TITLE: What benefits do real numbers bring to the theory of rational numbers?
QUESTION [6 upvotes]: Complex numbers make it easier to find real solutions of real polynomial equations. Algebraic topology makes it easier to prove theorems of (very) elementary topology (e.g. the invariance of domain theorem).
In that sense, what are theorems purely about rational numbers whose proofs are greatly helped by the introduction of real numbers? 
By "purely" I mean: not about Cauchy sequences, Dedekind cuts, etc. of rational numbers. (This is of course a meta-mathematical statement and therefore imprecise by nature.)
"No, there is no such thing, because..." would also be a valuable answer.

REPLY [2 votes]: Real numbers together with all the other completions of the rationals known as the $p$-adics are very useful indeed in finding rational solutions of quadratic forms. A general principle known as Hasse principle asserts that a quadratic form in $n$ variables has rational solutions if and only if it has solutions in each completion.