Exercise 1.3 Class 9

#mathsclass9 #rajansir Exercise 1.3 Class 9 Exercise 1.3 Class 9 by @rajansir07 for Online classes Call- 9811757843 Exercise 1.3 of NCERT Class 9 Maths Chapter 1, "Number Systems," focuses on the decimal expansions of rational numbers. It guides students in converting fractions into their decimal forms and identifying whether these decimals are terminating or non-terminating repeating. Below are the solutions to the exercise problems: Question 1: Write the following in decimal form and state the type of decimal expansion each has: (i) Solution: Dividing 36 by 100: This is a terminating decimal expansion. (ii) Solution: Dividing 1 by 11: This is a non-terminating repeating decimal expansion. (iii) Solution: Converting the mixed fraction to an improper fraction: Dividing 33 by 8: This is a terminating decimal expansion. (iv) Solution: Dividing 3 by 13: This is a non-terminating repeating decimal expansion. (v) Solution: Dividing 2 by 11: This is a non-terminating repeating decimal expansion. (vi) Solution: Dividing 329 by 400: This is a terminating decimal expansion. Question 2: You know that . Can you predict what the decimal expansions of , , , , and are without actually performing the division? If so, how? Solution: Yes, we can predict the decimal expansions by observing the repeating block in . The repeating sequence is 142857. Multiplying this sequence by 2, 3, 4, 5, and 6 respectively, and adjusting for decimal placement, we get: Each fraction has a non-terminating repeating decimal expansion. Question 3: Express the following in the form , where p and q are integers and q ≠ 0: (i) 0.6 Solution: Let . Multiplying both sides by 10: Therefore, . (ii) 0. Solution: Let . Multiplying both sides by 100: Subtracting the original equation from this: Therefore, . (iii) 0. Solution: Let . Multiplying both sides by 1000: Subtracting the original equation from this: Therefore, . Question 4: Express 0. in the form . Are you surprised by your answer? Discuss. 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 have 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 terminates. 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 the prime factorisation 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 and non-recurring. Solution: We know that all irrational numbers are non-terminating and non-recurring. The three numbers with decimal expansions that are non-terminating and 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 and non-recurring, 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; hence, it is a rational number. (v) 1.101001000100001… Solution: Since the number 1.101001000100001… is non-terminating and non-repeating (non-recurring), it is an irrational number.