Systems of dyadic wavelets on the positive half-line â+equipped with the operation of binary summation are studied. Several problems concerning approximation properties of such wavelets are solved. In particular, explicit formulae for the order of approximation of smooth functions and of binary-smooth functions on R+ (smooth in the dyadic metric on the binary half-line) are obtained. The dyadic approximations with best approximation properties are characterized. The relation between the smoothness of wavelets and their order of approximation is analysed in various function spaces. Â© 2007 RAS(DoM) and LMS.