This seems like a fairly simple question and I hope that it's just not my own stupidity guiding it. What i'm asking is that you could show any decimal as a ratio of rationals but in any situation how do you determine this.
Here is an example ratio 4/3=1.333333333..., now how could I determine the lowest number ratio of rationals for something like x/y=1.37. How could I determine "x" or "y" when they are unknown but they themselves cannot be decimals.
In general, you want to find the simple continued fraction expansion of the number. Given a number in finite decimal expansion, you will be able to continue only so far and then at the end, you will probably have a relatively big partial quotient. You probably want to ignore that and convert the rest of the continued fraction into a quotient of integers.
Take the number $x=1.37$, for example. The simple continued fraction expansion of $x$ is $x=1+1/(2+1/(1+1/(2+1/(2+1/(1+1/3)))))=137/100.$ However, the last partial quotient of $3$ may not be good, and replacing $1/3$ with $0$ gives $37/27=1.370370\dots$ instead. Take the number $x=1.33$ as a simple example. The simple continued fraction expansion of $x$ is $x=1+1/(3+1/33)=133/100.$ However, we suspect $33$, and replacing $1/33$ with $0$ gives $1+1/3=4/3$ which is probably what we wanted.
