The book presents a new methodology that allows one to represent infinite and infinitesimal numbers by a finite number of symbols and to execute arithmetical operations with all of them in the same manner as we are used to do with finite quantities. The new approach is not related to the non-standard analysis and infinity is considered in the book in a coherent way different from those proposed by Georg Cantor, Abraham Robinson, and John Conway. However, the new approach does not contradict Cantor, it evolves his theory and supplies new more powerful tools to deal with different infinite and infinitesimal quantities.

From the methodological point of view, the principle of Ancient Greeks ‘The part is less than the whole’ is adopted and applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The new positional system with the infinite radix introduced in the book allows one to consider infinite, finite, and infinitesimal numbers as particular cases of a unique framework. The new viewpoint gives detailed answers to many questions and paradoxes regarding infinite and infinitesimal quantities. Particularly, applications of the new approach to limit theory, measure theory, and set theory are given.

The book is mainly addressed to mathematicians, computer scientists, philosophers, physicists, and students. However, it is written in a popular way in order to allow any person having a high school education and interests in the foundations of these sciences to understand it easily.

Book Reviews

published in International Scientific Journals

·    Computational Management Science, 2007, vol. 4(1), pages 85-86, written by Prof. D. Trigiante

·    Cybernetics & Systems Analysis, 2006, vol. 42(6), page 183, written by Prof. V.P. Shylo

·    Optimization Methods and Software, 2006, vol. 21(6), pages 995–996, written by Prof. O. Prokopyev

·    International Journal of Unconventional Computing, 2006, vol. 2(2), pages 193-194, written by the Editor-in-Chief Prof. A. Adamatzky

·    Journal of Global Optimization, 2006, vol. 34, pages 157–158, written by the Editor-in-Chief Prof. P.M. Pardalos

The Infinity Computer (European patent EP 1728149, Russian patent 2395111, US patent 7,860,914)

The main difference of the new approach with respect to non-standard analysis theories is its strong computational character opening a new exiting area in the theory and practice of computations – Infinity Computing. European patent 1728149 issued 03.06.2009 introducing “Computer system  for storing infinite, infinitesimal, and finite quantities and executing arithmetical operations with them” describes the Infinity Computer able to execute computations with infinite, finite, and infinitesimal numbers numerically (not symbolically) in a unique framework. Russian patent 2395111 has been granted 20.07.2010 and USA patent 7,860,914 has been awarded 28.12.2010.

There exist the first software prototype of the Infinity Computer and the Infinity Calculator able to execute arithmetical operations with numbers having finite parts and/or infinitesimal and infinite parts of different orders.

Here you can see a screenshot of the Infinity Calculator with an example of division and a brief comment.

The interested investors are invited to contact the author.

New applications

·    M.C. Vita, S. De Bartolo, C. Fallico, M. Veltri, Usage of infinitesimals in the Menger’s Sponge model of porosity

·    L. D’Alotto, Cellular Automata Using Infinite Computations

·    M. Margenstern, An application of Grossone to the study of a family of tilings of the hyperbolic plane

·    S. De Cosmis, R. De Leone, The use of Grossone in Mathematical Programming and Operations Research

·    D.I. Iudin, Ya.D. Sergeyev, M. Hayakawa,  Interpretation of percolation in terms of infinity computations

About the author                                                                                                          Top of this page

Yaroslav D. Sergeyev is Distinguished Professor at the University of Calabria, Rende, Italy. He is also Professor (part-time) at the N.I. Lobatchevsky State University, Nizhni Novgorod, Russia and Affiliated Researcher at the Institute of High Performance Computing and Networking, Rende, Italy. He was awarded his Ph.D. (1990) from the N.I. Lobatchevsky State University and his D.Sc. degree from the M.V. Lomonosov State University of Moscow (1996). His research interests include numerical analysis, set theory, number theory, parallel computing, and fractals. His list of publications contains more than 150 items including 4 books and more than 60 papers in prestigious international journals. Additional information about the author can be found here.

News

·    Ph.D. course on Infinity Computing in New York

·    Лекция на телеканале «Культура» "Системы записи чисел и их влияние на прогресс науки"

·    Article in TASS-TELECOM, (in Russian)

·    Article in “New Scientist: Russian Edition”, (in Russian)

·    Article of Prof. G. Lolli for “Newton”, (in Italian)

·    Articoli divulgativi del Dott. Walter Caputo

Selected papers                                                                                                                                                                                                     Top of this page

Presentations                                                                                                              Top of this page

The Infinity Computer technology and its applications together with the respective mathematical and methodological issues have been presented at the following congresses:

·  invited lecture, the 12^th International Symposium “Frontiers of Fundamental Physics”, Udine, Italy, November 21-23, 2011;

·  invited plenary lecture, the 7th International Asian Summer School “Optimization of Complex Systems”, Tashkent, Uzbekistan, October 17-27, 2011

·  tutorial, Summer School “Achievements and Applications of Contemporary Informatics, Mathematics and Physics”, Kiev, Ukraine, August 12-19, 2011

·  invited plenary lecture, the 4th International conference "Mathematics, its Applications and Mathematical Education" Russia, Ulan-Ude, Baikal, June 27 - July 1, 2011

