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Chapter 1Numerical computations with infinite andinfinitesimal numbers: Theory and applications

Yaroslav D. Sergeyev

Abstract A new computational methodology for executing calculations with in-finite and infinitesimal quantities is described in this chapter. It is based on theprinciple ‘The part is less than the whole’ introduced by Ancient Greeks and ap-plied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes(finite and infinite). It is shown that it becomes possible to write down finite, infi-nite, and infinitesimal numbers by a finite number of symbols as particular casesof a unique framework not related to non-standard analysis theories. The InfinityComputer working with numbers of a new kind is described (its simulator has al-ready been realized). The concept of accuracy of mathematical languages and itsimportance for a number of theoretical and practical issues regarding computationsis discussed. Numerous examples dealing with divergent series, infinite sets, proba-bility, limits, fractals, etc. are given.

Key words: Numerical infinities and infinitesimals, numbers and numerals, InfinityComputer, numerical analysis, infinite sets, divergent series, fractals.

1.1 Introduction

In different periods of human history, mathematicians and physicists in order tosolve theoretical and applied problems existing in their times developed mathemati-cal languages that use different approaches to the ideas of infinity and infinitesimals

Yaroslav D. SergeyevUniversity of Calabria, Via P. Bucci, Cubo 42-C, 87030 Rende, Italy; N.I. Lobatchevsky StateUniversity, Nizhni Novgorod, Russia; Institute of High Performance Computing and Networkingof the National Research Council of Italy, Rende, Italy. e-mail: [email protected] research was partially supported by the project “High accuracy supercomputations and solvingglobal optimization problems using the information approach” of the Russian Federal Program“Scientists and Educators in Russia of Innovations”, project 14.B37.21.0878.

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2 Yaroslav D. Sergeyev

(see [1, 2, 5, 12, 14, 16, 19, 20, 25, 28, 51]) and references given therein). To em-phasize importance of the subject it is sufficient to mention that the ContinuumHypothesis related to infinity has been included by David Hilbert as the ProblemNumber One in his famous list of 23 unsolved mathematical problems (see [16])that have influenced strongly development of Mathematics in the XX-th century.However, arithmetics developed for working with infinities are quite different withrespect to the finite arithmetic we are used to deal with. Moreover, very often theyleave undetermined many operations where infinite numbers take part (for exam-ple, ∞−∞, ∞

∞ , sum of infinitely many items, etc.) or use representation of infinitenumbers based on infinite sequences of finite numbers.

Many approaches describing manipulations with infinities and infinitesimals arerather old: ancient Greeks following Aristotle distinguished the potential infinityfrom the actual infinity; John Wallis (see [51]) credited as the person who has in-troduced the infinity symbol, ∞, has published his work Arithmetica infinitorum in1655; the foundations of analysis we use nowadays have been developed more than200 years ago with the goal to develop mathematical tools allowing one to solveproblems that were emerging in the world at that remote time; Georg Cantor (see[2]) has introduced his cardinals and ordinals more than 100 years ago, as well. Asa result, mathematical languages that we use now while work with infinities andinfinitesimals do not reflect numerous achievements made by Physics of the XXthcentury1. Let us illustrate this observation by a couple of examples.

We know from the modern Physics that the same object can be viewed as eitherdiscrete or continuous in dependence on the instrument used for the observation (wesee a table continuous when we look at it by eye and we see it discrete (consisting ofmolecules, atoms, etc.) when we observe it at a microscope). In addition, physicistsdo not give some absolute results of their observations in sense that together withthe result of the observation they always supply the accuracy of the instrument usedfor this observation.

In Mathematics, both facts are absent: each mathematical object (e.g., function)is either discrete or continuous and nothing is said about the accuracy of the ob-servation of the mathematical objects and about tools used for these observations.The mathematical notion of continuity itself is from XIXth century. Many of math-ematical notions have an absolute character and the ideas of relativity are almost notpresent in them. The ideas of the influence of the instrument of an observation onthe object of the observation are almost absent in Mathematics, as well.

In some sense, there exists a gap between the physical achievements made in thelast two hundred years (especially during the XXth century) and their mathematicalmodels that continue to be written using the mathematical language developed two

1 Even the brilliant efforts of the creator of the non-standard analysis Robinson that were made inthe middle of the XX-th century have been also directed to a reformulation of the classical analysis(i.e., analysis created two hundred years before Robinson) in terms of infinitesimals and not to thecreation of a new kind of analysis that would incorporate new achievements of Physics. In fact, hewrote in paragraph 1.1 of his famous book [28]: ‘It is shown in this book that Leibniz’s ideas canbe fully vindicated and that they lead to a novel and fruitful approach to classical analysis and tomany other branches of mathematics’ (the words classical analysis have been emphasized by theauthor of this chapter).

1 Numerical computations with infinite and infinitesimal numbers 3

centuries ago on the basis of (among other things) physical ideas of that remote timethat now are absolutely outdated.

As was already mentioned, in relation to the concepts of infinite and infinitesimalwe have an analogous situation. In fact, the point of view on infinity accepted nowa-days takes its origins from the famous ideas of Cantor (see [2]) who has shown thatthere exist infinite sets having different number of elements. This has been done dur-ing the second half of the XIXth century. Infinitesimals have been developed evenearlier when, in the early history of calculus, arguments involving infinitesimalsplayed a pivotal role in the differential calculus developed by Leibniz and Newton(see [19, 25]). At that time the notion of an infinitesimal, however, lacked a precisemathematical definition and in order to provide a more rigorous foundation for thecalculus infinitesimals were gradually replaced by the d’Alembert-Cauchy conceptof a limit (see [4, 7]).

The creation of a rigorous mathematical theory of infinitesimals on which itwould be possible to construct Calculus remained an open problem until the end ofthe 1950s when Robinson (see [28]) has introduced his famous non-standard anal-ysis approach. He has shown that non-archimedean ordered field extensions of thereals contained numbers that could serve the role of infinitesimals and their recip-rocals could serve as infinitely large numbers. Robinson then has derived the theoryof limits, and more generally of calculus, and has found a number of importantapplications of his ideas in many other fields of Mathematics (see [28]).

It is important to emphasize that in his approach Robinson used Cantor’s math-ematical tools and terminology (cardinal numbers, countable sets, continuum, one-to-one correspondence, etc.) incorporating so advantages and disadvantages of Can-tor’s approach into non-standard analysis. In particular, we are reminded that it iswell known that Cantor’s approach leads to some situations that often are called bynon mathematicians ‘paradoxes’. The most famous and simple of them is, probably,Hilbert’s paradox of the Grand Hotel. In a normal hotel having a finite number ofrooms no more new guests can be accommodated if it is full. Hilbert’s Grand Hotelhas an infinite number of rooms (of course, the number of rooms is countable, be-cause the rooms in the Hotel are numbered). Due to Cantor, if a new guest arrives atthe Hotel where every room is occupied, it is, nevertheless, possible to find a roomfor him. To do so, it is necessary to move the guest occupying room 1 to room 2, theguest occupying room 2 to room 3, etc. In such a way room 1 will be ready for thenewcomer and, in spite of our assumption that there are no available rooms in theHotel, we have found one.

This result is very difficult to be fully realized by anyone who is not a mathe-matician since in our every day experience in the world around us the part is alwaysless than the whole and if a hotel is complete, there are no places in it. In order tounderstand how it is possible to tackle the problem of infinity in such a way thatHilbert’s Grand Hotel would be in accordance with the principle ‘the part is lessthan the whole’ let us consider a study published in Science by Peter Gordon (see


[13]) where he describes a primitive tribe living in Amazonia - Piraha - that uses avery simple numeral system2 for counting: one, two, many.

