Exercise 1.1 Exercise 1.2 Exercise 1.3 Exercise 1.4
Exercise 1.4
Question 1
Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:
(i)
Sol :
The denominator is in the form of
Hence, the decimal expansion of is terminating.
(ii)
Sol :
The denominator is in the form of
Hence, the decimal expansion of is terminating.
(iii)
Sol :
Since,the denominator is not in the form of and it also contains 7 and 13 as its factors , its decimal expansion will be non-terminating repeating .
(iv)
Sol :
The denominator is of the form .
Hence , the decimal expansion of is terminating .
(v)
Sol :
Since,the denominator is not in the form of and it also contains 7 as its factors , its decimal expansion will be non-terminating repeating .
(vi)
Sol :
The denominator is of the form .
Hence , the decimal expansion of is terminating .
(vii)
Sol :
The denominator is of the form but it also has 75 as its factor, the decimal expansion of
is non-terminating repeating .
(viii)
Sol :
The denominator is of the form 5n
Hence ,the decimal expansion of is terminating .
(ix)
Sol :
The denominator is of the form
Hence , the decimal expansion of is terminating .
(x)
Sol :
The denominator is in the form of but it contains 3 as its factors , the decimal expansion of
is non-terminating repeating .
Question 2
Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.
Sol :
(i)
(ii) (working)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
Question 3
The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form , what can you say about the prime factor of q ?
(i)
Sol :
Since this number has a terminating decimal expansion, it is a rational number of the form and q is of the form
.
i.e., the prime factors of q will be either 2 or 5 or both .
(ii)
Sol :
The decimal expansion is neither terminating nor recurring. Therefore, the given number is an irrational number.
(iii)
Sol :
Since, the decimal expansion is non-terminating recurring, the given number is a rational number of the form and q is not of the form
i.e., the prime factors of q will also have a factor other than 2 or 5 .