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numbers, I thought I would count something else. 

So I counted up all the possible fractions, and realised just how many the are just between 0 and 1.  For instance

0.0

0.1

0.01

0.001

0.0001

Etc 

and one has only just started.

But counting is very deceptive, when comparing large 

( ie infinite) masses of numbers.  There are actually exactly the same number of integers as there are fractions. 

If you represent an arbitrary fraction as 

A + B/C    

where you write A,B and C as same length integers by padding with zeros, then you can map that fraction to the integer ABC, formed by joining the three equal length integers together.

There are gaps of course where the number of digits is not of equal length, but who cares.  There clearly are at least as many fractions as integers.

M