A positive proof of Dehn's theorem

Yuval Filmus

Dehn proved that if a rectangle can be tiled by rectangles whose sides are commensurable, then the tiled rectangle is also commensurable. His proof, as described in Proofs from the book, applies a homomorphism which results in possibly negative side lengths. We modify his proof so that all side lengths are positive.

The crucial ingredient is the following lemma: for each finite set of positive reals there is a basis (over the rationals) of positive reals such that every element in the set is a non-negative integral combination of base elements. We provide two proofs of this lemma, one due to us and one due to Avinoam Braverman.