·  invited lecture, the International Workshop “New Worlds of Computation”, Orléans, France, May 23-24, 2011;

·  invited one-hour lecture, International Workshop of Natural Computing of the Ninth International Conference on Cellular Automata for Research and Industry, Ascoli Piceno, Italy, September, 21-24, 2010;

·  plenary lecture, the Second International Conference on Mathematical Physics and Its Applications, Samara, Russia, August 29 - September 4, 2010;

·  keynote lecture, the 3^rd Yalta Optimization Conference, Yalta, Ukraine, August, 2-4, 2010;

·  invited plenary lecture, International Conference on Optimization, Simulation and Control, Ulaanbaatar, Mongolia, July 25-28, 2010;

·  keynote lecture, the 2010 World Congress in Computer Science, Computer Engineering, and Applied Computing, Las Vegas, USA, July 12-15, 2010;

·  joint keynote lecture, the 2010 International Conference on Scientific Computing and Workshop on Cellular Automata, Theory and Applications, Las Vegas, USA, July 14, 2010;

·  tutorial, International workshop “Infinite and Infinitesimal in Mathematics, Computing, and Natural Sciences”, Cetraro, Italy, May 17-21, 2010;

·  invited plenary lecture, International conference "Optimization and Applications" (OPTIMA2009), Petrovac, Montenegro, September 21-25, 2009;

·  tutorial, the international workshop “Optimization and Control with Applications”, Harbin, China, June 6-11, 2009

·  invited one hour lecture, the First World Congress on Global Optimization in Engineering & Science, Hunan, China, June 1-5, 2009;

·  invited one hour lecture, the International Workshop “New Worlds of Computation”, Orléans, France, January 12, 2009

·  keynote lecture, Yalta Conference on Discrete and Global Optimization, Yalta, Ukraine, July, 31 – August, 2, 2008;

·  keynote lecture, the general plenary session of the World Multi-Conference WMSCI 2008 held jointly with the conferences RMCI 2008, MEI 2008, EIC 2008, and BMIC 2008, Orlando, Florida, June, 29 - July, 2, 2008;

·  invited one hour lecture, the World Congress of Nonlinear Analysts WCNA-2008, Orlando, Florida, July, 2-9, 2008;

·  tutorial, the 48th international workshop on Nonsmooth Analysis, Optimization and Applications, Erice, Italy, under the auspices of the International School of Mathematics “G. Stampacchia”,  May, 9–17, 2008;

·  invited plenary lecture, International Conference of Numerical Analysis and Applied Mathematics ICNAAM 2007, September 16-20, 2007;

·  invited lecture, the 8th Mediterranean Workshop and Topical Meeting "Novel Optical Materials and Applications", Cetraro, Italy, June 3-9, 2007;

·  tutorial, the International Workshop on Learning and Intelligent Optimization, Andalo, Italy, February 12-18, 2007;

·  invited one hour lecture, the 15th Euromicro Conference on Parallel, Distributed and Network-based Processing, Naples, February 7-9, 2007;

·  invited plenary lecture opening the MATHESIS Congress, Trento, Italy, November 2-4, 2006;

·  invited plenary lecture, the Third International Conference of Applied Mathematics, Plovdiv, Bulgaria, August 12-18, 2006;

·  invited plenary lecture, International Conference “Applied Optimization and Metaheuristic Innovations”, Yalta, Ukraine, July 17-23, 2006;

·  invited lecture, the second International Workshop on Variational Analysis and Partial Differential Equations, Erice, Italy, under the auspices of the International School of Mathematics “G. Stampacchia”, July 5-14, 2006;

·  round table “Infinity Computer and Calculus”, the 8th Congress of SIMAI (La Societa' Italiana di Matematica Applicata e Industriale), Ragusa (Sicily), May 22-26, 2006;

·  invited plenary lecture, the XXI Autumn Meeting of the Polish Information Processing Society, Katowice, Poland, December 5-9, 2005;

·  invited plenary lecture, the International Workshop on Global Optimization, Almeria, Spain, September 18-22, 2005;

·  tutorial, the International Conference on Complementarity, Duality, and Global Optimization in Science and Engineering, Blacksburg, Virginia, USA, August 15-17, 2005;

·  invited plenary lecture, International Conference on Difference Equations, Special Functions and Applications, Munich, July 25-30, 2005;

·  invited plenary lecture, International Conference “Numerical Analysis: the State of the Art”, Rende (CS), Italy, May, 19-21, 2005;

·  invited semi-plenary lecture at the International conference on Selected Problems of Modern Mathematics, dedicated to the 200th anniversary of K.G. Jacobi and the 750th anniversary of the Koenigsberg foundation, Kaliningrad, April, 4-8, 2005;

·  tutorial, the Workshop "Numerical Methods and Mathematical Software", Montecatini Terme (PT), January 31 - February 1, 2005;

·  invited plenary lecture, the VI-th International Congress on Mathematical Modelling, Nizhni Novgorod, Russia, September, 21-26, 2004;

·  opening invited lecture, the congress “Infinity in Mathematics, Physics, and Philosophy”, Pisa, Italy, 26.03.2004.

Results described in the book and in the patent have been also presented at the following universities, research institutes, and organizations: