Let $E$ be the smallest set of real numbers such that:
(1) $1\in E$,
(2) $x \in E \implies x/n\in E$ $\;$($n=1,2, \;...\;$),
(3) $x,y \in E \implies |x-y| \in E$,
(4) $x,y \in E \implies \sqrt{x^2+y^2} \in E$.
The idea is that $E$ is the set of distances generated by euclidean straight-edge and compass constructions from a unit segment.