аЯрЁБс>ўџ 46ўџџџ3џџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџьЅСU@јПЛbjbjЌ›Ќ›"$ЮёЮёf TџџџџџџˆОООООООвккк8,> вˆ іVlllllll  $~ Rа6- Ољllљљ- ООllB с с с љXОlОl с љ с Rс 3 ОО3 lJ рWf­ЧкQ @3 ѓ X 0ˆ 3 ‘ :3 ввОООО3 ОG ЌlvтTс 6Dzlll- - ввфЖ$Ы ввЖBOOK REVIEW Yaroslav D. Sergeyev (2003), Arithmetic of infinity, Edizioni Orizzonti Meridionali, 112 pp., ISBN 88-89064-01-3. The author of this small book is a young but already world-renowned Russian mathematician, working at the University of Calabria, Italy, where he has been invited some years ago for the prestigious position of Distinguished Full Professor. The field where Ya.D. Sergeyev is considered one of the major authorities is applied mathematics and, particularly, theory and methods of global optimization and parallel computing. However, his scientific interests are significantly wider and contain also foundations of mathematics and number theory. The book deals indeed with one of the fundamental topics in mathematics: an extension of numeral systems in order to include infinite and infinitesimal quantities in a coherent way and to allow us to execute usual arithmetic operations with them in the same manner as we are used to do with finite numbers. From the applied point of view, the most important result consists of the possibility to include in the newly proposed numeral system numeral systems currently used in computers and, as a result, to reinforce them significantly. This remarkable achievement distinguishes positively the new approach from the traditional non-standard analysis and leads to the clear understanding that the study of infinite and infinitesimal quantities becomes not purely theoretical but also applied science. From the theoretical point of view, the results presented in the book are even more notable because the usage of the new numeral system allows us to treat many problems in a simple unified way. For example, it gives possibility not only to classify infinite sets but to calculate directly the number of their elements, to write down in a compact form expressions including infinite and infinitesimal quantities substituting so limits by precise infinite and infinitesimal numbers. The new approach gives also solutions to several famous paradoxes to which the previous numeral systems led inevitably (Hilbert’s paradox of the Grand Hotel, Thompson Lamp Paradox, etc.). From the philosophical point of view, the book re-introduces the philosophical principle of Ancient Greeks `the part is less than the whole'. Its conciliation with the principle of the one-to-one correspondence is one of the most valuable results of the book (it is also discussed in detail in other publications of the author). The basic idea of the book appears to be simple, if you know already what it is. Analogously, at their time, appeared simple ideas laying in the basis of other extensions of numeral systems (e.g., introduction of complex numbers, of negative numbers, etc.). It is simple in the same way as other really innovative ideas are simple. I am sure that the new approach presented in this book will have a very deep impact both on Mathematics and Computer Science. The book is deliberately written in a clear and popular manner. The reading is pleasant and very often surprising. Thus, it can be easily read not only by specialists but also by students and all people interested in Mathematics and its foundations. Finally, it is worthy to mention that there exists a webpage dedicated to the book giving some additional information on the topic:  HYPERLINK "http://" http://www.grossone.com Prof. Donato Trigiante University of Florence Computational Management Science, 2007, Volume 4(1), pages 85-86, Springer-Verlag  €Њ­Ўкм3 6 7 : ; o q s ƒ „ … † ˆ ’ Ÿ   Б єцицЫОЏ Џ ‘ ‚ ‚s ‘d ‘U‘U‘ d h†S†hvPKCJaJmH sH h†S†hx+ŽCJaJmH sH h†S†h# CJaJmH sH h†S†hŸ[ЫCJaJmH sH h†S†haBCJaJmH sH h†S†hDOиCJaJmH sH h†S†hѓ|йCJaJmH sH hѓ|йhaBCJmH sH hѓ|йhaBCJmH sH hЁ'PhvPK5CJmH sH hЁ'PhaB5CJmH sH h†S†5CJmH sH   €r s Ђ Є У Ф ЋЌvw'()*fИЙКЛњњњњњњњњњњњђњњњъъъъњшццњ$a$gdvPK$a$gd ^$a$fКўўБ М ы ь њ  ' 2 6 J K Ѓ Є ы я № љ Y ` e h  І Ќ е ж ; \ m w ˆ К Т Ф ц ч KSU–Ї9ётётётёгёгёФёгёЕёІёІё—г—ё—ˆ—ˆ—ˆ—ёгІгyjyjyh†S†h!|CJaJmH sH h†S†hВдCJaJmH sH h†S†h7]\CJaJmH sH h†S†hД.QCJaJmH sH h†S†hл&єCJaJmH sH h†S†h# CJaJmH sH h†S†haBCJaJmH sH h†S†hx+ŽCJaJmH sH h†S†hDOиCJaJmH sH h†S†h})ВCJaJmH sH )9ЖРФЭ:;я№ёђѓ§8BCHЊЋЌЛЧќQRtuvwŠётётётёгёФтётётёЕІЕІ—ˆІyІyjyт[Іjh†S†haBCJaJmH sH h†S†hЁ'PCJaJmH sH h†S†h|=CJaJmH sH h†S†hx+ŽCJaJmH sH h†S†hnCJaJmH sH h†S†hнAЌCJaJmH sH h†S†hЃCCJaJmH sH h†S†hВдCJaJmH sH h†S†hŸ[ЫCJaJmH sH h†S†h!|CJaJmH sH h†S†h ^CJaJmH sH  Š‘Ыацѕії  &()*@AGOVefётётётгФВгЁгЁт’ФƒteVeVGh†S†haBCJaJmH sH h†S†h#'›CJaJmH sH h†S†h]mXCJaJmH sH h†S†hj(CJaJmH sH h†S†h!7CJaJmH sH h†S†hvPKCJaJmH sH  h†S†hЁ'P0JCJaJmH sH #jh†S†hЁ'PCJUaJh†S†hЁ'PCJaJmH sH jh†S†hЁ'PCJUaJh†S†hЁ'PCJaJmH sH h†S†hIWДCJaJmH sH f†‹ŒŽЈИЙКЛёфзфзфгЯРh†S†haBCJaJmH sH hзlhЫJ?hзlhзlCJmH sH hзlhЫJ?CJmH sH hзlhЫJ?6CJmH sH  PАƒ. 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