Question

How do i turn reapeating decimals into fractions?

Answer 1

Easy example that shows
`0.999\ldots=1`:

Let
`x=0.999\ldots=0.\overline9`.

Then
`10x=9.999\ldots=9.\overline9`.

So
`10x-x=9.\overline9-0.\overline9`, or
`9x=9`, and so
`x=1`.

The basic idea is to find the period of the repeating decimal, move the
`n` digits belonging to one period over to the left of the decimal point by multiplying by
`10^n`, then subtract the original repeating decimal from this new number, and finally divide by
`10^n-1`.

A slightly more complicated example:

Let
`x=0.142857142857142857\ldots=0.\overline{142857}`.

Then
`10^6x=142857.\overline{142857}`.

Then
`10^6x-x=142857.\overline{142857}-0.\overline{142857}`, or


`999999x=142857\implies x=(142857)/(999999)=\frac17`