For Piraha, all quantities larger than two are just ‘many’ and such operations as2+2 and 2+1 give the same result, i.e., ‘many’. Using their weak numeral systemPiraha are not able to see, for instance, numbers 3, 4, 5, and 6, to execute arith-metical operations with them, and, in general, to say anything about these numbersbecause 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 traditionalcalculus

∞+1 = ∞, ∞+2 = ∞

and in the context of Cantor’s infinite cardinals3 we also have

ℵ0 +1 = ℵ0, ℵ1 +1 = ℵ1. (1.1)

These observations lead us to the following idea: Probably our difficulty in workingwith infinity is not connected to the nature of infinity but is a result of inadequatenumeral systems used to express numbers.

In this chapter, we describe a new methodology for treating infinite and infinites-imal quantities (examples of its usage can be found in [31, 32, 33, 34, 35, 36, 37,39, 41]). It has a strong numerical character and is closer to the point of view on theworld accepted by modern Physics4. In particular, it incorporates the following twoideas borrowed from the modern Physics: relativity and interrelations holding be-tween the object of an observation and the tool used for this observation. The latteris directly related to connections between numeral systems used to describe math-ematical objects and the objects themselves. Numerals that we use to write down

2 We remind that numeral is a symbol or group of symbols that represents a number. The differencebetween numerals and numbers is the same as the difference between words and the things theyrefer to. A number is a concept that a numeral expresses. The same number can be represented bydifferent numerals. For example, the symbols ‘3’, ‘three’, and ‘III’ are different numerals, but theyall represent the same number.3 In connection with Cantor’s ℵ0 and ℵ1 it makes sense to remind another Amazonian tribe –Munduruku (see [27]) who fail in exact arithmetic with numbers larger than 5 but are able tocompare and add large approximate numbers that are far beyond their naming range. Particularly,they use the words ‘some, not many’ and ‘many, really many’ to distinguish two types of largenumbers. Their arithmetic with ‘some, not many’ and ‘many, really many’ reminds strongly therules Cantor uses to work with ℵ0 and ℵ1, respectively. For instance, compare ‘some, not many’+‘many, really many’ = ‘many, really many’ with ℵ0 +ℵ1 = ℵ1.4 As it was already mentioned, in 1900, at the second Mathematical Congress in Paris, DavidHilbert has presented his 23 problems for the XXth century promoting the abstract philosophy inMathematics that was close to Kant. However, before this event, at the first Congress three yearsearlier Henri Poincare has given a general talk emphasizing the connection of Mathematics withPhysics sharing this point of view with Fourier, Laplace, and many others. Clearly, in this disputebetween Poincare and Hilbert the present chapter is closer to the position of Poincare.


numbers, functions, etc. are among our tools of investigation and, as a result, theystrongly influence our capabilities to study mathematical objects.

Since new numeral systems appear very rarely, in each concrete historical pe-riod people tend to think that any number can be expressed by the current numeralsystem and the importance of numeral systems for Mathematics is very often un-derestimated (especially by pure mathematicians). However, if we observe the sit-uation in the historical prospective we can immediately see limitations that variousnumeral systems induce. In order to illustrate this assertion, it is sufficient to thinkabout Piraha. We can also remind the Roman numeral system that does not allowone to express zero and negative numbers. In this system, the expression III-X is anindeterminate form. As a result, before appearing the positional numeral system andinventing zero (by the way, the second event was several hundred years later withrespect to the first one) mathematicians were not able to create theorems involvingzero and negative numbers and to execute computations with them. Thus, develop-ing new (more powerful than existing ones) numeral systems can help a lot both intheory and practice of computations.

If we compare the usage of numeral systems in Mathematics when one works,on the one hand, with finite quantities and, on the other hand, with infinities andinfinitesimals then we can see immediately an important difference. In our every-day activities with finite numbers the same finite numerals are used for differentpurposes (e.g., the same numeral 6 can be used to express the number of elementsof a set, to indicate the position of an element in a finite sequence, and to executepractical computations). In contrast, when we face the necessity to work with infini-ties or infinitesimals, the situation changes drastically. In fact, in this case differentnumerals are used to work with infinities and infinitesimals in different situations:

• ∞ in standard analysis;• ω for working with ordinals;• ℵ0,ℵ1, ... for dealing with cardinalities;• non-standard numbers using a generic infinitesimal h in non-standard analysis,

etc.

In particular, since the mainstream of the traditional Mathematics very often doesnot pay a great attention to the distinction between numbers and numerals (in thisoccasion it is necessary to recall constructivists who studied this issue), many the-ories dealing with infinite and infinitesimal quantities have a symbolic (not numer-ical) character. For instance, many versions of non-standard analysis are symbolic,since they have no numeral systems to express their numbers by a finite number ofsymbols (the finiteness of the number of symbols is necessary for organizing nu-merical computations). Namely, if we consider a finite n than it can be taken n = 7,or n = 108 or any other numeral used to express finite quantities and consisting of afinite number of symbols. In contrast, if we consider a non-standard infinite m thenit is not clear which numerals can be used to assign a concrete value to m.

Analogously, in non-standard analysis, if we consider an infinitesimal h then itis not clear which numerals consisting of a finite number of symbols can be used toassign a value to h and to write h = ... In fact, very often in non-standard analysis


texts, a generic infinitesimal h is used and it is considered as a symbol, i.e., onlysymbolic computations can be done with it. Approaches of this kind leave unclearsuch issues, e.g., whether the infinite 1/h is integer or not or whether 1/h is thenumber of elements of an infinite set. Another problem is related to comparison ofvalues. When we work with finite quantities then we can compare x and y if theyassume numerical values, e.g., x = 4 and y = 6 then, by using rules of the numeralsystem the symbols 4 and 6 belong to, we can compute that y > x. If one wishesto consider two infinitesimals h1 and h2 then it is not clear how to compare thembecause numeral systems that can express infinitesimals are not provided by non-standard analysis techniques.

The approach developed in [31, 37, 43] proposes a numeral system that uses thesame numerals for several different purposes for dealing with infinities and infinites-imals: in analysis for working with functions that can assume different infinite, fi-nite, and infinitesimal values (functions can also have derivatives assuming differentinfinite or infinitesimal values); for measuring infinite sets; for indicating positionsof elements in ordered infinite sequences; in probability theory, etc. It is importantto emphasize that the new numeral systems avoids situations like that of Piraha and(1.1) providing results ensuring that if a is a numeral written in this system thenfor any a (i.e., a can be finite, infinite, or infinitesimal) it follows a+ 1 > a. Thenew methodology has allowed the author to introduce the Infinity Computer (seethe patent [41]) working numerically with infinite and infinitesimal numbers.

In order to see the place of the new approach in the historical panorama of ideasdealing with infinite and infinitesimal, see [21, 22, 40, 42, 47]. The new method-ology has been successfully applied for studying percolation (see [17, 50]), Eu-clidean and hyperbolic geometry (see [23, 29]), fractals (see [36, 38, 46, 50]), nu-merical differentiation and optimization (see [6, 39, 44, 53]), infinite series (see[40, 45, 52]), the first Hilbert problem, Riemann zeta function, and Turing machines(see [42, 45, 47]), cellular automata (see [8]), etc.

