Converting Infinite Periodic Decimal into Proper Fraction

If we need to multiply infinite decimal fraction by 10, 100, 1000 and so on we will shift a dot at one, two or three and so on digits to the right such as it do in the finite decimal fraction.

For example, `0.1(23)*100=0.1232323...*100=12.323232...=12.(32)` .

Example 1. Converting into proper fraction number: 0.(13).

Suppose `x=0.(13)=0.131313...` . Let′s multiply this purely periodic fraction `x` by such number that dot will shift exactly at the period to the right. As the period has two digits, we need to shift the dot to the right at the two periods and for this we multiply `x` by 100, then `100x=0.131313...*100=13.1313...=13.(13)`. Then we subtract `x` from `100x` and we will obtain `100x-x=13.(13)-0.(13)`. So, `99x=13` , whence we obtain `x=13/99`.

Example 2. Converting into proper fraction number: 0.2(54).

Suppose `x=0.2(54)`.Let′s shift the dot in such mixed periodic fraction to the right so as to obtain purely periodic fraction. For this we multiply `x` by 10 and we will obtain `10x=2.(54)`.

Suppose `y=2.(54)`.Let′s convert such purely periodic fraction into proper fraction, as we made earlier. We have: `100y=254.(54); 100y-y=254.(54)-2.54` ;

`99y=252; y=252/99=28/11` .

So, `10x=28/11` , from which `x=28/(11*10)=14/55`.