Decimal Digits in rational fraction

Question

Given any positive rational fraction F of 16 maximum digits beyond the decimal point
    maximum size 0.9999999999999999
    minimum size 0.0000000000000001
where 1 > F > 0 and where F is the subset of numbers that are multiples
of 0.0000000000000001...

Is there an algorithm to determine the number of significant digits (D) of the
fraction similar to the method of using log10(I) to determine the significant
decimal digits of an integer?

For example:  if F = 0.5000   then D = 1
              if F = 0.7500   then D = 2
              if F = 0.6660   then D = 3  etc. ??

I would like to multiply said fraction F by 10^D in order to create an integer I such that the integer I can be printed using simple binary to decimal methods as an alternative to using a table of 1600 constants, or performing 16 multiplications by 10 ( and analyzing each for the last case of significance ).

Answer 1

You can just check the lowest digit and see if it is zero. If not, you have $16$ digits. If so, check the next to lowest and keep going up until you find a non-zero digit.

Answer 2

Since each number you are considering is of the form $F=\dfrac{m}{10^n}$ where $1 \le m \le 10^n-1$, use the standard algorithm for converting $F$ into a continued fraction.

Then look at the convergents and choose the first that is a good enough approximation.

This is not optimal, but is quite close.

Also look up finding the closest fraction to another fraction withing a given tolerance.