This book discusses methods for quickly performing various types of calculations, describes the implementation of fast algorithms both in the form of logical circuits - a mathematical model of real electronic microcircuits, and in the form of computer programs. Questions are also being explored about how to measure the complexity of a particular computational algorithm and estimate its running time on a computer. Most of the material in the book is accessible to everyone who is familiar only with a school mathematics course, but an experienced reader can also find something new for themselves in this book. The book is written on the basis of lectures that the author gave at different times to students of the Physics and Mathematics School named after. A.N. Kolmogorov at Moscow State University, at the Small and Large Mechanics and Mathematics, as well as at the faculties of information security and computer science of the Russian State University for the Humanities.