Taking Rational Numbers at Random

Abstract

In this article, some prescriptions to define a distribution on the set Q0 of all rational numbers in [0,1] are outlined. We explored a few properties of these distributions and the possibility of making these rational numbers asymptotically equiprobable in a suitable sense. In particular, it will be shown that in the said limit—albeit no absolutely continuous uniform distribution can be properly defined in Q0—the probability allotted to every single q∈Q0 asymptotically vanishes, while that of the subset of Q0 falling in an interval [a,b]⊆Q0 goes to b−a. We finally present some hints to complete sequencing without repeating the numbers in Q0 as a prerequisite to laying down more distributions on it.

Keywords

• rational numbers
• discrete distributions
• randomness