Question

How does one express the recurring decimal $0.125125125......$ as a rational number or real number.

Answer 1

Hint: For solving this particular question , we first consider $x = 0.\overline {125} $ as our first equation then multiply both the side of the first equation by thousand and then consider the resulting equation as our second equation. Lastly, subtract equation one from equation two then you will get the desired result.

Complete step-by-step solution:
We have given the recurring number or repeating number $0.125125125......$ .
We can write the given repeating number $0.125125125......$ as $0.\overline {125} $ .
Now, let us consider $x = 0.\overline {125} ........................(1)$ ,
Now multiply the equation one by $1000$. We will get the following,
$
   \Rightarrow x \times 1000 = 0.\overline {125} \times 1000 \\
   \Rightarrow 1000x = 125.\overline {125} \\
 $
Now consider $1000x = 125.\overline {125} .................(2)$
Now we subtract equation one from equation two.
$ \Rightarrow 1000x - x = 125.\overline {125} - 0.\overline {125} $
$
   \Rightarrow 999x = 125 \\
   \Rightarrow x = \dfrac{{125}}{{999}} \\
 $
Hence, we have the required rational number or real number.

Additional Information: A real number could be a number within the style of “pq” where ‘p’ and q’ are the integers and ‘q’ isn't adequate zero. Both ‘p’ and ‘q’ may be negative moreover as positive. we have also seen how rational numbers are often converted to both terminating and non-terminating decimal numbers. Now, non-terminating decimal numbers are further classified into two types which are recurring and non-recurring decimal numbers.

Note: Recurring numbers: Recurring numbers are those numbers that carry on repeating the identical value after the mathematical notation. These numbers are called repeating decimals.
Non- recurring numbers: Non- recurring numbers are those, which do not repeat their values after the percentage point. they are also referred to as non-terminating and non-repeating decimal numbers.