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Write the following in decimal form and say what kind of decimal expansion each has : (i) (ii)
(iii)
(iv)
(v)
(vi)
(i) 
Terminating
(ii) 
Non-terminating repeating
(iii) 
Terminating
(iv) 
Non-terminating repeating
(v) 
Non-terminating repeating
(vi) 
Terminating
You know that = 0.
. Can you predict what the decimal expansions of
,
,
,
,
are, without actually doing the long division? If so, how? [Hint : Study the remainders while finding the value of
carefully.]
Yes. It can be done as follows.
Express the following in the form , where p and q are integers and
(i)
(ii)
(iii)
(i) 
Let x = 0.666…
10x = 6.666…
10x = 6 + x
9x = 6
(ii) 
Let x = 0.777…
10x = 7.777…
10x = 7 + x
(iii) 
Let x = 0.001001…
1000x = 1.001001…
1000x = 1 + x
999x = 1
Express 0.99999 .... in the form . Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.
Let x = 0.9999…
10x = 9.9999…
10x = 9 + x
9x = 9
x = 1
What can the maximum number of digits be in the repeating block of digits in the decimal expansion of ? Perform the division to check your answer.
It can be observed that,
There are 16 digits in the repeating block of the decimal expansion of 
.
Look at several examples of rational numbers in the form (
), 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?
Terminating decimal expansion is present when the denominator q of rational number 
is either of 2, 4, 5, 8, 10, and so on…
A terminating decimal maybe found in the situation where prime factorisation of the denominator of the given fractions has the power of 2 only or 5 only or both.
Write three numbers whose decimal expansions are non-terminating non-recurring.
3 numbers whose decimal expansions are non-terminating non-recurring are as follows.
(i) 0.505005000051509 ..
(ii) 0.7207200720007200007200000…
(iii) 0.080080008000080000080000008…
Find three different irrational numbers between the rational numbers and
.
3 irrational numbers are as follows.
0.73073007300073000073…
0.75075007500075000075…
0.79079007900079000079…
Classify the following numbers as rational or irrational : (i) (ii)
(iii) 0.3796 (iv) 7.478478... (v) 1.101001000100001...
(i) 
As the decimal expansion of this number is non-terminating non-recurring, therefore, it is an irrational number.
(ii) 
It is a rational number as it can be represented in 
form.
(iii) 0.3796
As the decimal expansion of this number is terminating, therefore, it is a rational number.
(iv) 7.478478 … 
As the decimal expansion of this number is non-terminating recurring, therefore, it is a rational number.
(v) 1.10100100010000 …
As the decimal expansion of this number is non-terminating non-repeating, therefore, it is an irrational number.
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