The rest of the chapter is structured as follows. An introduction to the newmethodology is given in Section 1.2. It allows us to introduce in Section 1.3 a newinfinite unit of measure that is then used as the radix of a new positional numeralsystem. Section 1.4 shows that this system gives a possibility to express finite, in-finite, and infinitesimal numbers in a unique framework and to execute arithmeti-cal operations with all of them. Section 1.5 discusses first applications of the newmethodology. Section 1.6 establishes relations of the new methodology to some ofthe results of Cantor. New computational possibilities for mathematical modelingsupplied by the new approach are discussed in Section 1.7. A quantitative analysis offractals executed by using infinite and infinitesimal numbers is given in Section 1.8.Concepts of continuity in Physics and Mathematics from the point of view of thenew methodology are discussed in Section 1.9. Finally, Section 1.10 concludes thechapter.

We close this Introduction by emphasizing that the new approach is not a con-traposition to the ideas of Cantor, Levi-Civita, and Robinson. In contrast, it is intro-duced as an applied evolution of their ideas. The problem of infinity is consideredfrom positions of applied Mathematics and theory and practice of computations –


fields being among the main scientific interests (see, e.g., monographs [48, 49]) ofthe author. The new computational methodology introduces the notion of the accu-racy of mathematical languages and shows that different tools (numeral systems)can express different sets of numbers (and other mathematical objects) with differ-ent accuracies. It can be shown that Cantor’s alephs and new numerals have differentaccuracies and cases where the new tools are more accurate can be provided. Thus,the traditional approaches and the new one do not contradict one another, they arejust different instruments having different accuracies for observations of mathemat-ical objects.

1.2 A new computational methodology and accuracy of numeralsystems

The aim of this section is to introduce a new methodology that would allow oneto work with infinite and infinitesimal quantities in the same way as one workswith finite numbers. Evidently, it becomes necessary to define what does it meanin the same way. Usually, in modern Mathematics, when it is necessary to define aconcept or an object, logicians try to introduce a number of axioms describing theobject. However, this way is fraught with danger because of the following reasons.First of all, when we describe a mathematical object or concept we are limited bythe expressive capacity of the language we use to make this description. A morerich language allows us to say more about the object and a weaker language – less(remind Piraha that are not able to say a word about number 4). Thus, developmentof the mathematical (and not only mathematical) languages leads to a continuousnecessity of a transcription and specification of axiomatic systems. Second, thereis no any guarantee that the chosen axiomatic system defines ‘sufficiently well’ therequired concept and a continuous comparison with practice is required in order tocheck the goodness of the accepted set of axioms. However, there cannot be againany guarantee that the new version will be the last and definitive one. Finally, thethird limitation latent in axiomatic systems has been discovered by Godel in his twofamous incompleteness theorems (see [11]).

In this chapter, we introduce a different, significantly more applied and less am-bitious view on axiomatic systems related only to utilitarian necessities to makecalculations. We start by introducing three postulates that will fix our methodolog-ical positions with respect to infinite and infinitesimal quantities and Mathematics,in general. In contrast to the modern mathematical fashion that tries to make all ax-iomatic systems more and more precise (decreasing so degrees of freedom of thestudied part of Mathematics), we just define a set of general rules describing howpractical computations should be executed leaving so as much space as possible forfurther, dictated by practice, changes and developments of the introduced mathe-matical language. Speaking metaphorically, we prefer to make a hammer and to useit instead of describing what is a hammer and how it works.


Usually, when mathematicians deal with infinite objects (sets or processes) it issupposed (even by constructivists (see, for example, [24])) that human beings areable to execute certain operations infinitely many times. For example, in a fixed nu-meral system it is possible to write down a numeral with any number of digits. How-ever, this supposition is an abstraction (courageously declared by constructivists in[24]) because we live in a finite world and all human beings and/or computers fin-ish operations they have started. In this chapter, this abstraction is not used and thefollowing postulate is adopted.

Postulate 1. We postulate existence of infinite and infinitesimal objects but ac-cept that human beings and machines are able to execute only a finite numberof operations.

Thus, we accept that we shall never be able to give a complete description ofinfinite processes and sets due to our finite capabilities. Particularly, this means thatwe accept that we are able to write down only a finite number of symbols to expressnumbers. However, we do not agre with finitists who deny infinite mathematicalobjects. We accept their existence and shall try to study them using our finite capa-bilities.

The second postulate is adopted following the way of reasoning used in naturalsciences where researchers use tools to describe the object of their study and theused instrument influences the results of the observations. When a physicist uses aweak lens A and sees two black dots in his/her microscope he/she does not say: Theobject of the observation is two black dots. The physicist is obliged to say: the lensused in the microscope allows us to see two black dots and it is not possible to sayanything more about the nature of the object of the observation until we change theinstrument - the lens or the microscope itself - by a more precise one. Suppose thathe/she changes the lens and uses a stronger lens B and is able to observe that theobject of the observation is viewed as ten (smaller) black dots. Thus, we have twodifferent answers: (i) the object is viewed as two dots if the lens A is used; (ii) theobject is viewed as ten dots by applying the lens B. Which of the answers is correct?Both. Both answers are correct but with the different accuracies that depend on thelens used for the observation.

The same happens in Mathematics studying natural phenomena, numbers, andobjects that can be constructed by using numbers. Numeral systems used to ex-press numbers are among the instruments of observations used by mathematicians.The usage of powerful numeral systems gives the possibility to obtain more preciseresults in Mathematics in the same way as usage of a good microscope gives thepossibility of obtaining more precise results in Physics. However, even for the bestexisting tool the capabilities of this tool will be always limited due to Postulate 1(we are able to write down only a finite number of symbols when we wish to de-scribe a mathematical object) and due to Postulate 2 we shall never tell, what is,for example, a number but shall just observe it through numerals expressible in achosen numeral system.


Postulate 2. We shall not tell what are the mathematical objects we deal with;we just shall construct more powerful tools that will allow us to improve ourcapacities to observe and to describe properties of mathematical objects.

This Postulate means that we emphasize that mathematical results are not ab-solute, they depend on mathematical languages used to formulate them, i.e., therealways exists an accuracy of the description of a mathematical result, fact, object,etc. imposed by the mathematical language used to formulate this result. For in-stance, the result of Piraha 2+ 2 = ‘many’ is not wrong, it is just inaccurate. Theintroduction of a stronger tool (in this case, a numeral system that contains a nu-meral for a representation of the number four) allows us to have a more preciseanswer.

The concept of the accuracy allows us to look at paradoxes in a new way: paradoxis a situation where the accuracy of the used language is not sufficient to describe thephenomenon we are interested in. For instance, the answers of Piraha 2+1= ‘many’and 2+ 2 = ‘many’ can be viewed as a paradox because from these two recordsone could conclude that 2+ 1 = 2+ 2. This paradox shows us the borderline thatseparates the zone where the language has the high precision from the region wherethe language cannot be applied because it does not allow one to distinguish differentobjects within ‘many’. Analogously, the records ‘many’ + 1= ‘many’, ∞+ 1 = ∞,1+ω = ω = ω+1, (1.1), etc. can also be viewed as situations where the accuracyof the used numeral systems is not sufficient.

It is necessary to comment upon another important aspect of the distinction be-tween a mathematical object and a mathematical tool used to observe this object.Postulates 1 and 2 impose us to think always about the possibility to execute a math-ematical operation by applying a numeral system. They tell us that there always existsituations where we are not able to express the result of an operation. Let us con-sider, for example, the operation of construction of the successive element widelyused in number and set theories. In the traditional Mathematics, the aspect whetherthis operation can be executed is not taken into consideration, it is supposed that it isalways possible to execute the operation k = n+1 starting from any integer n. Thus,there is no any distinction between the existence of the number k and the possibilityto execute the operation n+ 1 and to express its result, i.e. to have a numeral thatcan express k.

