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Number Systems : Exercise 1.1 (Mathematics NCERT Class 9th)


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Q.1     Is zero a rational number ? Can you write it in the form {p \over q}, where p and q are integers and q \ne 0 ?
Sol.

Yes, zero is a rational number. 
Zero can be written in any of the following forms : 
{0 \over 1},{0 \over { - 1}},{0 \over 2},{0 \over { - 2}} and so on.
Thus, 0 can be written as {p \over q}, where p = 0 and q is any non- zero integer.
Hence , 0 is a rational number.


 Q.2      Find six rational numbers between 3 and 4.
Sol.

We know that between two rational numbers x and y, such that x < y, there is a rational number {{x + y} \over 2}. That is , x < {{x + y} \over 2} < y
There are two rational numbers 3 and 4 , such that 3 < 4 , a rational number between 3 and 4 is {1 \over 2}\left( {3 + 4} \right)i.e.,\,{7 \over 2}
Therefore 3 < {7 \over 2} < 4
Now ,two rational numbers 3 and{7 \over 2} such that, a rational number between 3 and {7 \over 2}is

{1 \over 2}\left( {3 + {7 \over 2}} \right) = {1 \over 2} \times {{6 + 7} \over 2} = {{13} \over 4}
There are two rational numbers{7 \over 2} and 4 , a rational number between {7 \over 2} and 4 is

{1 \over 2}\left( {{7 \over 2} + 4} \right) = {1 \over 2} \times {{7 + 8} \over 2} = {{15} \over 4}
Therefore 3 < {{13} \over 4} < {7 \over 2} < {{15} \over 4} < 4
There are two rational numbers 3 and {13 \over 4}  ,a rational number between 3 and {{13} \over 4}is

{1 \over 2}\left( {3 + {{13} \over 4}} \right) = {1 \over 2} \times {{12 + 13} \over 4} = {{25} \over 8}
There are two rational numbers{15 \over 4} and 4 , a rational number between {{15} \over 4} and 4 is

{1 \over 2}\left( {{{15} \over 4} + 4} \right) = {1 \over 2} \times {{15 + 16} \over 4} = {{31} \over 8}
There are two rational numbers {31 \over 8} and 4 , a rational number between {31 \over 8} and 4 is

{1 \over 2}\left( {{{31} \over 8} + 4} \right) = {1 \over 2} \times {{31 + 32} \over 8} = {{63} \over {16}}
Therefore 3 < {{25} \over 8} < {{13} \over 4} < {7 \over 2} < {{15} \over 4} < {{31} \over 8} < {{63} \over {16}} < 4
Hence, six rational numbers between 3 and 4 are :
{{25} \over 8},{{13} \over 4},{7 \over 2},{{15} \over 4},{{31} \over 8}and\,{{63} \over {16}}
ALTERNATIVE METHOD
Since we want 6 rational number between 3 and 4, so we write
3 = {{3 \times 7} \over {1 \times 7}} = {{21} \over 7} \,\,\,{\mkern 1mu} and{\mkern 1mu} \,\,4 = {{4 \times 7} \over {1 \times 7}} = {{28} \over 7}
We know that 21 < 22 < 23 < 24 < 25 < 26 < 27 < 28
 \Rightarrow  {{21} \over 7} < {{22} \over 7} < {{23} \over 7} < {{24} \over 7}  < {{25} \over 7} < {{26} \over 7} < {{27} \over 7} < {{28} \over 7}
Hence, six rational numbers between 3 = {{21} \over 7}\,and\,4 = {{28} \over 7}are
{{22} \over 7},{{23} \over 7},{{24} \over 7},{{25} \over 7},{{26} \over 7} \,and\,\,{\mkern 1mu} {{27} \over 7}


Q.3      Find five rational numbers between {3 \over 5}and{4 \over 5}.
Sol.

Since we want 5 rational numbers between {3 \over 5}and{4 \over 5}, so we write.
{3 \over 5} = {{3 \times 6} \over {5 \times 6}} = {{18} \over {30}} and {4 \over 5} = {{4 \times 6} \over {5 \times 6}} = {{24} \over {30}}
We know that 18 < 19 < 20 < 21 < 22 < 23 < 24
 \Rightarrow    {{18} \over {30}} < {{19} \over {30}} < {{20} \over {30}} < {{21} \over {30}} < {{22} \over {30}}  < {{23} \over {30}} < {{24} \over {30}}
Hence , 5 rational numbers between {3 \over 5} = {{18} \over {30}}\,and\,{4 \over 5} = {{24} \over {30}}are:
{{19} \over {30}},{{20} \over {30}},{{21} \over {30}},{{22} \over {30}},{{23} \over {30}}and\,{{24} \over {30}}


Q.4     State whether the following statements are true or false. Give reasons for your answers.
          
(i) Every natural number is a whole number.
                (ii) Every integer is a whole number.
                (iii) Every rational number is a whole number.
Sol.

(i) True : Every natural number lies in the collection of whole numbers.
(ii) False : -3 is not a whole number.
(iii) False {3 \over 5} is not a whole number.



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