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CBSE 10th Mathematics | Decimal Representation Of Rational Numbers

Decimal Representation Of Rational Numbers    

 

Theorem:

Let \displaystyle x=\frac{p}{q} be a rational number such that \displaystyle q\ne 0 and prime factorization of q is of the form \displaystyle {{2}^{n}}\times {{5}^{m}} where m, n are non-negative integers then x has a decimal representation which terminates.

    For example :     \displaystyle 0.275=\frac{275}{{{10}^{3}}}=\frac{{{5}^{2}}\times 11}{{{2}^{3}}\times {{5}^{3}}}=\frac{11}{{{2}^{3}}\times 5}=\frac{11}{40}

 

Theorem:

 

Let \displaystyle x=\frac{p}{q} be a rational number such that \displaystyle q\ne 0 and prime factorization of q is not of the form \displaystyle {{2}^{m}}\times {{5}^{n}}, where m, n are non-negative integers, then x has a decimal expansion which is non-terminating repeating.

    For example :     \displaystyle \frac{5}{3}=1.66666...

    

 

Solved Examples Based on Decimal Representation Of Rational Numbers    

 

Question:

 

Without actually calculating, state whether the following rational numbers have a terminating or non-terminating repeating decimal expansion.

        (i)     \displaystyle \frac{27}{343}            (ii)    \displaystyle \frac{19}{1600}        (iii)    \displaystyle \frac{129}{{{2}^{2}}\times {{5}^{5}}\times {{3}^{2}}}    

Hint: If the denominator is of the form \displaystyle {{2}^{m}}\times {{5}^{n}} for some non negative integer m and n, then rational number has terminating decimal otherwise non terminating.

 

Solution:     

 

(i)    \displaystyle \frac{27}{343}=\frac{27}{{{7}^{3}}}

            Since \displaystyle q={{7}^{3}} which is not of the form \displaystyle {{2}^{m}}\times {{5}^{n}}.

            \    It has non terminating decimal representation.

        (ii)    \displaystyle \frac{19}{1600}=\frac{19}{{{2}^{6}}\times {{5}^{2}}}

            Since q = \displaystyle {{2}^{6}}\times {{5}^{2}} which is of the form \displaystyle {{2}^{m}}\times {{5}^{n}}.

            \    It has a terminating decimal representation.

        (iii)    \displaystyle \frac{129}{{{2}^{2}}\times {{5}^{5}}\times {{3}^{2}}}

             Since \displaystyle q={{2}^{2}}\times {{5}^{5}}\times {{3}^{2}} is not of the form \displaystyle {{2}^{m}}\times {{5}^{n}}. It has a non-terminating decimal representation.

 

Question:

 

What can you say about the prime factorization of the denominators of the following rationales:

        (i)     36.12345            (ii)    \displaystyle 36.\overline{5678}

 

Solution:     

 

(i)    Since 36.12345 has terminating decimal expansion. So, its denominator is of the form \displaystyle {{2}^{m}}\times {{5}^{n}} where m, n are non-negative integers.

        (ii)    Since \displaystyle 36.\overline{5678}has non terminating repeating decimal expansion. So, its denominator has factors other than 2 or 5.

 

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