Postulates 1 and 2 emphasize this distinction and tell us that: (i) in order to exe-cute the operation it is necessary to have a numeral system allowing one to expressboth numbers, n and k; (ii) for any numeral system there always exists a number kthat cannot be expressed in it. For instance, for Piraha k = 3, for Munduruku k = 6.Even for modern powerful numeral systems there exist such a number k (for in-stance, nobody is able to write down a numeral in the decimal positional systemhaving 10100 digits). Hereinafter we shall always emphasize the triad – researcher,object of the investigation, and tools used to observe the object – in various mathe-matical and computational contexts paying a special attention to the accuracy of theobtained results.


Another important issue related to Postulate 2 consists of the fact that, from ourpoint of view, axiomatic systems do not define mathematical objects but just deter-mine formal rules for operating with certain numerals reflecting some (not all) prop-erties of the studied mathematical objects using a certain mathematical language L.We are aware that the chosen language L has its accuracy and there always can exista richer language L that would allow us to describe the studied object better. As hasalready been discussed above, any language has a limited expressibility, in partic-ular, there always exist situations where the accuracy of the answers expressible inthis language is not sufficient.

Numerals that we use to write down numbers, functions, etc. are among our toolsof the investigation and, as a result, they strongly influence our capabilities to studymathematical objects. This separation (having an evident physical spirit) of math-ematical objects from the tools used for their description is crucial for our studybut it is used rarely in contemporary Mathematics. In fact, the idea of finding anadequate (absolutely the best) set of axioms for one or another field of Mathematicscontinues to be among the most attractive goals for contemporary mathematicians.Usually, when it is necessary to define a concept or an object, logicians try to in-troduce a number of axioms defining the object. However, this way is fraught withdanger because of the following reasons.

First, when one describes a mathematical object or concept he or she is limitedby the expressive capacity of the language that is used to make this description. Aricher language allows one to say more about the object and a weaker language –less. Thus, development of the mathematical (and not only mathematical) languagesleads to a continuous necessity of a transcription and specification of axiomatic sys-tems. Second, there is no guarantee that the chosen axiomatic system defines ‘suf-ficiently well’ the required concept and a continuous comparison with practice isrequired in order to check the goodness of the accepted set of axioms. However,there cannot be again any guarantee that the new version will be the last and defini-tive one. Finally, the third limitation has been discovered by Godel in his two famousincompleteness theorems (see [11]).

It should be emphasized that in both in Philosophy and Linguistics, the relativityof the language (the instrument) with respect to the world around (the object ofstudy) is a well known thing. It is sufficient to mention Wittgenstein: ‘The limitsof my language are the limits of my mind. All I know is what I have words for.’In Linguistics, it is sufficient to remind the Sapir–Whorf thesis (see [3, 30]), alsoknown as the ‘linguistic relativity thesis’. As becomes clear from its name, the thesisdoes not accept the idea of the universality of language and postulates that the natureof a particular language influences the thought of its speakers. The thesis challengesthe possibility of perfectly representing the world with language, because it impliesthat the mechanisms of any language condition the thoughts of its speakers.

Thus, due to Postulate 2, our point of view on axiomatic systems is significantlymore applied with respect to the modern mathematical fashion that tries to make allaxiomatic systems more and more precise (decreasing so degrees of freedom of thestudied part of Mathematics). We just define a set of general rules describing howpractical computations should be executed leaving so as much space as possible for


further, dictated by practice, changes and developments of the introduced mathe-matical language. Speaking metaphorically, we prefer to make a hammer and to useit instead of trying to define what the hammer is and how it works.

For example, from this applied point of view, axioms for real numbers are con-sidered together with a particular numeral system S used to write down numeralsand are viewed as practical rules (associative and commutative properties of mul-tiplication and addition, distributive property of multiplication over addition, etc.)describing operations with the numerals. The completeness property is interpretedas a possibility to extend S with additional symbols (e.g., e, π,

√2, etc.) taking care

of the fact that the results of computations with these symbols agree with the factsobserved in practice. As a rule, the assertions regarding numbers that cannot be ex-pressed in a numeral system are avoided (e.g., it is not supposed that real numbersform a field).

Finally, before we switch our attention to Postulate 3, it should be noticed thekey difference distinguishing our approach from the constructivism. Constructivistsassert that it is necessary to construct (in some sense) a mathematical object toprove that it exists. Following Physics, we do not discuss the questions of existenceof mathematical objects at all. We discuss just what can be observed through ourtools (languages, numeral systems, etc.).

Let us now start to introduce the last Postulate. We want to treat infinite andinfinitesimal numbers in the same manner as we are used to deal with finite ones,i.e., by applying the philosophical principle of Ancient Greeks ‘The part is less thanthe whole’. This principle, in our opinion, very well reflects organization of theworld around us but is not incorporated in many traditional infinity theories whereit is true only for finite numbers. The reason of this traditional discrepancy (as theexample with Piraha advices) is related to the accuracy of numeral systems used towork with infinity.

Postulate 3. We adopt the principle ‘The part is less than the whole’ to allnumbers (finite, infinite, and infinitesimal) and to all sets and processes (finiteand infinite).

Due to this Postulate, the traditional point of view on infinity accepting suchresults as ∞− 1 = ∞ should be substituted in a way that ∞− 1 < ∞. One of themotivations pro this substitution has already been discussed in detail in connectionwith the numerals of Piraha. We can introduce another simple argument. Supposethat we are at a point A and at another point, B, being infinitely far from A thereis an object. Let us see what will happen if we shall change our position and willmove, let say, one meter forward in the direction of the point B. The traditionalnumeral system using the symbol ∞ will not be able to register this movement ina quantitative way because ∞− 1 = ∞. This numeral system allows us to say onlythat the object was infinitely far before the movement and remains to be infinitelyfar after the movement, i.e., the accuracy of the answer is very low. In practice,due to this traditional way of doing, we are forced to negate the finite movement


that we have executed. Hereinafter, our goal will be to avoid similar situations bythe introduction of a new numeral system that instead of the traditional numerals∞,ℵ0,ω,ℵ1, etc. would use a new kind of numerals satisfying Postulates 1 – 3introduced above.

Due to Postulates 1 – 3, such concepts as bijection, numerable and continuumsets, cardinal and ordinal numbers cannot be used in this chapter because they be-long to theories working with different assumptions. It can seem at first glance thatPostulate 3 contradicts Cantor’s one-to-one correspondence principle. However, asit will be shown hereinafter, this is not the case. Instead, the situation is similar tothe example from Physics described above where we have considered two lenseshaving different accuracies. We have here just two different instruments (numeralsystems) having different accuracies: Cantor’s approach and the new one based onPostulates 1 – 3. Analogously, in the finite case, when we observe a garden with 123trees, then our answer, i.e., 123 trees, and the answer of Piraha, i.e., many trees, areboth correct, but the accuracy of our answer is higher.

It is important to notice that the adopted Postulates impose also the style of ex-position of results in the chapter: we first introduce new mathematical instruments,then show how to use them in several areas of Mathematics, introducing each itemas soon as it becomes indispensable for the problem under consideration.

Let us introduce now the new way of counting by studying a situation arising inpractice and related to the necessity to operate with extremely large quantities (see[31] for a detailed discussion). Imagine that we are in a granary and the owner asksus to count how much grain he has inside it. In this occasion, nobody counts thegrain seed by seed because the number of seeds is enormous.

