The paper is aimed to elaborate the floating point multiresolution, considering convergence that allows some more fractions than otherwise. It implies a calculation concerning infinite strings of digits, which is not implementable in the standard representation, but requires a dyadic one. Such a view is much more convenient for regarding convergence because of specific norm whose logarithm follows the multiresolution scale. Arithmetic operations are performed in almost the same manner as the standard floating point method. Conversions from one representation to another are discussed in details. The main advantage of the method concerns an opportunity of representing constructible angles in the Euclidean plane, which is significant inter alia for computational geometry. A basic application also concerns two's complement representation of negative numbers, which is accurate only if one implies convergence in regard to the norm. In that respect, it offers a consistent realization of methods the computer science already provides.