Arithmetic of infinity

Extracted from Review of Professor Panos. M. Pardalos published in Journal of Global Optimization, 2006, volume 34. This small book deals with two concepts that are among the most fundamental in mathematics and in general in all natural sciences: concept of number and concept of infinity. The goal of the book is to construct a new type of arithmetic that would allow us to introduce and to treat infinite and infinitesimal numbers in the same manner as we are used to do with finite ones, i.e., by applying the philosophical principle of Ancient Greeks “the part is less than the whole”. The introduced viewpoint leads to many new powerful tools in analysis, computer science, set theory, measure theory, etc. For instance, it allows one to substitute classification of infinite sets with respect to cardinal numbers by a simple procedure for direct calculating the number of elements of infinite sets. It becomes possible to substitute limits by arithmetical expressions and to calculate these expressions in different infinite or infinitesimal points even when limits do not exist or when indeterminate forms appear. Divergent series can be calculated and divergent processes can be studied at different infinite points. It is important to notice an extremely clear exposition of the material. The excellent didactical style of the author is well known to the readers of his previous fundamental book in global optimization and parallel computations published by Kluwer. Since also in this book the author was able to combine successfully new deep ideas and a very intuitive presentation, I recommend this book not only to professional mathematicians but also to amateurs. In conclusion, this is a valuable book for both students and researchers interested in mathematics, physics, computer science, and philosophy. The book opens new horizons in these fields and I strongly recommend it to anyone who at least once asked himself about the nature of infinite and infinitesimal.