To overcome this difficulty, people take sacks, fill them in with seeds, and countthe number of sacks. In this situation, we suppose that: (i) the number of seeds ineach sack is the same but it is so huge that we are not able to count seed by seed howmany they are and (ii) in any case the resulting number would not be expressible byavailable numerals.

Then, if the granary is huge and it becomes difficult to count the sacks, thentrucks or even big train waggons are used. In this model, we suppose that all sackscontain the same number of seeds, all trucks – the same number of sacks, and allwaggons – the same number of trucks, however, these numbers are so huge that itbecomes impossible to determine them. At the end of the counting of this type weobtain a result in the following form: the granary contains 14 waggons, 54 trucks,18 sacks, and 47 seeds of grain. Note, that if we add, for example, one seed to thegranary, we can count it and see that the granary has more grain. If we take out onewaggon, we again are able to say how much grain has been subtracted.

Thus, in our example it is necessary to count large quantities. They are finite butit is impossible to count them directly by using an elementary unit of measure, u0,(seeds in our example) because the quantities expressed in these units would be toolarge. Therefore, people are forced to behave as if the quantities were infinite.

To solve the problem of ‘infinite’ quantities, new units of measure, u1,u2, andu3, are introduced (units u1 – sacks, u2 – trucks, and u3 – waggons). The new unitshave the following important peculiarity: all the units ui+1 contain a certain number


Ki of units ui but this number, Ki, is unknown. Naturally, it is supposed that Ki isthe same for all instances of the units ui+1. Thus, numbers that it was impossible toexpress using only the initial unit of measure are perfectly expressible in the newunits we have introduced in spite of the fact that the numbers Ki are unknown.

This key idea of counting by introduction of new units of measure will be usedin the chapter to deal with infinite quantities together with the idea of separate countof units with different exponents used in traditional positional numeral systems.

1.3 A new way of counting and the infinite unit of measure

The infinite unit of measure is expressed by the numeral ¬ called grossone and isintroduced as the number of elements of the set, N, of natural numbers. Remindthat the usage of a numeral indicating totality of the elements we deal with is notnew in Mathematics. It is sufficient to mention the theory of probability (axioms ofKolmogorov) where events can be defined in two ways. First, as union of elementaryevents; second, as a sample space, Ω, of all possible elementary events (or its partsΩ/2,Ω/3, etc.) from which some elementary events have been excluded (or addedin case of parts of Ω). Naturally, the latter way to define events becomes particularlyuseful when the sample space consists of infinitely many elementary events.

Grossone is introduced by describing its properties (similarly, in order to passfrom natural to integer numbers a new element – zero – is introduced by describingits properties) postulated by the Infinite Unit Axiom (IUA) consisting of three parts:Infinity, Identity, and Divisibility. This axiom is added to axioms for real numbers(remind that we consider axioms in sense of Postulate 2). Thus, it is postulated thatassociative and commutative properties of multiplication and addition, distributiveproperty of multiplication over addition, existence of inverse elements with respectto addition and multiplication hold for grossone as for finite numbers5. Let us intro-duce the axiom and then give comments on it.

Infinity. Any finite natural number n is less than grossone, i.e., n < ¬.Identity. The following relations link ¬ to identity elements 0 and 1

0 ·¬ = ¬ ·0 = 0, ¬−¬ = 0,¬

¬= 1, ¬0 = 1, 1¬ = 1, 0¬ = 0. (1.2)

Divisibility. For any finite natural number n sets Nk,n,1 ≤ k ≤ n, being the nthparts of the set, N, of natural numbers have the same number of elements indicatedby the numeral ¬

n where

5 It is important to emphasize that we speak about axioms of real numbers in sense of Postulate 2,i.e., axioms define formal rules of operations with numerals in a given numeral system. Therefore,if we want to have a numeral system including grossone, we should fix also a numeral system toexpress finite numbers. In order to concentrate our attention on properties of grossone, this pointwill be investigated later.


Nk,n = k,k+n,k+2n,k+3n, . . ., 1 ≤ k ≤ n,n∪

k=1

Nk,n = N. (1.3)

The first part of the introduced axiom – Infinity – is quite clear. In fact, we wantto describe an infinite number, thus, it should be larger than any finite number. Thesecond part of the axiom – Identity – tells us that ¬ behaves itself with identity el-ements 0 and 1 as all other numbers. In reality, we could even omit this part of theaxiom because, due to Postulate 3, all numbers should be treated in the same wayand, therefore, at the moment we have told that grossone is a number, we have fixedusual properties of numbers, i.e., the properties described in Identity, associative andcommutative properties of multiplication and addition, distributive property of mul-tiplication over addition, existence of inverse elements with respect to addition andmultiplication. The third part of the axiom – Divisibility – is the most interesting, itis based on Postulate 3. Let us first illustrate it by an example.

Example 1. If we take n = 1, then N1,1 = N and Divisibility tells that the set, N, ofnatural numbers has ¬ elements. If n = 2, we have two sets N1,2 and N2,2

N1,2 = 1, 3, 5, 7, . . . ,

N2,2 = 2, 4, 6, . . . (1.4)

and they have ¬2 elements each. Pay attention that we are not able to count the num-

ber of elements of the sets N, N1,2, and N2,2 one by one because due to Postulate 1we are able to execute only a finite number of operations and these sets are infinite.To define their number of elements we apply Postulate 3 and determine the numberof the elements of the parts using the whole.

Then, if n = 3, we have three sets

N1,3 = 1, 4, 7, . . . ,

N2,3 = 2, 5, . . . ,

N3,3 = 3, 6, . . .

(1.5)

and they have ¬3 elements each. Note that in formulae (1.4), (1.5) we have added

extra spaces writing down the elements of the sets N1,1,N1,2,N1,3,N2,3,N3,3 just toemphasize Postulate 3 and to show visually that N1,1 ∪N1,2 = N and N1,3 ∪N2,3 ∪N3,3 = N. ⊓⊔

We emphasize again that to introduce ¬n we do not try to count elements k,k+

n,k+2n,k+3n, . . . one by one in (1.3). In fact, we cannot do this due to Postulate 1.By using Postulate 3, we construct the sets Nk,n,1 ≤ k ≤ n, by separating the whole,i.e., the set N, in n parts (this separation is highlighted visually in formulae (1.4) and(1.5)). Again due to Postulate 3, we affirm that the number of elements of the nth


part of the set, i.e., ¬n , is n times less than the number of elements of the whole set,

i.e., than ¬.In terms of our granary example ¬ can be interpreted as the number of seeds in

the sack. In that example, the number K0 of seeds in each sack was fixed and finitebut impossible to be expressed in units u0, i.e., seeds, by counting seed by seedbecause we have supposed that sacks were very big and the corresponding numberwould not be expressible by available numerals. In spite of the fact that K0 andK1,K2, . . . were inexpressible and unknown, by using new units of measure (sacks,trucks, etc.) it was possible to count easier and to express the required quantities.Now our sack has the infinite but again fixed number of seeds. It is fixed becauseit has a strong link to a concrete set – it is the number of elements of this set,precisely, of the set of natural numbers. This number is inexpressible by existingnumeral systems with the same high accuracy as we do it with finite small sets6

and we introduce a new number – grossone – expressible by a new numeral – ¬.Then, we apply Postulate 3 and say that if the sack contains ¬ seeds, its nth partcontains n times less quantity, i.e., ¬

n seeds. Note that, since the numbers ¬n have

been introduced as numbers of elements of sets Nk,n, they are integer.The new unit of measure allows us to calculate easily the number of elements of

sets being union, intersection, difference, or product of other sets of the type Nk,n.Due to our accepted methodology, we do it in the same way as these measurementsare executed for finite sets. Let us consider two simple examples (a general rule fordetermining the number of elements of infinite sets having a more complex structurewill be given in Section 1.5) showing how grossone can be used for this purpose.

