Ncert Class 9 Maths Chapter 1 Exercise 1.3 Solutions

Decimal Expansion which ends after the division is called terminating expansion. And when a fraction is divided such that the value repeats itself, it is a recurring expansion. When the expansion never ends it is called non-terminating expansion.
For example: 0.2342516782943257 is neither repeating nor terminating expansion.
\( \frac{36}{100}=0.36 \)
(Since, decimal expansion has finite no. of figures. Hence, it is terminating).

Decimal Expansion which ends after the division is called terminating expansion. And when a fraction is divided such that the value repeats itself, it is a recurring expansion. When the expansion never ends it is called non-terminating expansion.
For example: 0.2342516782943257 is neither repeating nor terminating expansion.
\( \frac{1}{11}=0.09090909 \ldots \)
\( =0 \overline{18} \) (Since decimal expansion repeats itself so it is a non-terminating recurring decimal expansion).

Decimal Expansion which ends after the division is called terminating expansion. And when a fraction is divided such that the value repeats itself, it is a recurring expansion. When the expansion never ends it is called non-terminating expansion.
For example: 0.2342516782943257 is neither repeating nor terminating expansion.
\( 4 \frac{1}{8}=\frac{33}{8} \)
\( =4.125 \)
(Since, decimal expansion has finite no. of figures. Hence, it is terminating).

Decimal Expansion which ends after the division is called terminating expansion. And when a fraction is divided such that the value repeats itself, it is a recurring expansion. When the expansion never ends it is called non-terminating expansion.
For example: 0.2342516782943257 is neither repeating nor terminating expansion.
\(\frac{3}{13} \) \(=0.230769230769\)
\( =0 . \overline{230769}\)
(Since decimal expansion repeats itself so it is a non-terminating recurring decimal expansion).

Decimal Expansion which ends after the division is called terminating expansion. And when a fraction is divided such that the value repeats itself, it is a recurring expansion. When the expansion never ends it is called non-terminating expansion.
For example: 0.2342516782943257 is neither repeating nor terminating expansion.
\( \frac{2}{11}=0.1818181818 \ldots \)
\(=0 . \overline{18}\)
(Since decimal expansion repeats itself so it is a non-terminating recurring decimal expansion)

Decimal Expansion which ends after the division is called terminating expansion. And when a fraction is divided such that the value repeats itself, it is a recurring expansion. When the expansion never ends it is called non-terminating expansion.
For example: 0.2342516782943257 is neither repeating nor terminating expansion.
\( \frac{329}{400}=0.8225 \)
(Since, decimal expansion has finite no. of figures. Hence, it is terminating.)

Yes, we can do this by the following method:
\(\frac{2 }{ 7}=2 \times \frac{1 }{ 7}=2 \times 0.\overline{142857}=0.\overline{285714}\)
\(\frac{3 }{ 7}=3 \times \frac{1 }{ 7}=3 \times 0.\overline{142857}=0.\overline{428571}\)
\(\frac{4 }{ 7}=4 \times \frac{1 }{ 7}=4 \times 0.\overline{142857}=0.\overline{571428}\)
\(\frac{5 }{ 7}=5 \times \frac{1 }{ 7}=5 \times 0.\overline{142857}=0.\overline{714285}\)
\(\frac{6 }{ 7}=6 \times \frac{1 }{ 7}=6 \times 0.\overline{142857}=0.\overline{857142}\)

\( 0 . \overline{6}=0.666 \ldots \)
Let \( \mathrm{x}=0.666666 \ldots \)
Multiplying both sides by 10
\(10 x=6.666=6+0.666\)
\(10 x=6+x\)
\(9 x=6\)
\(x=\frac{2}{3}\)

\( \mathrm{x}=0 . \overline{47}=0.4777777 \)
Let \( \mathrm{x}=0.4777777 \)
Now multiplying both sides by \( 10,10 \mathrm{x}=4.77777777 \ldots \ldots\) eq(1) Now multiplying eq(1) by 100 we get \( 100 x=47.777777 \ldots\ldots\)
eq(2) Now eq(2) - eq(1) \( 100 x-10 x=47-490 x=43 x=\frac{43}{90} \)

\( 0 . \overline{001}=0.001001 \ldots \)
Let \( \mathrm{x}=0.001001 \ldots \) eq(1)
Multiplying both sides by 1000
\(1000 x=1.001001 \ldots=1+0.001001\ldots\text{eq}(2)\)
we can see that,
\(1000 x=1+x\)
\( 999 x=1 \)
\(x=\frac{1}{999}\)

Let \( x=0.99999 \ldots\ldots\) (1)
multiply both sides by \( 1010 \mathrm{x}=9.9999 \) \( \ldots \ldots\)(2)
subtract (1) from (2).
\( 9 x=9 \)
As \( x=\frac{9 }{ 9} \)
or \( x=1 \)
Therefore, on converting \( 0.99999 \ldots \) in the \( \frac{\mathrm{p} }{ \mathrm{q}} \) form, we get the answer as 1.
The difference between 1 and 0.999999 is 0.000001 which is negligible. Hence, 0.999 is too much near to 1. Therefore, 1 as an answer can be justified

\( \frac{1 }{ 17}=0.0\overline{588235294117647} \)
There are 16 digits in the repeating numbers of the decimal expansion of \( \frac{1 }{ 17} \).
Division check:

After 0.0588235294117647 digits will start repeating itself.

