Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Grossone Infinity Computing

  • Yaroslav D. Sergeyev
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_685-1

Introduction

There exists an important distinction between numbers and numerals. A numeral is a symbol (or a group of symbols) that represents a number. A number is a concept that a numeralexpresses. The same number can be represented by different numerals. For example, the symbols “10,” “ten,” “IIIIIIIIII,” “X,” “=,” and “Ĩ” are different numerals, but they all represent the same number. (The last two numerals, = and Ĩ, are probably less known. The former belongs to the Maya numeral system where one horizontal line indicates five and two lines one above the other indicate ten. Dots are added above the lines to represent additional units. For instance, = means eleven in this numeral system. The latter symbol, Ĩ, belongs to the Cyrillic numeral system derived from the Cyrillic script. This numeral system was developed in the late tenth century and was used by South and East Slavic peoples. The system was used in Russia as late as the early eighteenth century when it was replaced with...

Keywords

Numbers and numerals Grossone-based numerals Numerical infinities and infinitesimals Infinite sets 
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References

  1. Amodio P, Iavernaro F, Mazzia F, Mukhametzhanov MS, Sergeyev YaD (2016) A generalized Taylor method of order three for the solution of initial value problems in standard and infinity floating-point arithmetic. Math Comput Simul. in pressGoogle Scholar
  2. Butterworth B, Reeve R, Reynolds F, Lloyd D (2008) Numerical thought with and without words: evidence from indigenous Australian children. Proc Natl Acad Sci U S A 105(35):13179–13184ADSCrossRefGoogle Scholar
  3. Cantor G (1955) Contributions to the founding of the theory of trans nite numbers. Dover Publications, New YorkGoogle Scholar
  4. Cococcioni M, Pappalardo M, Sergeyev YaD (2016) Towards lexicographic multiobjective linear programming using grossone methodology. In: Sergeyev YaD, Kvasov DE, Dell’ Accio F, Mukhametzhanov MS (eds) Proceedings of the 2nd international conference “numerical computations: theory and algorithms”, vol 1776. AIP Publishing, New York, p 090040Google Scholar
  5. Conway JH, Guy RK (1996) The book of numbers. Springer, New YorkCrossRefMATHGoogle Scholar
  6. D’Alotto L (2012) Cellular automata using infinite computations. Appl Math Comput 218(16):8077–8082MathSciNetMATHGoogle Scholar
  7. D’Alotto L (2013) A classification of two-dimensional cellular automata using infinite computations. Ind J Math 55:143–158MathSciNetMATHGoogle Scholar
  8. D’Alotto L (2015) A classification of one-dimensional cellular automata using infinite computations. Appl Math Comput 255:15–24MathSciNetMATHGoogle Scholar
  9. De Cosmis S, De Leone R (2012) The use of grossone in mathematical programming and operations research. Appl Math Comput 218(16):8029–8038MathSciNetMATHGoogle Scholar
  10. De Leone R (2017) Nonlinear programming and grossone: quadratic programming and the role of constraint qualifications. Appl Math Comput. in pressGoogle Scholar
  11. Gödel K (1940) The consistency of the continuum-hypothesis. Princeton University Press, PrincetonMATHGoogle Scholar
  12. Gordon P (2004) Numerical cognition without words: evidence from Amazonia. Science 306:496–499ADSCrossRefGoogle Scholar
  13. Hardy GH (1910) Orders of in nity. Cambridge University Press, CambridgeGoogle Scholar
  14. Hilbert D (1902) Mathematical problems: lecture delivered before the international congress of mathematicians at Paris in 1900. Bull Am Math Soc 8:437–479CrossRefMATHGoogle Scholar
