Cantor proved the enumerability of rational numbers by creating a chart like the one below
the order crated by the zig-zag allowed the rational number to be put into one to one corespondence with natural numbers, but there are some repeats, which will be skipped.
Figuring that the reducable fractions should be eliminated, I wrote a program in java that counted the nonreduceable fractions and divided this number by the total number of numerator/denominator pairs. It can easily be seen that three down and three across on the chart has 7 out of nine nonreducable fractions. 4 has 11 out of 16. I ran the program for 100,000 down and across, came back a half hour later and got a number very close to the reciprical of phi, the golden number.
Does the ratio of nonreducable fractions to numerator pairs really approach this number? How can this be varified?