1. Write the following in decimal form and say what kind of decimal expansion each has : (i) 36/100 Solution: = 0.36 (Terminating) (ii)1/11 Solution: Solution: = 4.125 (Terminating) (iv) 3/13 Solution: (v) 2/11 Solution: (vi) 329/400 Solution: = 0.8225 (Terminating) 2. You know that 1/7 = 0.142857. Can you predict what the decimal expansions of 2/7, 3/7, 4/7, 5/7, 6/7 are, without actually doing the long division? If so, how? [Hint: Study the remainders while finding the value of 1/7 carefully.] Solution: 3. Express the following in the form p/q, where p and q are integers and q 0. (i) Solution: Assume that x = 0.666… Then,10x = 6.666… 10x = 6 + x 9x = 6 x = 2/3 Solution: = (4/10)+(0.777/10) Assume that x = 0.777… Then, 10x = 7.777… 10x = 7 + x x = 7/9 (4/10)+(0.777../10) = (4/10)+(7/90) ( x = 7/9 and x = 0.777…0.777…/10 = 7/(9×10) = 7/90 ) = (36/90)+(7/90) = 43/90 Solution: Assume that x = 0.001001… Then, 1000x = 1.001001… 1000x = 1 + x 999x = 1 x = 1/999 4. Express 0.99999…. in the form p/q . Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense. Solution: Assume that x = 0.9999…..Eq (a) Multiplying both sides by 10, 10x = 9.9999…. Eq. (b) Eq.(b) – Eq.(a), we get 10x = 9.9999… x = 0.9999… 9x = 9 x = 1 The difference between 1 and 0.999999 is 0.000001 which is negligible. Hence, we can conclude that, 0.999 is too much near 1, therefore, 1 as the answer can be justified. 5. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 1/17 ? Perform the division to check your answer. Solution: 1/17 Dividing 1 by 17: There are 16 digits in the repeating block of the decimal expansion of 1/17. 6. Look at several examples of rational numbers in the form p/q (q ≠ 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy? Solution: We observe that when q is 2, 4, 5, 8, 10… Then the decimal expansion is terminating. For example: 1/2 = 0. 5, denominator q = 21 7/8 = 0. 875, denominator q =23 4/5 = 0. 8, denominator q = 51 We can observe that the terminating decimal may be obtained in the situation where prime factorization of the denominator of the given fractions has the power of only 2 or only 5 or both. 7. Write three numbers whose decimal expansions are non-terminating non-recurring. Solution: We know that all irrational numbers are non-terminating non-recurring. three numbers with decimal expansions that are non-terminating non-recurring are:√3 = 1.732050807568√26 =5.099019513592√101 = 10.04987562112 8. Find three different irrational numbers between the rational numbers 5/7 and 9/11. Solution: Three different irrational numbers are:0.73073007300073000073…0.75075007300075000075…0.76076007600076000076… 9. Classify the following numbers as rational or irrational according to their type: (i)√23 Solution: √23 = 4.79583152331… Since the number is non-terminating non-recurring therefore, it is an irrational number. (ii)√225 Solution: √225 = 15 = 15/1 Since the number can be represented in p/q form, it is a rational number. (iii) 0.3796 Solution: Since the number,0.3796, is terminating, it is a rational number. (iv) 7.478478 Solution: The number,7.478478, is non-terminating but recurring, it is a rational number. (v) 1.101001000100001… Solution: Since the number,1.101001000100001…, is non-terminating non-repeating (non-recurring), it is an irrational number.