  We introduce the ring of Fermat reals, an extension of the real field
containing nilpotent infinitesimals. The construction takes inspiration from
Smooth Infinitesimal Analysis (SIA), but provides a powerful theory of actual
infinitesimals without any need of a background in mathematical logic. In
particular, on the contrary with respect to SIA, which admits models only in
intuitionistic logic, the theory of Fermat reals is consistent with classical
logic. We face the problem to decide if the product of powers of nilpotent
infinitesimals is zero or not, the identity principle for polynomials, the
definition and properties of the total order relation. The construction is
highly constructive, and every Fermat real admits a clear and order preserving
geometrical representation. Using nilpotent infinitesimals, every smooth
functions becomes a polynomial because in Taylor's formulas the rest is now
zero. Finally, we present several applications to informal classical
calculations used in Physics: now all these calculations become rigorous and,
at the same time, formally equal to the informal ones. In particular, an
interesting rigorous deduction of the wave equation is given, that clarifies
how to formalize the approximations tied with Hook's law using this language of
nilpotent infinitesimals.
