How do you approximate #sqrt3#?
Use a variant of Newton Raphson to construct a sequence of rational approximations and stop when you have enough significant digits.
In the case of finding an approximation for
If you apply this using decimals then you quickly run into choices as to where to truncate your decimals, etc.
If you apply it using fractions, then you frequently find yourself multiplying up the numerator and denominator to simplify the fractions.
Alternatively, notice that if
It can happen (in early iterations) that