Example 2. Let us determine the number of elements of the set Ak,n = Nk,n\a,a ∈ Nk,n,n ≥ 1. Due to the IUA, the set Nk,n has ¬

n elements. The set Ak,n has

been constructed by excluding one element from Nk,n. Thus, the set Ak,n has ¬n −1

elements. The granary interpretation can be also given for the number ¬n − 1: the

number of seeds in the nth part of the sack minus one seed. For n = 1 we have ¬−1interpreted as the number of seeds in the sack minus one seed. ⊓⊔

Divisibility and Example 2 show us that in addition to the usual way of counting,i.e., by adding units, that has been well formalized in Mathematics, there exist alsothe way to count by taking parts of the whole and by subtracting units or parts ofthe whole. The following example shows a little bit more complex situation (othermore sophisticated examples will be given later after the reader will got accustomedwith the concept of grossone).

6 First, this quantity is inexpressible by numerals used to count the number of elements of finitesets because N is infinite. Second, traditional numerals existing to express infinite numbers do nothave the required high accuracy (remind that we would like to be able to register the alteration ofthe number of elements of infinite sets even when one element has been excluded). For example,by using Cantor’s alephs we say that cardinality of the sets N and N\1 is the same – ℵ0. Thisanswer is correct but its accuracy is low – we are not able to register the fact that one elementwas excluded from the set N. Analogously, we can say that both of the sets have ‘many’ elements.Again, this answer is correct but its accuracy is low.


Example 3. Let us consider the following two sets

B1 = 4,9,14,19,24,29,34,39,44,49,54,59,64,69,74,79, . . .,

B2 = 3,14,25,36,47,58,69,80,91,102,113,124,135, . . .

and determine the number of elements in the set B = (B1 ∩B2)∪ 3,4,5,69. Itfollows immediately from the IUA that B1 = N4,5,B2 = N3,11. Their intersection

B1 ∩B2 = N4,5 ∩N3,11 = 14,69,124, . . .= N14,55

and, therefore, due to the IUA, it has ¬55 elements. Finally, since 69 belongs to the

set N14,55 and 3, 4, and 5 do not belong to it, the set B has ¬55 + 3 elements. The

granary interpretation: this is the number of seeds in the 55th part of the sack plusthree seeds. ⊓⊔

One of the important differences of the new approach with respect to the non-standard analysis consists of the fact that ¬ ∈ N because grossone has been intro-duced as the quantity of natural numbers. Similarly, the number 5 being the numberof elements of the set

A = 1,2,3,4,5 (1.6)

is the largest element in this set. The new numeral ¬ allows one to write down theset, N, of natural numbers in the form

N= 1,2, . . .¬

2−2,

¬

2−1,

¬

2,¬

2+1,

¬

2+2, . . . ¬−2, ¬−1, ¬. (1.7)

Note that traditional numeral systems did not allow us to see infinite natural numbers

. . .¬

2−2,

¬

2−1,

¬

2,¬

2+1,

¬

2+2, . . . ¬−2,¬−1,¬. (1.8)

It is important to emphasize that in the new approach the set (1.7) is the same setof natural numbers

N= 1,2,3, . . . (1.9)

we are used to deal with and infinite numbers (1.8) also take part of N. Both records,(1.7) and (1.9), are correct and do not contradict each other. They just use two dif-ferent numeral systems to express N. Traditional numeral systems do not allow us tosee infinite natural numbers that we can observe now thanks to ¬. Similarly, Pirahaare not able to see finite natural numbers greater than 2. In spite of this fact, thesenumbers (e.g., 3 and 4) belong to N and are visible if one uses a more powerfulnumeral system. Thus, we have the same object of observation – the set N – that canbe observed by different instruments – numeral systems – with different accuracies(see Postulate 2).

This example illustrates also the fact that when we speak about sets (finite orinfinite) it is necessary to take care about tools used to describe a set (remind Postu-late 2). In order to introduce a set, it is necessary to have a language (e.g., a numeral


system) allowing us to describe its elements and the number of the elements in theset. For instance, the set A from (1.6) cannot be defined using the mathematicallanguage of Piraha.

Analogously, the words ‘the set of all finite numbers’ do not define a set com-pletely from our point of view, as well. It is always necessary to specify whichinstruments are used to describe (and to observe) the required set and, as a conse-quence, to speak about ‘the set of all finite numbers expressible in a fixed numeralsystem’. For instance, for Piraha ‘the set of all finite numbers’ is the set 1,2 andfor Munduruku ‘the set of all finite numbers’ is the set A from (1.6). As it happensin Physics, the instrument used for an observation bounds the possibility of the ob-servation. It is not possible to say how we shall see the object of our observation ifwe have not clarified which instruments will be used to execute the observation.

Now the following obvious question arises: Which natural numbers can we ex-press by using the new numeral ¬? Suppose that we have a numeral system, S , forexpressing finite natural numbers and it allows us to express KS numbers (not nec-essary consecutive) belonging to a set NS ⊂ N. Note that due to Postulate 1, KS isfinite. Then, addition of ¬ to this numeral system will allow us to express also infi-nite natural numbers i¬

n ± k ≤ ¬ where 1 ≤ i ≤ n, k ∈ NS , n ∈ NS (note that since¬n are integers, i¬

n are integers too). Thus, the more powerful system S is used to ex-press finite numbers, the more infinite numbers can be expressed but their quantityis always finite, again due to Postulate 1. The new numeral system using grossoneallows us to express more numbers than traditional numeral systems thanks to theintroduced new numerals but, as it happens for all numeral systems, its abilities toexpress numbers are limited.

Example 4. Let us consider the numeral system, P , of Piraha able to express onlynumbers 1 and 2 (the only difference will be in the usage of numerals ‘1’ and ‘2’instead of original numerals I and II used by Piraha). If we add to P the new numeral¬, we obtain a new numeral system (we call it P ) allowing us to express only tennumbers represented by the following numerals

1,2︸︷︷︸f inite

, . . .¬

2−2,

¬

2−1,

¬

2,

¬

2+1,

¬

2+2︸︷︷︸

in f inite

, . . . ¬−2,¬−1,¬︸︷︷︸in f inite

. (1.10)

The first two numbers in (1.10) are finite, the remaining eight are infinite, and dotsshow natural numbers that are not expressible in P . As a consequence, P does notallow us to execute such operation as 2+2 or to add 2 to ¬

2 +2 because their resultscannot be expressed in it. Of course, we do not say that results of these operationsare equal (as Piraha do for operations 2+2 and 2+1). We just say that the resultsare not expressible in P and it is necessary to take another, more powerful numeralsystem if we want to execute these operations. ⊓⊔

Note that crucial limitations discussed in Example 4 hold for sets, too. As aconsequence, the numeral system P allows us to define only the sets N1,2 and N2,2among all possible sets of the form Nk,n from (1.3) because we have only two finite


numerals, ‘1’ and ‘2’, in P . This numeral system is too weak to define other setsof this type, for instance, N4,5, because numbers greater than 2 required for thesedefinition are not expressible in P . These limitations have a general character and arerelated to all questions requiring a numerical answer (i.e., an answer expressed onlyin numerals, without variables). In order to obtain such an answer, it is necessary toknow at least one numeral system able to express numerals required to write downthis answer.

