In the past two decades, so-called non-standard analysis has emerged. His proposed approach to the justification of mathematical analysis is based on the assumption of the existence, in addition to the usual real numbers, of "infinitely large numbers" and "infinitely small numbers." The full rationale for this approach is quite complex and relies on constructions of mathematical logic. The purpose of the book is not to give a complete justification, but only by postulating the necessary facts, to explain on the available examples what is the essence of non-standard analysis.