Question

Write the condition to be satisfied by q so that a rational number $\frac{p}{q}$ has a non terminating decimal expansions.

Answer 1

SOLUTION :  

The condition to be satisfied by q so that a rational number p/q has a non terminating decimal expansion is not of the form 2^m × 5ⁿ, where m and n are non negative integers.

For e .g :  

1/13  

Here, prime factors of 13 are not of the form 2^m × 5ⁿ,so it will not have a terminating decimal expansion.

1/13 = 0.076923076923……. = 0.076923(bar on whole number)

Hence, 1/13 has non terminating repeating decimal expansion.

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Answer 2

$\textbf{\huge{\underline{ Hello Mate! }}}$

To get the condition where p/q is non terminating decimal expansion first we need to find when a rational number have $\textsf{\red{terminating decimal expansion.}}$

Whenever the dinominators are in form of $2^m × 5^n$ then the rational number have terminating decimal expansions.

For example:

⅜ = 3 / 2³ = ( 3 × 5³ )/( 2³ × 5³ )

= ( 3 × 125 )/( 10³ ) = 375 / 1000 = 0.375

But if the rational number is not in such form, then it will have non terminating decimal expansions.

$\textsf{\blue{For example, 7 / 9 = 0.7777........}}$

Hence in order to get non terminating decimal expansions, the denominator of rational number should not be in form of $2^m × 5^n$

$\textbf{Have great future ahead!}$