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We present a characterization of the completeness of the field of realnumbers in the form of a \emph{collection of several equivalent statements}borrowed from algebra, real analysis, general topology, and non-standardanalysis. We also discuss the completeness of non-Archimedean fields andpresent several examples of such fields. As an application, we exploit thecharacterization of the reals to argue that the Leibniz-Euler infinitesimalcalculus in the $17^\textrm{th}$-$18^\textrm{th}$ centuries was already arigorous branch of mathematics -- at least much more rigorous than mostcontemporary mathematicians prefer to believe. By advocating our particularhistorical point of view, we hope to provoke a discussion on the importance ofmathematical rigor in mathematics and science in general. This article isdirected to all mathematicians and scientists who are interested in thefoundations of mathematics and the history of infinitesimal calculus.

Ordered Fields, the Purge of Infinitesimals from Mathematics and the Rigorousness of Infinitesimal Calculus