Farey sequences can be used to count this set of numbers, which can be expressed as the set of all algebraic numbers of degree one.  
Members of this set of numbers form the widest steps on the Devil's Staircase. G.H. Hardy allegedly offered his chair at Oxford to the person who could prove that the Euler-Mascheroni constant was not a member of this set of numbers.  Farey sequences can be used to count this set of numbers, which can be expressed as the set of all algebraic numbers of degree one.  One way to define this set of numbers is to use Cauchy sequences, which do not necessarily converge for this set.  This set of numbers is acted upon by Dedekind cuts. Diophantine approximation uses this set of numbers to approximate the reals. For 10 points, name this set of numbers, indicated by a doublestruck Q, which can be represented as a fraction of two integers.
Members of this set of numbers form the widest steps on the Devil's Staircase.
rational numbers