Question

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DECIMAL REPRESENTATION OF RATIONAL NUMBERS.

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Answer 1

$\huge{\pink{\green {\sf\boxed{EXPLANATION⟹}}}}$

$\bold{\underline {DECIMAL\: REPRESENTATION\: OF \:RATIONAL NUMBERS}}$

Ler us observe, introspect and acknowledge

We may write : ${\dfrac{1}{2}={0.05}} , {\dfrac{5}{4}={1.25}}, {\dfrac{-5}{8}={0.625}}$

So, ${\dfrac{1}{2}, \dfrac{5}{4} and </p><p>\dfrac{-5}{8}} {are\: terminating\: decimals.}$

And 1/3 = 0.3333..... = 0.3, 4/11 = 0.36363636.....=0.36

So, 1/3 , 4/11 are non - terminating and repeating (recurring) decimals.

Thus, every rational number can be expressed as a decimal - terminating or non - terminating (repeating/non repeating).

Now, let us define

$\bold{\underline {1.TERMINATING\: DECIMALS:}}$ If a rational number ${\dfrac{p}{q}}$ terminates comes to end then decimal so obtained is said to be terminating decimal.

AN IMPORTANT NOTE : A rational number $</em><em>\</em><em>dfrac</em><em>{</em><em>p</em><em>}</em><em>{</em><em>q</em><em>}</em><em> </em><em>$ is a terminating decimal only when prime factor of q are 2 and/or 5, i. e., q = ${</em><em>2</em><em>}</em><em>^</em><em>{</em><em>m</em><em>}</em><em>*</em><em>{</em><em>5</em><em>}</em><em>^</em><em>{</em><em>n</em><em>}</em><em>$

$\bold{\underline {1.RECCURING\:(REPEATING) \:DECIMALS:}}$ A decimal in which a digit are a set of finite number of digits repeats periodically is called a recurring or repeating decimal.

In recurring decimal, a bar is placed over the first clock of the repeating digits and other repeating blocks are omitted.