  15. Iudin DI, Sergeyev YaD, Hayakawa M (2012) Interpretation of percolation in terms of infinity computations. Appl Math Comput 218(16):8099–8111Google Scholar
  16. Iudin DI, Sergeyev YaD, Hayakawa M (2015) Infinity computations in cellular automaton forest-fire model. Commun Nonlinear Sci Numer Simul 20(3):861–870Google Scholar
  17. Leder GC (2015) Mathematics for all? The case for and against national testing. In: Cho SJ (ed) The proceedings of the 12th international congress on mathematical education: intellectual and attitudinal chalenges. Springer, New York, pp 189–207Google Scholar
  18. Leibniz GW, Child JM (2005) The early mathematical manuscripts of Leibniz. Dover Publications, New YorkGoogle Scholar
  19. Levi-Civita T (1898) Sui numeri transfiniti. Rend Acc Lincei Series 5a 113:7–91MATHGoogle Scholar
  20. Lolli G (2012) Infinitesimals and infinites in the history of mathematics: a brief survey. Appl Math Comput 218(16):7979–7988MathSciNetMATHGoogle Scholar
  21. Lolli G (2015) Metamathematical investigations on the theory of grossone. Appl Math Comput 255:3–14MathSciNetMATHGoogle Scholar
  22. Margenstern M (2011) Using grossone to count the number of elements of infinite sets and the connection with bijections. p-Adic Numbers Ultrametric Anal Appl 3(3):196–204MathSciNetCrossRefMATHGoogle Scholar
  23. Margenstern M (2012) An application of grossone to the study of a family of tilings of the hyperbolic plane. Appl Math Comput 218(16):8005–8018MathSciNetMATHGoogle Scholar
  24. Margenstern M (2015) Fibonacci words, hyperbolic tilings and grossone. Commun Nonlinear Sci Numer Simul 21(1–3):3–11ADSMathSciNetCrossRefMATHGoogle Scholar
  25. Mazzia F, Sergeyev YaD, Iavernaro F, Amodio P, Mukhametzhanov MS (2016) Numerical methods for solving ODEs on the Infinity Computer. In: Sergeyev YaD, Kvasov DE, Dell’Accio F, Mukhametzhanov MS (eds) Proceedings of the 2nd international conference “numerical computations: theory and algorithms”, vol 1776. AIP Publishing, New York, p 090033Google Scholar
  26. Montagna F, Simi G, Sorbi A (2015) Taking the Pirahã seriously. Commun Nonlinear Sci Numer Simul 21(1–3):52–69ADSMathSciNetCrossRefMATHGoogle Scholar
  27. Pica P, Lemer C, Izard V, Dehaene S (2004) Exact and approximate arithmetic in an amazonian indigene group. Science 306:499–503ADSCrossRefGoogle Scholar
  28. Rizza D (2016) Supertasks and numeral systems. In: Sergeyev YaD, Kvasov DE, Dell’Accio F, Mukhametzhanov MS (eds) Proceedings of the 2nd international conference “numerical computations: theory and algorithms”, vol 1776. AIP Publishing, New York, p 090005Google Scholar
  29. Robinson A (1996) Non-standard analysis. Princeton University Press, PrincetonCrossRefMATHGoogle Scholar
  30. Sergeyev YaD (2007) Blinking fractals and their quantitative analysis using infinite and infinitesimal numbers. Chaos Solitons Fractals 33(1):50–75Google Scholar
  31. Sergeyev YaD (2008) A new applied approach for executing computations with infinite and infinitesimal quantities. Informatica 19(4):567–596Google Scholar
  32. Sergeyev YaD (2009a) Evaluating the exact infinitesimal values of area of Sierpinski’s carpet and volume of Menger’s sponge. Chaos Solitons Fractals 42(5):3042–3046Google Scholar
  33. Sergeyev YaD (2009b) Numerical computations and mathematical modelling with infinite and infinitesimal numbers. J Appl Math Comput 29:177–195Google Scholar
  34. Sergeyev YaD (2009c) Numerical point of view on Calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains. Nonlinear Anal Ser A Theory Methods Appl 71(12):e1688–e1707Google Scholar
  35. Sergeyev YaD (2010a) Computer system for storing in nite, in nitesimal, and nite quantities and executing arithmetical operations with them. US Patent 7,860,914Google Scholar
