## Chapter 1 Number Systems Exercise 1.3

Question 1: Write the following in decimal from and say what kind of decimal expansion each has:

i) 36/100

ii) 1/11

iii) 4 1/8

iv) 3/13

v) 2/11

vi) 329/400

Answer:

i) 36/100 = 0.36 (Terminating because denominator contains the factors of powers of 2 and 5)

ii) 1/11 = 0.09 (Non-Terminating Repeating because the denominator does not contain the factors of powers of 2 and 5)

iii) 4 1/8 = 4.125 (Terminating because denominator contains the factors of powers of 2)

iv) 3/13 = 0.230… (Non-Terminating Repeating because denominator does not contain the factors of powers of 2 and 5)

v) 2/11 = 0.18 (Non-Terminating Repeating because denominator does not contain the factors of powers of 2 and 5)

vi) 329/400 = 0.8225 (Terminating because denominator contains the factors of powers of 2 and 5)

Question 2: We know that 1/7 = 0.142857. Can you predict what the decimal expansion 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]

Answer:

Since, 1/7 = 0.142857

2 × 1/7 = 2/7

2 × 0.142857 = 0.285714

3 × 1/7 = 3/7

3 × 0.142857 = 0.428571

4 × 1/7 = 4/7

4 × 0.142857 = 0.571428

5 × 1/7 = 5/7

5 × 0.142857 = 0.714285

6 × 1/7 = 6/7

6 × 0.142857 = 0.857142

Question 3: Express the following in the form of p/q, where p and q are integers and q ≠ 0.

i) 0.6

ii) 0.47

iii) 0.001

Answer:

i)

x = 0.666…

10x = 6.666…

x = 0.666…

9x = 6

x = 6/9

x = 2/3

ii)

x = 0.4777…

10x = 4.777…

100x = 47.777…

-10x = 4.777….

90x = 43

x = 43/90

iii)

x = 0.001001001…

1000x = 1.001001001…

-x = 0.001001001…

999x = 1

x = 1/999

Question 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.

Answer:

x = 0.999…

10x = 9.999…

-x = 0.999…

9x = 9

x = 9/9 or 1

Question 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.

Answer:

Question 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?

Answer: The prime factorisation of q has only powers of 2 or powers of 5 or both.

Question 7: Write three numbers whose decimal expansions are non-terminating non-recurring.

Answer:

0.956509267…

0.7200710201…

0.76700769907…

Question 8: Find three different irrational numbers between the rational numbers 5/7 and 9/11.

Answer:

5/7 × 11/11 = 55/77

9/11 × 7/7 = 63/77

55/77 = 0.7142199…

63/77 = 0.818181…

Hence, the rational numbers between 5/7 and 9/11 are

0.724130…

0.796120…

0.80010001…

Question 9: Classify the following numbers as rational or irrational according to their type:

i) √23

ii) √225

iii) 0.3796

iv) 7.478478…

v) 1.101001000100001…

Answer:

i) √23 = 4.795…

As the number is Non-Terminating Non-Recurring, it is an irrational number.

ii) √225 = 15

As the number can be represent in the p/q form, it is an rational number.

iii) 0.3796

As the number is terminating, it is an rational number.

iv) 7.478478…

As the number is Non-Terminating Non-Recurring, it is an irrational number.

v) 1.101001000100001…

As the number is Non-Terminating Non-Recurring, it is an irrational number.

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