A basis for the real numbers 
, considered as a vector space
over the rationals 
,
i.e., a set of real numbers 
such that every real number 
has a unique representation of the form
 |
where 
is rational and 
depends on 
.
The axiom of choice is equivalent to the statement: "Every vector space has a vector space basis," and this is the only justification for the existence of a Hamel basis.