We are ready now to formulate the following important result being a direct con-sequence of the accepted methodological postulates.

Theorem 1. The set N is not a monoid under addition.

Proof. Due to Postulate 3, the operation ¬+1 gives us as the result a number greaterthan ¬. Thus, by definition of grossone, ¬+1 does not belong to N and, therefore,N is not closed under addition and is not a monoid. ⊓⊔

This result also means that adding the IUA to the axioms of natural numbersdefines the set of extended natural numbers indicated as N and including N as aproper subset

N= 1,2, . . . ,¬−1,¬,¬+1, . . . ,¬2 −1,¬2,¬2 +1, . . .. (1.11)

The extended natural numbers greater than grossone are also linked to sets of num-bers and can be interpreted in the terms of grain.

Example 5. Let us determine the number of elements of the set

Cm = (a1,a2, . . . ,am−1,am) : ai ∈ N,1 ≤ i ≤ m, 2 ≤ m ≤ ¬.

The elements of Cm are m-tuples of natural numbers. It is known from combinatorialcalculus that if we have m positions and each of them can be filled in by one of lsymbols, the number of the obtained m-tuples is equal to lm. In our case, since Nhas grossone elements, l = ¬. Thus, the set Cm has ¬m elements. In the particularcase, m = 2, we obtain that the set

C2 = (a1,a2) : ai ∈ N, i ∈ 1,2,

being the set of couples of natural numbers, has ¬2 elements. These couples areshown below

(1,1), (1,2), . . . (1,¬−1), (1,¬),(2,1), (2,2), . . . (2,¬−1), (2,¬),. . . . . . . . . . . . . . .

(¬−1,1), (¬−1,2), . . . (¬−1,¬−1), (¬−1,¬),(¬,1), (¬,2), . . . (¬,¬−1), (¬,¬).

Another interesting particular case is the set

C¬ = (a1,a2, . . . ,a¬−1,a¬) : ai ∈ N,1 ≤ i ≤ ¬


having ¬¬ elements.Note that we can also give the granary interpretation for the numbers of the type

¬m: if we accept that the numbers Ki from page 13 are such that Ki = ¬,1 ≤ i ≤m−1, then ¬2 can be viewed as the number of seeds in the truck, ¬3 as the numberof seeds in the train waggon, etc. ⊓⊔

The set, Z, of extended integer numbers can be constructed from the set, Z, ofinteger numbers by a complete analogy and inverse elements with respect to additionare introduced naturally. For example, 7¬ has its inverse with respect to additionequal to −7¬.

It is important to notice that, due to Postulates 1 and 2, the new system of count-ing cannot give answers to all questions regarding infinite sets. What can we say, forinstance, about the number of elements of the sets N and Z? The introduced numeralsystem based on ¬ is too weak to give answers to these questions. It is necessary tointroduce in a way a more powerful numeral system by defining new numerals (forinstance, , ®, etc).

We conclude this section by the following remark. The IUA introduces a newnumber – the quantity of elements in the set of natural numbers – expressed by thenew numeral ¬. However, other numerals and sets can be used to state the idea ofthe axiom. For example, the numeral ¶ can be introduced as the number of elementsof the set, E, of even numbers and can be taken as the base of a numeral system.In this case, the IUA can be reformulated using the numeral ¶ and numerals usingit will be used to express infinite numbers. For example, the number of elements ofthe set, O, of odd numbers will be expressed as |O|= |E|= ¶ and |N|= 2· ¶. Weemphasize through this note that infinite numbers (similarly to the finite ones) canbe expressed by various numerals and in different numeral systems.

1.4 Arithmetical operations in the new numeral system

We have already started to write down simple infinite numbers and to execute arith-metical operations with them without concentrating our attention upon this question.Let us consider it systematically.

1.4.1 Positional numeral system with infinite radix

Different numeral systems have been developed to describe finite numbers. In posi-tional numeral systems, fractional numbers are expressed by the record

(anan−1 . . .a1a0.a−1a−2 . . .a−(q−1)a−q)b (1.12)

where numerals ai,−q≤ i≤ n, are called digits, belong to the alphabet 0,1, . . . ,b−1, and the dot is used to separate the fractional part from the integer one. Thus, the


numeral (1.12) is equal to the sum

anbn +an−1bn−1 + . . .+a1b1 +a0b0 +a−1b−1 + . . .+a−(q−1)b−(q−1)+a−qb−q.

(1.13)Record (1.12) uses numerals consisting of one symbol each, i.e., digits ai ∈ 0,1,. . . ,b− 1, to express how many finite units of the type bi belong to the number(1.13). Quantities of finite units bi are counted separately for each exponent i andall symbols in the alphabet 0,1, . . . ,b−1 express finite numbers.

To express infinite and infinitesimal numbers we shall use records that are similarto (1.12) and (1.13) but have some peculiarities. In order to construct a number Cin the new numeral positional system with base ¬, we subdivide C into groupscorresponding to powers of ¬:

C = cpm¬pm + . . .+ cp1¬p1 + cp0¬p0 + cp−1¬p−1 + . . .+ cp−k ¬p−k . (1.14)

Then, the record

C = cpm¬pm . . .cp1¬p1 cp0¬p0cp−1¬p−1 . . .cp−k ¬p−k (1.15)

represents the number C, where all numerals ci = 0, they belong to a traditionalnumeral system and are called grossdigits. They express finite positive or negativenumbers and show how many corresponding units ¬pi should be added or subtractedin order to form the number C. Grossdigits can be expressed by several symbolsusing positional systems, the form Q

q where Q and q are integer numbers, or in anyother finite numeral system.

Numbers pi in (1.15) called grosspowers can be finite, infinite, and infinitesimal(the introduction of infinitesimal numbers will be given soon), they are sorted in thedecreasing order

pm > pm−1 > .. . > p1 > p0 > p−1 > .. . p−(k−1) > p−k

with p0 = 0.In the traditional record (1.12), there exists a convention that a digit ai shows how

many powers bi are present in the number and the radix b is not written explicitly.In the record (1.15), we write ¬pi explicitly because in the new numeral positionalsystem the number i in general is not equal to the grosspower pi. This gives possi-bility to write, for example, such a number as 7.6¬244.5 34¬32 having grosspowersp2 = 244.5, p1 = 32 and grossdigits c244.5 = 7.6,c32 = 34 without indicating gross-digits equal to zero corresponding to grosspowers less than 244.5 and greater than32. Note also that if a grossdigit cpi = 1 then we often write ¬pi instead of 1¬pi .

The term having p0 = 0 represents the finite part of C because, due to (1.2), wehave c0¬0 = c0. The terms having finite positive grosspowers represent the simplestinfinite parts of C. Analogously, terms having negative finite grosspowers representthe simplest infinitesimal parts of C. For instance, the number ¬−1 = 1

¬is infinites-

imal. It is the inverse element with respect to multiplication for ¬:


¬−1 ·¬ = ¬ ·¬−1 = 1. (1.16)

Note that all infinitesimals are not equal to zero. Particularly, 1¬

> 0 because it is aresult of division of two positive numbers. It also has a clear granary interpretation.Namely, if we have a sack containing ¬ seeds, then one sack divided by the numberof seeds in it is equal to one seed. Vice versa, one seed, i.e., 1

¬, multiplied by the

number of seeds in the sack, ¬, gives one sack of seeds.All of the numbers introduced above can be grosspowers, as well, giving so a

possibility to have various combinations of quantities and to construct terms havinga more complex structure7.

