Which numbers simplify your problem?

Abstract

An extension of the ring of scalar quantities, from the usual field of real numbers to a non-Archimedean, sometimes permits to simplify some problems which, at a first sight, may seem not correlated with infinitesimal and infinite numbers. We present four simple cases, each one at the level of possibility for the creativity of a motivated student. The ring of Fermat reals and its applications to physics and differential geometry, the ring of Colombeau generalized numbers and its applications to the foundations of generalized functions, the Levi-Civita field and the derivation of complicated computer functions and the Surreals numbers as a universal non-Archimedean ring. The definition of each one of these rings is strongly motivated at elementary level and some open problems and ideas are introduced in the first two cases.