for more information see

· Pythagoras International Prize for Mathematics (Premio Internazionale Pitagora per la Matematica), Croton, Italy, 2010

· Achievement Award from the World Congress in Computer Science, Computer Engineering, and Applied Computing, USA, 2015

· Honorary Fellowship , the highest distinction of the European Society of Computational Methods in Sciences, Engineering and Technology, 2015

Numerical Infinity and the Infinity Computer: Books, patents , research papers , presentations , visitors

Lolli G. (2015) Metamathematical investigations on the theory of Grossone , Applied Mathematics and Computation , 255, 3-14.

Sergeyev Ya.D. (2017) Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems , EMS Surveys in Mathematical Sciences , 4(2), 219320

A nice animation showing grossone-based treatment of the paradox of Hilberts Hotel has been created by Prof. D. Rizza from the University of East Anglia, Norwich, UK at his educational site https://www.numericalinfinities.com

Paperback edition is available at European Amazon sites: UK , DE , FR , IT , ES

# 1 in eBook Kindle > eBook in lingua straniera > eBook in inglese > Scienze, tecnologia e medicina > Matematica

Revised electronic edition is available at Amazon sites: COM , UK , DE , FR , IT , ES , CA , JP , BR , IN , MX

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.

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 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.

· The Infinity Computer presented at the Johnson Space Center, NASA, Houston

· The paper of D. Rizza is among the most read in the Journal : Rizza D. (2018) A Study of Mathematical Determination through Bertrands Paradox , Philosophia Mathematica , 26(3), 375395.

The interested investors are invited to contact the author .

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.

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.

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

Yaroslav D. Sergeyev Ph.D., D.Sc., D.H.C. is Distinguished Professor at the University of Calabria , Rende, Italy and Head of Numerical Calculus Laboratory at the same university. He is also Professor (part-time) at the Lobachevsky State University , Nizhni Novgorod, Russia and Affiliated Researcher at the Institute of High Performance Computing and Networking , Rende, Italy. His research interests include numerical analysis, global optimization (he is President of the International Society of Global Optimization ), philosophy and foundations of mathematics, set theory, number theory, parallel computing, and fractals. His list of publications contains more than 250 items including 6 books and more than 100 papers in prestigious international journals. He is a member of editorial boards of 6 international journals. He was awarded several research prizes (The 2017 Khwarizmi International Award; Pythagoras International Prize in Mathematics, Italy, 2010; EUROPT Fellow, 2016; Outstanding Achievement Award from the 2015 World Congress in Computer Science, Computer Engineering, and Applied Computing, USA; Honorary Fellowship being the highest distinction of the European Society of Computational Methods in Sciences, Engineering and Technology, 2015; Lagrange Lecture, Turin University, Italy, 2010; MAIK Prize for the best scientific monograph published in Russian, Moscow, 2008, etc.). He was Chairman of 9 international conferences. Additional information about Prof. Sergeyev can be found here .

About the author Top of this page

61. Sergeyev Ya.D. (2006) Misuriamo linfinito: Un semplice modo per insegnare i concetti delle grandezze infinite , Periodico di Matematiche, vol. 6(2), 11-26, (In Italian).

60. Sergeyev Ya.D. (2005) A few remarks on philosophical foundations of a new applied approach to Infinity , Scheria , vol. 26-27, pp. 63-72.

59. Sergeyev Ya.D. (2008) Measuring fractals by infinite and infinitesimal numbers , Mathematical Methods, Physical Methods & Simulation Science and Technology , vol. 1(1), 217-237 .

58. Sergeyev Ya.D. (2008) A new applied approach for executing computations with infinite and infinitesimal quantities , Informatica , 19(4), 567-596.

57. Sergeyev Ya.D. (2009) Numerical computations and mathematical modelling with infinite and infinitesimal numbers , Journal of Applied Mathematics and Computing , 29, 177-195.

55. Sergeyev Ya.D. (2013) Numerical computations with infinite and infinitesimal numbers: Theory and applications, in "Dynamics of Information Systems: Algorithmic Approaches", Eds. Alexey Sorokin and Panos M. Pardalos, Springer, New York, 2013, pp. 1-66.

54. Lepellere M.A., Piccinini L.C., Taverna M. (2018) From linguistic representation to fuzzy mathematics in grown up people , Proc. Int. Conf. Society. Integration. Education. Vol. III, May 25th -26th, 2018. 555-565.

53. Iannone P., Rizza D., Thoma A. (2018) Investigating secondary school students epistemologies through a class activity concerning infinity , in E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.). Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, 131-138). Umeå, Sweden: PME.

52. Sergeyev Ya.D. (2015) Letter to the Editor , The Mathematical Intelligencer , 37(4), 2-3.

51. Sergeyev Ya.D. (2015) The Olympic medals ranks, lexicographic ordering and numerical infinities , The Mathematical Intelligencer , 37(2), 4-8.

