Book review

The book under review is written by an internationally known scientist in numerical analysis. The readers of the journal ‘Optimization Methods and Software’must know ProfessorYaroslav Sergeyev particularly well, thanks to his interesting papers on parallel computations and global optimization. The book deals with a very unusual subject for computational mathematics – infinity. Surprisingly, the author shows that the subject having been considered as extremely theoretical can be viewed as an applied one. The author not only introduces new infinite and infinitesimal numbers, but also provides a practical way of manipulating both infinite and infinitesimal quantities, which are usually avoided in computational mathematics. The first part of the book is quite traditional. The author gives a nicely written description of classical views on infinity. In the second part, he considers some paradoxes related to these views as well as new interesting observations appealing to simple, but well-chosen examples from every day life. In the third and the most interesting part of the book, the author develops his ideas and in addition to the usual unit of measure used for counting finite elements, introduces a new infinite unit of measure. In doing so, the author applies a quite natural methodology. First of all, he explicitly accepts that human beings and their machines are able to execute only a finite number of operations. Therefore, he accepts that we will never be able to give a complete description of infinite processes and sets because of our finite capabilities. Thus, following natural sciences, he does not discuss the mathematical objects he is dealing with. He just constructs more powerful tools (numeral systems used to express numbers are among the instruments of observations used by mathematicians) that allow him to improve his capacities of observation and description properties of mathematical objects. This position is very well known in physics. When a physicist sees a black dot in his microscope he cannot say: ‘The object of observation is the black dot.’ He is obliged to say: ‘The lens used in the microscope allows us to see the black dot. It is not possible to say anything more regarding the nature of the object of observation unless we replace the instrument, the lens or the microscope itself by a more precise one.’ In his constructions, the author applies the philosophical principle of ancient Greeks ‘The part is less than the whole,’ which is true for finite numbers, but is not incorporated in many