The Infinity Computer and Numerical CalculusDipartimento di Elettronica, Informatica e
Sistemistica, Università della Calabria,
87030 Rende (CS), Italy and N.I. Lobatchevsky State University,
Nizhni Novgorod, Russia |
There exist different ways to generalize traditional arithmetic
for finite numbers to the case of infinite and infinitesimal
numbers (see
Belaga [Version 2007], Benci and Di Nasso [2003], Cantor [1955], Cohen [1966], Conway and Guy [1996], Hardy [1910], Kanamori [1996], Mayberry [2001], Robinson [1996]
and references given therein). However, arithmetics developed for
infinite numbers are quite different with respect to the finite
arithmetic we are used to deal with. Moreover, very often they
leave undetermined many operations where infinite numbers take
part (for example, ∞−∞, ∞/∞, sum
of infinitely many items, etc.) or use representation of infinite
numbers based on infinite sequences of finite numbers. These
crucial difficulties
did not allow people to create
computers working with infinite and infinitesimal quantities.
In this survey lecture, we describe a new methodology (see
Sergeyev [2003], Sergeyev [2004a], Sergeyev [2004b], Sergeyev [2005], Sergeyev [2006b], Sergeyev [2006a], Sergeyev [2007], Sergeyev [2008c], Sergeyev [2008a], Sergeyev [2008b], Sergeyev [25 July 2008])
having a strong numerical character for treating infinite and
infinitesimal quantities based on the principle ‘the part is less
than the whole’. In order to understand how it is possible to
affront the problem of infinity in this new way let us consider
a study published in Science by Peter Gordon (see
Gordon [2004]) where he describes a primitive tribe living in
Amazonia - Pirahã - that uses a very simple numeral
system1 for counting: one, two, many. For Pirahã, all
quantities bigger than two are just ‘many’ and such operations as
2+2 and 2+1 give the same result, i.e., ‘many’. Using their weak
numeral system Pirahã are not able to see, for instance,
numbers 3, 4, 5, and 6, to execute arithmetical operations with
them, and, in general, to say anything about these numbers because
in their language there are neither words nor concepts for that.
Moreover, the weakness of their numeral system leads to such
results as
‘many’+ 1= ‘many’, ‘many’ +
2 = ‘many’,
|
which are very familiar to us in the context of views on infinity
used in the traditional calculus
This observation leads us to the following idea: Probably our
difficulty in working with infinity is not connected to the nature
of infinity but is a result of inadequate numeral systems used to
express infinite numbers.
The lecture introduces a new positional system with
infinite radix allowing one to write down finite, infinite, and
infinitesimal numbers as particular cases of a unique framework
(see the survey Sergeyev [2008b] and also
Sergeyev [2003], Sergeyev [2004b], Sergeyev [2005], Sergeyev [2006a], Sergeyev [2007]). The new numeral
system gives possibility to introduce a new type of a computer –
the Infinity Computer – able to operate not only with finite
numbers but also with infinite and infinitesimal ones (the
European Patent Office has expressed its positive opinion with
respect to the patent Sergeyev [2004a]).
The new
approach has a strong applied character and is not related to the
non-standard analysis. It both gives possibilities to execute
calculations of a new type and simplifies fields of Mathematics
where usage of the infinity and/or infinitesimals is necessary
(e.g., divergent series, limits, derivatives, integrals, measure
theory, probability theory, etc.).
Applications that can be treated by computers are determined by
their computational abilities. In the following there are listed
both operations that the Infinity Computer can execute and
traditional computers are not able to perform and some of new
areas of applications. It becomes possible:
-
0cm
0cm
0cm
0cm
to introduce
notions of numbers of elements for infinite sets in a way
compatible with the notion used traditionally for finite sets
significantly evolving traditional approaches able to distinguish
numerable sets from continuum only (the principle ‘The part is
less than the whole’ is used for this purpose);
- to substitute
symbols +∞ and −∞ by spaces of positive and negative
infinite numbers, to represent them in the memory of the Infinity
Computer and to execute arithmetical operations with all of them
using this new type of a computer, as we are used to do with usual
finite numbers on traditional computers;
- to substitute qualitative descriptions of the type ‘a number
tends to zero’ by precise infinitesimal numbers, to represent them
in the memory of the Infinity Computer, and to execute
arithmetical operations with them using the Infinity Computer as
we are used to do with usual finite numbers using traditional
computers;
- to calculate divergent limits and series, providing as results
explicitly written different infinite numbers, to be possibly used
in further calculations on the Infinity Computer;
- to calculate indeterminate forms (e.g., difference and division
of divergent series) that can give as results different
infinitesimal, finite or infinite numbers;
- to calculate on the Infinity Computer sums of divergent series
and improper integrals of various types that can give as results
different explicitly written infinite numbers;
- to evaluate functions and their derivatives at infinitesimal,
finite, and infinite points (infinite and infinitesimal values of
functions and their derivatives can be also explicitly
calculated);
- to study divergent processes at different infinite points;
- to introduce notions of lengths, areas, and volumes of fractal
objects obtained after infinite numbers of steps and compatible
with traditional lengths, areas, and volumes of non-fractal
objects and to calculate them in a unique framework.
