An alternative but equivalent description of the Dehn invariant involves the choice of a Hamel basis, an infinite subset formula_7 of the real numbers such that every real number can be expressed uniquely as a sum of finitely many rational multiples of elements of formula_7. Thus, as an additive group, formula_9 is isomorphic to formula_10, the direct sum of copies of formula_11 with one summand for each element of formula_7. If formula_7 is chosen carefully so that (or a rational multiple of ) is one of its elements, and formula_14 is the rest of the basis with this element excluded, then the tensor product formula_15 is the (infinite dimensional) real vector space formula_16. The Dehn invariant can be expressed by decomposing each dihedral angle formula_5 into a finite sum of basis elements