We show how to represent every real-valued function as a series of rectangular pulse functions whose values are integer multiples of factorials or reciprocal factorials to any given power  The functions of the bases used to get these representations depend onthe represented function, while the corresponding coefficients are independent of it. The number of terms in such a representation is finite at every rational values of the function, and infinite otherwise. Also, we exploit the similarity of this expansion with that of Taylor to introduce a Diophantine operation analogous to the differentiation of functions. Defined in terms of the fractional part function and called fractionization, it applies to any real-valued function. Basic fractionization rules and examples of fractional equations are presented.

Diophantine representations of functions, fractionization of functions, fractional equations, measure of irrationality.