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Number Systems

    Exercise 1.3

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9th class > Number Systems > Exercise 1.3

Exercise 1.3

1. Write the following in decimal form and say what kind of decimal expansion each has.

(i)36100 : which is .

(ii)111 : which is .

(iii)418 : which is .

(iv)313 : which is .

(v)211 : which is .

(vi)329400 : which is .

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 going the long division? If so,how?

Hint: Study the remainders while finding the value of 1/7 carefully.

Instruction

To find the decimal expansions of 27, 37 , 47 , 57 , 67 we observe that each fraction is a multiple of 17.
So, 27 = 2 × 17, 37 = × 17, 47 = × 17, 57 = × 17, 67 = × 17.
Since, 17 = 0.142857 , multiply it with each of these fractions.
27 × 0.142857 = , 37 × 0.142857 = , 47 × 0.142857 = .
57 × 0.142857 = and 67 × 0.142857 = .

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

(i)0.6 = .

(ii)0.47 = .

(iii)0.001 = .

4. Express 0.99999 .... in the form pq. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

We see that: 0.9999999 = . (Put in pq form)

Thus, we see that no matter whatever the number of intervals we take, 0.99999... always lies closer to .

Hence, we can say that 0.99999 = 1 which is algebraically proven.

5. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of pq ?Perform the division to check your answer.

Take, 117 which is equal to .

The maximum number of digits in the quotient are .

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?

25 : while 3100 :

2716 : meanwhile 3350 : .

We observe that the denominators of the above rational numbers are in the form of 2a × 5b, where a and b are whole numbers.

Hence if q is in the form 2a × 5b then pq is a terminating decimal.

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

All irrational numbers are non-terminating and non-repeating.

Example : , and .

8. Find three different irrational numbers between the rational numbers 57 and 911.

Instruction

For 57 = while 911 = .
, and are all than 57.
The upper limit of 0.81818 is than 911.
Three different irrational numbers between the rational numbers 57 and 911 are 0.7644513, 0.736546 and 0.7465664.

9. Classify the following numbers as rational or irrational :

(i) 23 : .

(ii)225 : .

(iii) 0.379 : .

(iv) 7.478478... :.

(v) 1.101001000100001... : .