Decimal notation

Hello,

To convert $0.038\overline{146}$ into a fraction, we can use a relatively simple operation, which can be used for solving a variety of problems. We set $x = 0.038\overline{146}$, but we also see that mutitplying x by 1000 produces a similar repeating decimal, $1000x = 38.\overline{146}$. We see that these repeating decimals, are both some relatively simple number + the repeating decimal 0.000146146.... To cancel out, we subtract x from 1000x, yielding $1000x-x=38.\overline{146} - 0.038\overline{146}$, so $999x=38.108$, and after simplification, x=9527/249750

We can find three infinite sums and add these fractions to 38 /1000

(1*10^(-4)) /  ( 1 -10^(-3) )  = 1/9990

(4*10^(-5)) / (1 -10^(-3)) = 1/24975

(6*10^(-6)) / (1-10^(-3))  = 1/166500

38/1000 + 1/9990 + 1/24975 + 1/166500  =  9527  / 249750