor to get where a and b are coprime.

So, b = a.

Squaring on both sides and rearranging, we get 2b2 = a2. Therefore, 2 divides a2.

Now, by Theorem 1.3, it follows that 2 divides a.

So, we can write a = 2c for some integer c.

Substituting for a, we get 2b2 = 4c2, that is, b2 = 2c2.

This means that 2 divides b2, and so 2 divides b (again using Theorem 1.3 with p = 2).

Therefore, a and b have at least 2 as a common factor.

But this contradicts the fact that a and b have no common factors other than 1.

This contradiction has arisen because of our incorrect assumption that is rational.

So, we conclude that is irrational.

#### Example 9 :

Prove that is irrational.

#### Solution :

Let us assume, to the contrary, that is rational.

That is, we can find integers a and b (≠ 0) such that =

Suppose a and b have a common factor other than 1, then we can divide by the common factor, and assume that a and b are coprime.

So,

Squaring on both sides, and rearranging, we get 3b2 = a2.

Therefore, a2 is divisible by 3, and by Theorem 1.3, it follows that a is also divisible by 3.

So, we can write a = 3c for some integer c.

Substituting for a, we get 3b2 = 9c2, that is, b2 = 3c2.

This means that b2 is divisible by 3, and so b is also divisible by 3 (using Theorem 1.3 with p = 3).

Therefore, a and b have at least 3 as a common factor.

But this contradicts the fact that a and b are coprime.

This contradiction has arisen because of our incorrect assumption that is rational. So, we conclude that is irrational.

In Class IX, we mentioned that :

the sum or difference of a rational and an irrational number is irrational and

the product and quotient of a non-zero rational and irrational number is irrational.

We prove some particular cases here.

#### Example 10 :

Show that is irrational.

#### Solution :

Let us assume, to the contrary, that is rational.

That is, we can find coprime a and b (b ≠ 0) such that

Therefore,

Rearranging this equation, we get

Since a and b are integers, we get is rational, and so is rational.

But this contradicts the fact that is irrational.

This contradiction has arisen because of our incorrect assumption that 5 – is rational.

So, we conclude that is irrational.

#### Example 11 :

Show that is irrational.

#### Solution :

Let us assume, to the contrary, that is rational.

That is, we can find coprime a and b (b ≠ 0) such that

Rearranging, we get

Since 3, a and b are integers, is rational, and so is rational.

But this contradicts the fact that is irrational.

So, we conclude that is irrational.

#### Exercise 1.3

1. Prove that is irrational.

2. Prove that is irrational.

3. Prove that the following are irrationals :

(i) (ii) (iii)

#### 1.5 Revisiting Rational Numbers and their Decimal Expansions

In Class IX, you studied that rational numbers have either a terminating decimal expansion or a non-terminating repeating decimal expansion. In this section, we are going to consider a rational number, say , and explore exactly when the decimal expansion of is terminating and when it is non-terminating repeating

(or recurring). We do so by considering several examples.

Let us consider the following rational numbers :

(i) 0.375 (ii) 0.104 (iii) 0.0875 (iv) 23.3408.

Now (i) (ii)

(iii) (iv)

As one would expect, they can all be expressed as rational numbers whose denominators are powers of 10. Let us try and cancel the common factors between the numerator and denominator and see what we get :

(i) (ii)

(iii) (iv)

Do you see any pattern? It appears that, we have converted a real number whose decimal expansion terminates into a rational number of the form where p and q are coprime, and the prime factorisation of the denominator (that is, q) has only powers of 2, or powers of 5, or both. We should expect the denominator to look like this, since powers of 10 can only have powers of 2 and 5 as factors.

Even though, we have worked only with a few examples, you can see that any real number which has a decimal expansion that terminates can be expressed as a rational number whose denominator is a power of 10. Also the only prime factors of 10 are 2 and 5. So, cancelling out the common factors between the numerator and the denominator, we find that this real number is a rational number of the form where the prime factorisation of q is of the form 2n5m, and n, m are some non-negative integers.

