This book introduces a disruptive computational paradigm, explaining its motivations, breakthroughs, and wide-ranging applications. For the first time, it enables numerical computations with quantities containing finite, infinite, and infinitesimal components, performed on the patented Infinity Computer. Accessible yet rigorous, the book provides a clear introduction to the paradigm and presents pioneering theoretical results, innovative numerical algorithms, efficient software implementations, and practical case studies demonstrating the transformative potential of Infinity Computing across mathematics and computer science. The author is internationally recognized for his pioneering contributions, receiving prestigious awards and delivering more than one hundred plenary lectures and tutorials worldwide. This forward-looking volume opens new horizons for researchers, engineers, professors, and students interested in next-generation supercomputing, advanced numerical methods, and the foundations of mathematics and computer science.

Reviews

“Mathematicians have never been comfortable handling infinities… But an entirely new type of mathematics looks set to by-pass the problem… Today, Yaroslav Sergeyev, a mathematician at the University of Calabria in Italy solves this problem… ”

MIT Technology Review

“So, the idea that Sergeyev asks us to accept as natural is in fact a bold one, the boldest among a heap of bold proposals, but it works precisely for its boldness.”

G. Lolli, Soft Computing.

The numerical nature of these new infinities and infinitesimals is one of the key differences with respect to traditional theories (Cantor, Levi-Civita, Nelson, Robinson, etc.) giving advantages in practice and opening new horizons for computer science and applied mathematics.

J. Gillard, Optimization Letters

“These ideas and future hardware prototypes may be productive in all fields of science where infinite and infinitesimal numbers (derivatives, integrals, series, fractals) are used.”

A. Adamatzky, Editor-in-Chief of the International Journal of Unconventional Computing

“I am sure that the new approach … will have a very deep impact both on Mathematics and Computer Science.”

D. Trigiante, Computational Management Science.

“After an almost infinite analysis of the infinite, where special attention should be given to the great mathematicians Cauchy and Cantor, a new light, not only theoretical but also computational and applied, comes from the seminal work of Yaroslav Sergeyev, a Russian mathematician working now in the same Italian region where Pythagoras once worked.”

E. Spedicato, University of Bergamo, Italy

Yaroslav D. Sergeyev is a Distinguished Professor at the University of Calabria, Italy (appointed by direct invitation of the Italian Ministry of Education, Universities and Research in 2002), and Head of the Numerical Calculus Laboratory at the same university. He is Editor-in-Chief of Spinger’s Journal “Operations Research Forum”. He also works part-time as a Professor at Lobachevsky University of Nizhni Novgorod, Russia, and is an Affiliated Faculty at the Center for Applied Optimization, University of Florida, Gainesville, USA. For several decades, he was also an Affiliated Researcher at the Institute of High-Performance Computing and Networking of the Italian National Research Council.
He was awarded his Ph.D. (1990) from Lobachevsky Gorky State University and his D.Sc. degree (1996) from Lomonosov State University, Moscow (this degree is Habilitation for the Full Professorship in Russian universities). His research interests include infinity computing and calculus (the field he has founded), global optimization (he was President of the International Society of Global Optimization, 2017–2021), numerical computations, scientific computing, philosophy of computations, foundations of mathematics, set theory, number theory, fractals, parallel computing, and interval analysis.
He was awarded several research prizes (International Constantin Carathéodory Prize, International ICNAAM Research Excellence Award, International Prize of the city of Gioacchino da Fiore, all in 2023; Khwarizmi International Award, 2017; Pythagoras International Prize in Mathematics, 2010; EUROPT Fellow, 2016; Outstanding Achievement Award from the 2015 World Congress in Computer Science, Computer Engineering, and Applied Computing, USA; Honorary Fellowship, the highest distinction of the European Society of Computational Methods in Sciences, Engineering and Technology, 2015; The 2015 Journal of Global Optimization (Springer) Best Paper Award; Lagrange Lecture, Turin University, Italy, 2010 and Distinguished Lectures at Lancaster University, UK, 2020 and at Venice School of Management, Ca’ Foscari University, Venice, Italy, 2024; MAIK Prize for the best scientific monograph published in Russian, Moscow, 2008, etc.). In 2020, he was elected corresponding member of Accademia Peloritana dei Pericolanti in Messina, Italy.
Since 2020, he is included in the ranking “Top 2% highly cited authors in Scopus” produced by Stanford University, the list of top researchers in mathematics produced by Research.com, the list “Top Italian Scientists. Mathematics”, lists of highly cited Russian scientists in mathematics, computer science, and cybernetics, etc.
His list of publications contains more than 300 items (among them 6 authored and 12 edited books and more than 130 articles in international journals). He is a member of editorial boards of 12 international and 3 national journals and coeditor of 16 special issues. He delivered more than 100 plenary/keynote lectures and tutorials at prestigious international congresses. He was Chairman of 7 and Co-Chairman of 8 international conferences and a member of Scientific Committees of more than 110 international congresses.