We observe that when q is \( 2,4,5,8,10 \ldots \) then the decimal expansion is terminating. For example:
\(\frac{1}{2}=0.5 \text { denominator } \mathrm{q}=2^{1}\)
\(\frac{7}{8}=0.875, \text { denominator } \mathrm{q}=2^{3}\)
\(\frac{4}{5}=0.8, \text { denominator } \mathrm{q}=5^{1}\)
It can be observed that the terminating decimal can be obtained in a condition where prime factorization of the denominator of the given fractions has the power of 2 only or 5 only or both.

Numbers with non-terminating decimal expression means that they never completely divided.
For example, when you divide 10 by 3, the answer you get keeps on dividing and no single value can be obtained but this number is recurring because \( \frac{10 }{ 3}=2.666666666 \) and in this value 6 is repeating or recurring. So we have to write a number which is non terminating and non-repeating. Write any number which does not end and does not repeat. ( \( 0.123123456345876 \ldots \ldots \) or \( 2.123231245627549 \ldots \) there can be infinite such numbers) Three numbers whose decimal expansions are non-terminating non-recurring are:
\(0.304004000400003 \ldots\)
\(0.80506000900005 \ldots\)
\(0.7205200820008200008200000 \ldots\)
OR
All irrational numbers are non-terminating and non-repeating.
Example: \( \sqrt{2}, \sqrt{3} \) and \( \sqrt{5} \).

Irrational Numbers: Numbers that cannot be expressed in \(\frac{p}{q}\) form are called irrational number.
Any number that is non-repeating and non terminating is an irrational number. To find irrational numbers first we find a decimal expansion of given fractions.
\(\frac{5}{7}=0. \overline{714285}\)
\(\frac{9}{11}=0. \overline{81}\)
Now, for irrational numbers, we write any non-terminating and non-repeating decimal number between given numbers.
Three different irrational numbers can be:
\( 0.74074007400074000074 \ldots \)
\( 0.75075007500075000075 \ldots \)
\( 0.78078007800078000078 \ldots \)

Rational Number: A number that can be expressed in the form \( \frac{\mathrm{p} }{ \mathrm{q}} \), where \( \mathrm{q} \neq 0 \).
Irrational Number: A number which can not be expressed in the form \( \frac{\mathrm{p} }{ \mathrm{q}} \). The numbers that are non-repeating and non-terminating are called irrational.
\( \sqrt{23}=4.79583152331 \ldots \)
Since the decimal expansion is non-terminating and non-recurring hence, it is an irrational number

Rational Number: A number that can be expressed in the form \( \frac{\mathrm{p} }{ \mathrm{q}} \), where \( \mathrm{q} \neq 0 \).
Irrational Number: A number which can not be expressed in the form \( \frac{\mathrm{p} }{ \mathrm{q}} \). The numbers that are non-repeating and non-terminating are called irrational.
\( \sqrt{225}=15=\frac{15 }{ 1} \)
It is a rational number as it can be represented in the form of \( \frac{p }{ q} \)

Rational Number: A number that can be expressed in the form \( \frac{\mathrm{p} }{ \mathrm{q}} \), where \( \mathrm{q} \neq 0 \).
Irrational Number: A number which can not be expressed in the form \( \frac{\mathrm{p} }{ \mathrm{q}} \). The numbers that are non-repeating and non-terminating are called irrational.
0.3796
Since the decimal expansion is terminating number.
Therefore, it is a rational number

Rational Number: A number that can be expressed in the form \( \frac{\mathrm{p} }{ \mathrm{q}} \), where \( \mathrm{q} \neq 0 \).
Irrational Number: A number which can not be expressed in the form \( \frac{\mathrm{p} }{ \mathrm{q}} \). The numbers that are non-repeating and non-terminating are called irrational.
\( 7.478478=7.4\overline{78} \)
Since the decimal expansion is a non-terminating recurring number.
Therefore, it is a rational number

Rational Number: A number that can be expressed in the form \( \frac{\mathrm{p} }{ \mathrm{q} }\), where \( \mathrm{q} \neq 0 \).
Irrational Number: A number which can not be expressed in the form \( \frac{\mathrm{p} }{ \mathrm{q}} \). The numbers that are non-repeating and non-terminating are called irrational.
\( 1.101001000100001 \ldots \)
Since the number has decimal expansion as non-terminating and non-repeating.
Hence, it is an irrational number