49. Rizza, D. (2018) How to make an infinite decision, Bulletin of Symbolic Logic , 24(2), p.227.

48. Fiaschi L., Cococcioni M. (2018) Numerical asymptotic results in Game Theory using Sergeyev's Infinity Computing , International Journal of Unconventional Computing , 14(1), 1-25.

47. Rizza D. (2019) Numerical methods for infinite decision-making processes , International Journal of Unconventional Computing , 14(2), 139-158.

46. Sergeyev Ya.D. (2009) Numerical point of view on Calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains , Nonlinear Analysis Series A: Theory, Methods & Applications , 71(12), e1688-e1707.

45. Sergeyev Ya.D. (2011) On accuracy of mathematical languages used to deal with the Riemann zeta function and the Dirichlet eta function , p-Adic Numbers, Ultrametric Analysis and Applications , 3(2), 129-148.

44. A. Zhigljavsky (2012) Computing sums of conditionally convergent and divergent series using the concept of grossone , Applied Mathematics and Computation , 218, 80648076.

43. Sergeyev Ya.D. (2016) The difficulty of prime factorization is a consequence of the positional numeral system , International Journal of Unconventional Computing , Vol. 12 (5-6), 453463.

42. Sergeyev Ya.D. Numerical infinities applied for studying Riemann series theorem and Ramanujan summation , AIP Conference Proceedings 1978, 020004 (2018); doi: 10.1063/1.5043649

41. Rizza D. (2018) A Study of Mathematical Determination through Bertrands Paradox , Philosophia Mathematica , 26(3), 375395.

40. Sergeyev Ya.D. (2007) Blinking fractals and their quantitative analysis using infinite and infinitesimal numbers , Chaos, Solitons & Fractals , vol. 33(1), 50-75.

39. Sergeyev Ya.D. (2009) Evaluating the exact infinitesimal values of area of Sierpinski's carpet and volume of Menger's sponge , Chaos, Solitons & Fractals , 42, 30423046.

38. Sergeyev Ya.D. (2011) Using blinking fractals for mathematical modelling of processes of growth in biological systems , Informatica , 2011, 22(4), 559576.

37. Iudin D.I., Ya.D. Sergeyev, M. Hayakawa (2012) Interpretation of percolation in terms of infinity computations , Applied Mathematics and Computation , 218(16), 8099-8111.

36. Margenstern M. (2012) An application of Grossone to the study of a family of tilings of the hyperbolic plane , Applied Mathematics and Computation , 218(16), 8005-8018.

35. Vita M.C., S. De Bartolo, C. Fallico, M. Veltri (2012) Usage of infinitesimals in the Mengers Sponge model of porosity , Applied Mathematics and Computation , 218(16), 8187-8196.

34. Iudin D.I., Sergeyev Ya.D. Hayakawa M. (2015) Infinity computations in cellular automaton forest-fire model , Communications in Nonlinear Science and Numerical Simulation , 20(3), 861-870.

33. Margenstern M. (2015) Fibonacci words, hyperbolic tilings and grossone , Communications in Nonlinear Science and Numerical Simulation , 21(13), 3-11.

32. Sergeyev Ya.D. (2016) The exact (up to infinitesimals) infinite perimeter of the Koch snowflake and its finite area , Communications in Nonlinear Science and Numerical Simulation , 31(13):2129.

31. Margenstern M. (2016) Infinigons of the hyperbolic plane and grossone , Applied Mathematics and Computation , 278, 4553.

30. Caldarola F. (2018) The Sierpinski curve viewed by numerical computations with infinities and infinitesimals, Applied Mathematics and Computation , 318, 321328.

29. Caldarola, F. (2018) The exact measures of the Sierpiński d-dimensional tetrahedron in connection with a Diophantine nonlinear system , Communications in Nonlinear Science and Numerical Simulation , 63, 228-238.

28. Sergeyev Ya.D. (2013) Solving ordinary differential equations by working with infinitesimals numerically on the Infinity Computer , Applied Mathematics and Computation , 219(22), 1066810681.

27. Sergeyev Ya.D., Mukhametzhanov M.S., Mazzia F., Iavernaro F., Amodio P. (2016) Numerical methods for solving initial value problems on the Infinity Computer , International Journal of Unconventional Computing , 12(1), 323.

26. Mazzia F., Sergeyev Ya. D., Iavernaro F., Amodio P., and Mukhametzhanov M. S. (2016) Numerical methods for solving ODEs on the infinity computer , AIP Conference Proceedings 1776, 090033.