  36. Sergeyev YaD (2010b) Counting systems and the first Hilbert problem. Nonlinear Anal Ser A Theory Methods Appl 72(3–4):1701–1708Google Scholar
  37. Sergeyev YaD (2010c) Lagrange lecture: methodology of numerical computations with infinities and infinitesimals. Rendiconti del Seminario Matematico dell’Universita e del Politecnico di Torino 68(2):95–113Google Scholar
  38. Sergeyev YaD (2011a) Higher order numerical differentiation on the Infinity Computer. Optim Lett 5(4):575–585Google Scholar
  39. Sergeyev YaD (2011b) On accuracy of mathematical languages used to deal with the Riemann zeta function and the Dirichlet eta function. p-Adic Numbers Ultrametric Anal Appl 3(2):129–148Google Scholar
  40. Sergeyev YaD (2011c) Using blinking fractals for mathematical modelling of processes of growth in biological systems. Informatica 22(4):559–576Google Scholar
  41. Sergeyev YaD (2013a). Arithmetic of In nity, Edizioni Orizzonti Meridionali, CS, 2003, 2nd ednGoogle Scholar
  42. Sergeyev YaD (2013b) Numerical computations with infinite and infinitesimal numbers: theory and applications. In: Sorokin A, Pardalos PM (eds) Dynamics of information systems: algorithmic approaches. Springer, New York, pp 1–66Google Scholar
  43. Sergeyev YaD (2013c) Solving ordinary differential equations by working with infinitesimals numerically on the Infinity Computer. Appl Math Comput 219(22):10668–10681Google Scholar
  44. Sergeyev YaD (2015a) Numerical infinitesimals for solving ODEs given as a black-box. In: Simos TE, Tsitouras C (eds) AIP proceedings of the international conference on numerical analysis and applied mathematics 2014 (ICNAAM-2014), vol 1648. Melville, New York, p 150018Google Scholar
  45. Sergeyev YaD (2015b) The olympic medals ranks, lexicographic ordering, and numerical infinities. Math Intell 37(2):4–8Google Scholar
  46. Sergeyev YaD (2015c) Un semplice modo per trattare le grandezze infinite ed infinitesime. Matematica Nella Societa e Nella Cultura: Rivista Della Unione Matematica Italiana 8(1):111–147Google Scholar
  47. Sergeyev YaD (2016) The exact (up to infinitesimals) infinite perimeter of the Koch snowflake and its finite area. Commun Nonlinear Sci Numer Simul 31(1–3):21–29Google Scholar
  48. Sergeyev YaD, Garro A (2010) Observability of Turing machines: a refinement of the theory of computation. Informatica 21(3):425–454Google Scholar
  49. Sergeyev YaD, Garro A (2013) Single-tape and multi-tape Turing machines through the lens of the Grossone methodology. J Supercomput 65(2):645–663Google Scholar
  50. Sergeyev YaD, Mukhametzhanov MS, Mazzia F, Iavernaro F, Amodio P (2016) Numerical methods for solving initial value problems on the Infinity Computer. Int J Unconv Comput 12(1):3–23Google Scholar
  51. Vita MC, De Bartolo S, Fallico C, Veltri M (2012) Usage of infinitesimals in the Menger’s sponge model of porosity. Appl Math Comput 218(16):8187–8196MathSciNetMATHGoogle Scholar
  52. Wallis J (1656) Arithmetica in nitorumGoogle Scholar
  53. Zhigljavsky AA (2012) Computing sums of conditionally convergent and divergent series using the concept of grossone. Appl Math Comput 218(16):8064–8076MathSciNetMATHGoogle Scholar
  54. Žilinskas A (2012) On strong homogeneity of two global optimization algorithms based on statistical models of multimodal objective functions. Appl Math Comput 218(16):8131–8136MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media LLC 2017

Authors and Affiliations

  1. 1.University of CalabriaRendeItaly
  2. 2.Lobachevsky State UniversityNizhni NovgorodRussia
  3. 3.Institute of High Performance Computing and NetworkingC.N.RRomeItaly