Let us write our result formally:

#### Theorem 1.5 :

Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form where p and q are coprime, and the prime factorisation of q is of the form 2n5m, where n, m are non-negative integers.

You are probably wondering what happens the other way round in Theorem 1.5. That is, if we have a rational number of the form and the prime factorisation of q is of the form 2n5m, where n, m are non negative integers, then does have a terminating decimal expansion?

Let us see if there is some obvious reason why this is true. You will surely agree that any rational number of the form where b is a power of 10, will have a terminating decimal expansion. So it seems to make sense to convert a rational number of the form , where q is of the form 2n5m, to an equivalent rational number of the form where b is a power of 10. Let us go back to our examples above and work backwards.

(i)

(ii)

(iii)

(iv)

So, these examples show us how we can convert a rational number of the form , where q is of the form 2n5m, to an equivalent rational number of the form where b is a power of 10. Therefore, the decimal expansion of such a rational number terminates. Let us write down our result formally.

#### Theorem 1.6 :

Let x = be a rational number, such that the prime factorisation of q is of the form 2n5m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.

We are now ready to move on to the rational numbers whose decimal expansions are non-terminating and recurring. Once again, let us look at an example to see what is going on. We refer to Example 5, Chapter 1, from your Class IX textbook, namely, . Here, remainders are 3, 2, 6, 4, 5, 1, 3, 2, 6, 4, 5, 1, . . . and divisor is 7.

Notice that the denominator here, i.e., 7 is clearly not of the form 2n5m. Therefore, from Theorems 1.5 and 1.6, we know that will not have a terminating decimal expansion. Hence, 0 will not show up as a remainder (Why?), and the remainders will start repeating after a certain stage. So, we will have a block of digits, namely, 142857, repeating in the quotient of .

What we have seen, in the case of , is true for any rational number not covered by Theorems 1.5 and 1.6. For such numbers we have :

#### Theorem 1.7 :

Let x = , where p and q are coprimes, be a rational number, such that the prime factorisation 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 (recurring).

From the discussion above, we can conclude that the decimal expansion of every rational number is either terminating or non-terminating repeating.

#### Exercise 1.4

1. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

(i) (ii) (iii) (iv)

(v) (vi) (vii) (viii)

(ix) (x)

2. Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

3. The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form what can you say about the prime factors of q?

(i) 43.123456789 (ii) 0.120120012000120000. . . (iii)

#### 1.6 Summary

In this chapter, you have studied the following points:

1. Euclid’s division lemma :

Given positive integers a and b, there exist whole numbers q and r satisfying a = bq + r,

0 ≤ r < b.

2. Euclid’s division algorithm : This is based on Euclid’s division lemma. According to this, the HCF of any two positive integers a and b, with a > b, is obtained as follows:

#### Step 1 :

Apply the division lemma to find q and r where a = bq + r, 0 ≤ r < b.

#### Step 2 :

If r = 0, the HCF is b. If r ≠ 0, apply Euclid’s lemma to b and r.

#### Step 3 :

Continue the process till the remainder is zero. The divisor at this stage will be HCF (a, b). Also, HCF(a, b) = HCF(b, r).

3. The Fundamental Theorem of Arithmetic :

Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

4. If p is a prime and p divides a2, then p divides a, where a is a positive integer.

5. To prove that are irrationals.

6. Let x be a rational number whose decimal expansion terminates. Then we can express x in the form , where p and q are coprime, and the prime factorisation of q is of the form 2n5m, where n, m are non-negative integers.

7. Let x = be a rational number, such that the prime factorisation of q is of the form 2n5m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.

8. Let x = be a rational number, such that the prime factorisation of q is not of the form 2n 5m, where n, m are non-negative integers. Then x has a decimal expansion which is non-terminating repeating (recurring).

#### A Note to the Reader

You have seen that :

HCF (p, q, r) × LCM (p, q, r) ≠ p × q × r, where p, q, r are positive integers (see Example 8). However, the following results hold good for three numbers p, q and r :

LCM (p, q, r) =

HCF (p, q, r) =