25. Amodio, P., Iavernaro, F., Mazzia, F., Mukhametzhanov, M.S., Sergeyev, Ya.D. (2017) A generalized Taylor method of order three for the solution of initial value problems in standard and infinity floating-point arithmetic , Mathematics and Computers in Simulation , 141, 2439.

24. Sergeyev Ya.D., Garro A. (2010) Observability of Turing Machines: a refinement of the theory of computation , Informatica , 21(3), 425454.

23. DAlotto L. (2012) Cellular Automata Using Infinite Computations , Applied Mathematics and Computation , 218(16), 8077-8082.

22. Sergeyev Ya.D., Garro A. (2013) Single-tape and multi-tape Turing machines through the lens of the Grossone methodology , Journal of Supercomputing , 65(2), 645-663.

21. DAlotto L. (2013) A classification of two-dimensional cellular automata using infinite computations , Indian Journal of Mathematics , 55, 143-158.

20. Sergeyev Ya.D., Garro A. (2015) The Grossone methodology perspective on Turing machines , in "Automata, Universality, Computation", A. Adamatzky (ed.), Springer Series "Emergence, Complexity and Computation", Vol. 12, pp. 139-169.

19. DAlotto L. (2015) A classification of one-dimensional cellular automata using infinite computations , Applied Mathematics and Computation , 255, 15-24.

18. Sergeyev Ya.D. (2011) Higher order numerical differentiation on the Infinity Computer , Optimization Letters , 5(4), 575-585.

17. De Cosmis S., R. De Leone (2012) The use of Grossone in Mathematical Programming and Operations Research , Applied Mathematics and Computation , 218(16), 8029-8038.

16. Cococcioni M., Pappalardo M., Sergeyev Ya.D. (2018) Lexicographic multiobjective linear programming using grossone methodology: Theory and algorithm , Applied Mathematics and Computation , 318, 298311.

15. De Leone R. (2018) Nonlinear programming and grossone: Quadratic programming and the role of constraint qualifications , Applied Mathematics and Computation , 318, 290297.

14. Gaudioso M., Giallombardo G., Mukhametzhanov M.S. (2018) Numerical infinitesimals in a variable metric method for convex nonsmooth optimization , Applied Mathematics and Computation , 318, 312320.

13. De Leone R., Fasano G., Sergeyev Ya.D. (2018) Planar methods and grossone for the Conjugate Gradient breakdown in nonlinear programming , Computational Optimization and Applications , 71(1), 73-93.

12. Sergeyev Ya.D., Kvasov D.E., Mukhametzhanov M.S. (2018) On strong homogeneity of a class of global optimization algorithms working with infinite and infinitesimal scales , Communications in Nonlinear Science and Numerical Simulation , 59, 319-330.

11. De Leone R., Fasano G., Roma M., Sergeyev YaD., (2019) How Grossone can be helpful to iteratively compute negative curvature directions , R. Battiti et al. (Eds.): Proc. of the 12th International Conference LION 12, Lecture Notes in Computer Science , vol. 11353, Springer, 80183.

10. Sergeyev Ya.D. (2010) Counting systems and the First Hilbert problem , Nonlinear Analysis Series A: Theory, Methods & Applications , 72(3-4), 1701-1708.

9. Margenstern M. (2011) Using Grossone to count the number of elements of infinite sets and the connection with bijections , p-Adic Numbers, Ultrametric Analysis and Applications , 3(3), 196-204.

8. Lolli G. (2012) Infinitesimals and infinites in the history of Mathematics: A brief survey , Applied Mathematics and Computation , 218(16), 7979-7988.

7. Montagna F., Simi G., Sorbi A. (2015) Taking the Pirahã seriously , Communications in Nonlinear Science and Numerical Simulation , 21(13), 52-69.

6. Sergeyev Ya.D. (2015) Computations with grossone-based infinities , C.S. Calude, M.J. Dinneen (Eds.), Proc. of the 14th International Conference Unconventional Computation and Natural Computation, Lecture Notes in Computer Science , vol. 9252, Springer, 89-106.

5. Lolli G. (2015) Metamathematical investigations on the theory of Grossone , Applied Mathematics and Computation , 255, 3-14.

4. Sergeyev Ya.D. (2019) Independence of the grossone-based infinity methodology from non-standard analysis and comments upon logical fallacies in some texts asserting the opposite , Foundations of Science , 24(1), 153170.

Foundations and relations to traditional views on infinite and infinitesimal

2. Sergeyev Ya.D. (2017) Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems , EMS Surveys in Mathematical Sciences , 4(2), 219320.

1. Sergeyev Ya.D. (2010) Lagrange Lecture : Methodology of numerical computations with infinities and infinitesimals , Rendiconti del Seminario Matematico dell'Università e del Politecnico di Torino , 68(2), 95113.

Selected research papers Top of this page