▁equivalent ▁description ▁of ▁the ▁De hn ▁invariant ▁involves ▁the ▁choice ▁of ▁a ▁Ham el ▁basis , ▁an ▁infinite ▁subset ▁ ▁of ▁the ▁real ▁numbers ▁such ▁that ▁every ▁real ▁number ▁can ▁be ▁expressed ▁uniqu ely ▁as ▁a ▁sum ▁of ▁finit ely ▁many ▁rational ▁multi ples ▁of ▁elements ▁of ▁. ▁Thus , ▁as ▁an ▁add itive ▁group , ▁ ▁is ▁is omorphic ▁to ▁, ▁the ▁direct ▁sum ▁of ▁copies ▁of ▁ ▁with ▁one ▁summ and ▁for ▁each ▁element ▁of ▁. ▁If ▁ ▁is ▁chosen ▁carefully ▁so ▁that ▁ ▁( or ▁a ▁rational ▁multiple ▁of ▁) ▁is ▁one ▁of ▁its ▁elements , ▁and ▁ ▁is ▁the ▁rest ▁of ▁the ▁basis ▁with ▁this ▁element ▁ex cluded , ▁then ▁the ▁tensor ▁product ▁ ▁is ▁the ▁( in finite ▁dimensional ) ▁real ▁vector ▁space ▁. ▁The ▁De hn ▁invariant ▁can ▁be ▁expressed ▁by ▁decom pos ing ▁each ▁di hed ral ▁angle ▁ ▁into ▁a ▁finite ▁sum ▁of ▁basis ▁elements ▁ ▁where ▁ ▁is ▁rational , ▁ ▁is ▁one ▁of ▁the ▁real ▁numbers ▁in ▁the ▁Ham el ▁basis , ▁and ▁these ▁basis ▁elements ▁are ▁number ed ▁so ▁that ▁ ▁is ▁the ▁rational ▁multiple ▁of ▁ ▁that ▁belongs ▁to ▁ ▁but ▁not ▁. ▁With ▁this ▁decomposition , ▁the ▁De hn ▁invariant ▁is ▁ ▁where ▁each ▁ ▁is ▁the ▁standard ▁unit ▁vector ▁in ▁ ▁corresponding ▁to ▁the ▁basis ▁element ▁. ▁Note ▁that ▁the ▁sum ▁here ▁starts ▁at ▁, ▁to ▁o mit ▁the ▁term ▁corresponding ▁to ▁the ▁rational ▁multi ples ▁of ▁. ▁▁ ▁Although ▁the ▁Ham el ▁basis ▁form ulation ▁appears ▁to ▁involve