Applications of real numbers being a vector space over the rational numbers

Question

What applications are there of the fact that the real numbers form a vector space over the rational numbers?

Vector spaces over the rational numbers appear to have uses in number theory.

The motivation for this question comes from Dietrich's answer to this question: Applications of uncountability of the real numbers. That question asks why it is useful to know that the real numbers are uncountable. This was motivated by an open problem. The most popular answer to that question mentioned the fact that the real numbers form an infinite-dimensional vector space over the rational numbers. This has prompted me to ask why this fact is again useful.

Answer 1

There are nonlinear real functions that satisfy Cauchy's functional equation $f(x+y)=f(x)+f(y)$.

Answer 2

Probably the best known application is the construction of a Hamel basis (http://mathworld.wolfram.com/HamelBasis.html ), which allows you to solve the Cauchy functional equation (over the reals): $$f(x+y) = f(x) +f(y) $$ If $f$ is taken to be over the rationals then $f(x)=cx$ is a solution for each $c\in {\mathbb R}$, but if $f$ is taken over the reals then the Hamel basis is used to show that there are infinitely many solutions.