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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.

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The set of distances generated by euclidean straight-edge and compass constructions from a unit segment contains all rational numbers but the properties you gave do not imply that. –  lhf Dec 11 '12 at 10:47
    
Corrected now (I hope!). –  John Bentin Dec 11 '12 at 11:41

1 Answer 1

up vote 3 down vote accepted

The set of distances generated by Euclidean straight-edge and compass constructions from a unit segment is the same as the set of coordinates of constructible points. This is called the set of constructible numbers, as you expect.

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