Earlier I solved this question and it got me thinking.
So I decided to work it out, this is what I got.
Let be a number with digits, it can have both leading and trailing zeros.
Every recurring decimal can now be represented by the decimal :
To represent this as a fraction we need to see it as an infinite sum.
Now that we have it as an infinite sum we can use the geometric series formula where
So that means that :
But what if there's some decimal places in front of the recurring part?
Let be a number with digits, it can also have leading and trailing zeros.
The decimal is now :
Now we have to account for the decimal places between the decimal point and the recurring decimal. This is easy to do since we already have the equation for recurring decimals. The solution is simple - divide by to the power of the number of places ().
The represents the part of the decimal. It is simply the number divided by to the power of the number of decimal places () it takes up.
Let's simplify it a bit.
That's all for now, hope you found this note interesting.