"This ratio is ALWAYS an irrational number, as it's the result from the "DP/ Pi" or " MOD x Pi" lead/ turn.
Unfortunately real-world gears only have a rational number of teeth, so an acceptable compromise is needed.
Perhaps one of those ELS Whiz Kids out there (or Marv) will come up with an exact Pi leadscrew ratio one of these days?"
The best rational fraction approximation to pi using integers less than 1000 is the Tsu Chung Chi approximation 355/113. Its error is something like 8.5E-6 % !!
pi = 3.1415926535898...
355/113 = 3.14159292035... (0.0000085 %)
Ramanujan improved on it to the point where the error is beyond the range of most calculators..
Ramanujan's improvement to Chi: (355 / 113) * (1 - 0.0003 / 3533) = 3.14159265359 (below calculator range)]
103993 / 33102 = 3.141592653012 (-0.000000018 %)
is better but the numbers are too big to be practical for gears.
The fourth root approximation
(2143 / 22)^(1/4) = 3.14159265258 (-0.000000032 %)
provides truly astounding accuracy but is, of course, impractical for gear making.