what is number of 0.303003000300003..... decimal expansion
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0.6
(ii) 0.47
(iii) 0.001
Answer- (i) 0.6 = 0.666…
Let x = 0.666…
10x = 6.666…
10x = 6 + x
9x = 6
x = 2/3(ii) 0.47 = 0.4777…
= 4/10 + 0.777/10
Let x = 0.777…
10x = 7.777…
10x = 7 + x
x = 7/9
4/10 + 0.777…/10 = 4/10 + 7/90
= 36/90 +7/90 = 43/90
(iii) 0.001 = 0.001001…
Let x = 0.001001…
1000x = 1.001001…
1000x = 1 + x
999x = 1
x = 1/999
Q4. 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- Let x = 0.9999…
10x = 9.9999…
10x = 9 + x
9x = 9
x = 1
The difference between 1 and 0.999999 is 0.000001 which is negligible. Thus, 0.999 is too much near 1, Therefore, the 1 as answer can be justified.
Q5. 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- 1/17 = 0.0588235294117647
There are 16 digits in the repeating block of the decimal expansion of 1/17.
Division Check:

= 0.0588235294117647
Q6. 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- 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 observed that terminating decimal may be obtained in the situation where prime factorisation of the denominator of the given fractions has the power of 2 only or 5 only or both.
Q7. Write three numbers whose decimal expansions are non-terminating non-recurring.
Answer- Three numbers whose decimal expansions are non-terminating non-recurring are:
0.303003000300003…
0.505005000500005…
0.7207200720007200007200000…
Q8. Find three different irrational numbers between the rational numbers 5/7 and 9/11.
Answer- 5/7 = 0.714285
9/11 = 0.81
Three different irrational numbers are:
0.73073007300073000073…
0.75075007300075000075…
0.76076007600076000076…
Q9. Classify the following numbers as rational or irrational:
(i) √23
(ii) √225
(iii) 0.3796
(iv) 7.478478
(v) 1.101001000100001…
Answer- (i) √23 = 4.79583152331…
Since the number is non-terminating non-recurring therefore, it is an irrational number.
(ii) √225 = 15 = 15/1
Since the number is rational number as it can represented in p/q form.
(iii) 0.3796
Since the number is terminating therefore, it is an rational number.
(iv) 7.478478 = 7.478
Since the this number is non-terminating recurring, therefore, it is a rational number.
(v) 1.101001000100001…
Since the number is non-terminating non-repeating, therefore, it is an irrational number.
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(ii) 0.47
(iii) 0.001
Answer- (i) 0.6 = 0.666…
Let x = 0.666…
10x = 6.666…
10x = 6 + x
9x = 6
x = 2/3(ii) 0.47 = 0.4777…
= 4/10 + 0.777/10
Let x = 0.777…
10x = 7.777…
10x = 7 + x
x = 7/9
4/10 + 0.777…/10 = 4/10 + 7/90
= 36/90 +7/90 = 43/90
(iii) 0.001 = 0.001001…
Let x = 0.001001…
1000x = 1.001001…
1000x = 1 + x
999x = 1
x = 1/999
Q4. 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- Let x = 0.9999…
10x = 9.9999…
10x = 9 + x
9x = 9
x = 1
The difference between 1 and 0.999999 is 0.000001 which is negligible. Thus, 0.999 is too much near 1, Therefore, the 1 as answer can be justified.
Q5. 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- 1/17 = 0.0588235294117647
There are 16 digits in the repeating block of the decimal expansion of 1/17.
Division Check:

= 0.0588235294117647
Q6. 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- 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 observed that terminating decimal may be obtained in the situation where prime factorisation of the denominator of the given fractions has the power of 2 only or 5 only or both.
Q7. Write three numbers whose decimal expansions are non-terminating non-recurring.
Answer- Three numbers whose decimal expansions are non-terminating non-recurring are:
0.303003000300003…
0.505005000500005…
0.7207200720007200007200000…
Q8. Find three different irrational numbers between the rational numbers 5/7 and 9/11.
Answer- 5/7 = 0.714285
9/11 = 0.81
Three different irrational numbers are:
0.73073007300073000073…
0.75075007300075000075…
0.76076007600076000076…
Q9. Classify the following numbers as rational or irrational:
(i) √23
(ii) √225
(iii) 0.3796
(iv) 7.478478
(v) 1.101001000100001…
Answer- (i) √23 = 4.79583152331…
Since the number is non-terminating non-recurring therefore, it is an irrational number.
(ii) √225 = 15 = 15/1
Since the number is rational number as it can represented in p/q form.
(iii) 0.3796
Since the number is terminating therefore, it is an rational number.
(iv) 7.478478 = 7.478
Since the this number is non-terminating recurring, therefore, it is a rational number.
(v) 1.101001000100001…
Since the number is non-terminating non-repeating, therefore, it is an irrational number.
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