The Infinity Calculator using the Infinity Computer technology is
presented during the talk. Additional information can be
downloaded from the page http://www.theInfinitycomputer.com
References
-
Belaga [Version 2007]
-
E. Belaga.
Mathematical infinity, its inventors, discoveres, detractors,
defenders, masters, victims, users, and spectators.
Version 2007.
- Benci and Di Nasso [2003]
-
V. Benci and M. Di Nasso.
Numerosities of labeled sets: a new way of counting.
Advances in Mathematics, 173: 50–67, 2003.
- Cantor [1955]
-
G. Cantor.
Contributions to the founding of the theory of transfinite
numbers.
Dover Publications, New York, 1955.
- Cohen [1966]
-
P. Cohen.
Set Theory and the Continuum Hypothesis.
Benjamin, New York, 1966.
- Conway and Guy [1996]
-
J. Conway and R. Guy.
The Book of Numbers.
Springer-Verlag, New York, 1996.
- Gordon [2004]
-
P. Gordon.
Numerical cognition without words: Evidence from Amazonia.
Science, 306 (15 October): 496–499, 2004.
- Hardy [1910]
-
G. Hardy.
Orders of infinity.
Cambridge University Press, Cambridge, 1910.
- Kanamori [1996]
-
A. Kanamori.
The mathematical development of set theory from Cantor to Cohen.
Bull. Symbolic Logic, 2 (1): 1–71, 1996.
- Mayberry [2001]
-
J. Mayberry.
The Foundations of Mathematics in the Theory of Sets.
Cambridge University Press, Cambridge, 2001.
- Robinson [1996]
-
A. Robinson.
Non-standard Analysis.
Princeton Univ. Press, Princeton, 1996.
- Sergeyev [25 July 2008]
-
Y. Sergeyev.
Numerical computations and mathematical modelling with infinite and
infinitesimal numbers.
Journal of Applied Mathematics and Computing, 25 July 2008.
published online.
- Sergeyev [2006a]
-
Y. Sergeyev.
Misuriamo l’infinito.
Periodico di Matematiche, 6(2): 11–26,
2006a.
- Sergeyev [2005]
-
Y. Sergeyev.
A few remarks on philosophical foundations of a new applied approach
to Infinity.
Scheria, 26-27: 63–72, 2005.
- Sergeyev [2006b]
-
Y. Sergeyev.
Mathematical foundations of the Infinity Computer.
Annales UMCS Informatica AI, 4: 20–33,
2006b.
- Sergeyev [2003]
-
Y. Sergeyev.
Arithmetic of Infinity.
Edizioni Orizzonti Meridionali, CS, 2003.
- Sergeyev [2004a]
-
Y. Sergeyev.
Computer system for storing infinite, infinitesimal, and
finite quantities and executing arithmetical operations with them.
patent application 08.03.04, 2004a.
- Sergeyev [2008a]
-
Y. Sergeyev.
Modelling season changes in the infinite processes of growth of
biological systems.
Transactions on Applied Mathematics and Nonlinear Models, page
(to appear), 2008a.
- Sergeyev [2007]
-
Y. Sergeyev.
Blinking fractals and their quantitative analysis using infinite and
infinitesimal numbers.
Chaos, Solitons & Fractals, 33(1): 50–75, 2007.
- Sergeyev [2008b]
-
Y. Sergeyev.
A new applied approach for executing computations with infinite and
infinitesimal quantities.
Informatica, 19(4): 567–596, 2008b.
- Sergeyev [2008c]
-
Y. Sergeyev.
Measuring fractals by infinite and infinitesimal numbers.
Mathematical Methods, Physical Methods & Simulation Science
and Technology, 1(1): 217–237, 2008c.
- Sergeyev [2004b]
-
Y. Sergeyev.
http://www.theinfinitycomputer.com.
2004b.
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