|
The basis for this statement holds, but what the statement implies does not. That is, the numbers are a subset of the rationals, but it does not follow that we can't describe a rational with a number. In fact the rationals between [0, 1) have a well known numbering, [ 0/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5,
1/6, 5/6, 1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 1/8, 3/8, ... ]
where one increments the denominator and then goes through all numerators but keeps only numerators which have GCD 1 with the denominator (since if they share a factor they were already listed). |