Example 6. The left-hand expression below shows how to write down numbers inthe new numeral system and the right-hand shows how the value of the number iscalculated:

15¬1.4¬(−17.2045)¬37¬052.1¬−6 = 15¬1.4¬ −17.2045¬3 +7¬0 +52.1¬−6.

The number above has one infinite part having the infinite grosspower, one infinitepart having the finite grosspower, a finite part, and an infinitesimal part. ⊓⊔

Finally, numbers having a finite and infinitesimal parts can be also expressed inthe new numeral system, for instance, the number −3.5¬0(−37)¬−211¬−15¬+2.3

has a finite and two infinitesimal parts, the second of them has the infinite negativegrosspower equal to −15¬+2.3.

1.4.2 Arithmetical operations

We start the description of arithmetical operations for the new positional numeralsystem by the operation of addition (subtraction is a direct consequence of additionand is thus omitted) of two given infinite numbers A and B, where

7 At the first glance the record (1.14) (and, therefore, the numerals (1.15)) can remind numbersfrom the Levi-Civita field (see [20]) that is a very interesting and important precedent of alge-braic manipulations with infinities and infinitesimals. However, the two mathematical objects haveseveral crucial differences. They have been introduced for different purposes by using two math-ematical languages having different accuracies and on the basis of different methodological foun-dations. In fact, Levi-Civita does not discuss the distinction between numbers and numerals andworks with generic numbers while each numeral (1.15) represents a concrete number. His numbershave neither cardinal nor ordinal properties; they are build using a generic infinitesimal and only itsrational powers are allowed; he uses symbol ∞ in his construction; there is no any numeral systemthat would allow one to assign numerical values to his numbers; it is not explained how it wouldbe possible to pass from d a generic infinitesimal h to a concrete one (see also the discussion aboveon the distinction between numbers and numerals).In no way the said above should be considered as a criticism with respect to results of Levi-Civita.The above discussion has been introduced in this text just to underline that we are in front of twodifferent mathematical tools that should be used in different mathematical contexts.


A =K

∑i=1

aki¬ki , B =

M

∑j=1

bm j ¬m j , C =

L

∑i=1

cli¬li , (1.17)

and the result C =A+B is constructed by including in it all items aki¬ki from A such

that ki = m j,1 ≤ j ≤ M, and all items bm j ¬m j from B such that m j = ki,1 ≤ i ≤ K. If

in A and B there are items such that ki = m j, for some i and j, then this grosspowerki is included in C with the grossdigit bki +aki , i.e., as (bki +aki)¬

ki .

Example 7. We consider two infinite numbers A and B, where

A = 16.5¬44.2(−12)¬1217¬0, B = 6.23¬310.1¬015¬−4.1.

Their sum C is calculated as follows:

C = A+B = 16.5¬44.2 +(−12)¬12 +17¬0 +6.23¬3 +10.1¬0 +15¬−4.1 =

16.5¬44.2 −12¬12 +6.23¬3 +27.1¬0 +15¬−4.1 =

16.5¬44.2(−12)¬126.23¬327.1¬015¬−4.1. ⊓⊔

The operation of multiplication of two numbers A and B in the form (1.17) re-turns, as the result, the infinite number C constructed as follows:

C =M

∑j=1

C j, C j = bm j ¬m j ·A =

K

∑i=1

aki bm j ¬ki+m j , 1 ≤ j ≤ M. (1.18)

Example 8. We consider two infinite numbers

A = 1¬18(−5)¬2.4(−3)¬1, B =−1¬10.7¬−3

and calculate the product C = B ·A. The first partial product C1 is equal to

C1 = 0.7¬−3 ·A = 0.7¬−3(¬18 −5¬2.4 −3¬1) =

0.7¬15 −3.5¬−0.6 −2.1¬−2 = 0.7¬15(−3.5)¬−0.6(−2.1)¬−2.

The second partial product, C2, is computed analogously

C2 =−¬1 ·A =−¬1(¬18 −5¬2.4 −3¬1) =−¬195¬3.43¬2.

Finally, the product C is equal to

C =C1 +C2 =−1¬190.7¬155¬3.43¬2(−3.5)¬−0.6(−2.1)¬−2. ⊓⊔

In the operation of division of a number C by a number B from (1.17), we obtaina result A and a reminder R (that can be also equal to zero), i.e., C = A ·B+R. Thenumber A is constructed as follows. The first grossdigit akK and the correspondingmaximal exponent kK are established from the equalities


akK = clL/bmM , kK = lL −mM. (1.19)

Then the first partial reminder R1 is calculated as

R1 =C−akK ¬kK ·B. (1.20)

If R1 = 0 then the number C is substituted by R1 and the process is repeated witha complete analogy. The grossdigit akK−i , the corresponding grosspower kK−i andthe partial reminder Ri+1 are computed by formulae (1.21) and (1.22) obtained from(1.19) and (1.20) as follows: lL and clL are substituted by the highest grosspower niand the corresponding grossdigit rni of the partial reminder Ri that, in turn, substi-tutes C:

akK−i = rni/bmM , kK−i = ni −mM. (1.21)

Ri+1 = Ri −akK−i¬kK−i ·B, i ≥ 1. (1.22)

The process stops when a partial reminder equal to zero is found (this means thatthe final reminder R = 0) or when a required accuracy of the result is reached.

Example 9. Let us divide the number C = −10¬316¬042¬−3 by the number B =5¬37. For these numbers we have

lL = 3, mM = 3, clL =−10, bmM = 5.

It follows immediately from (1.19) that akK ¬kK =−2¬0. The first partial reminderR1 is calculated as

R1 =−10¬316¬042¬−3 − (−2¬0) ·5¬37 =

−10¬316¬042¬−3 +10¬314¬0 = 30¬042¬−3.

By a complete analogy we should construct akK−1¬kK−1 by rewriting (1.19) for R1.By doing so we obtain equalities

30 = akK−1 ·5, 0 = kK−1 +3

and, as the result, akK−1¬kK−1 = 6¬−3. The second partial reminder is

R2 = R1 −6¬−3 ·5¬37 = 30¬042¬−3 −30¬042¬−3 = 0.

Thus, we can conclude that the reminder R = R2 = 0 and the final result of divisionis A =−2¬06¬−3.

Let us now substitute the grossdigit 42 by 40 in C and divide this new numberC = −10¬316¬040¬−3 by the same number B = 5¬37. This operation gives usthe same result A2 = A = −2¬06¬−3 (where subscript 2 indicates that two partialreminders have been obtained) but with the reminder R = R2 = −2¬−3. Thus, weobtain C = B · A2 + R2. If we want to continue the procedure of division, we obtainA3 =−2¬06¬−3(−0.4)¬−6 with the reminder R3 = 0.28¬−6. Naturally, it follows


Fig. 1.1 Operation of multiplication executed at the Infinity Calculator

C = B · A3 + R3. The process continues until a partial reminder Ri = 0 is found orwhen a required accuracy of the result will be reached. ⊓⊔

A working software simulator of the Infinity Computer has been implementedand the first application – the Infinity Calculator – has been realized. Fig. 1.1 showsoperation of multiplication executed at the Infinity Calculator that works using theInfinity Computer technology. The left operand has two infinitesimal parts and theright operand has an infinite part and a finite one.

We conclude this section by emphasizing the following important issue: the In-finity Computer works with infinite, finite, and infinitesimal numbers numerically,not symbolically (see [41]).

1.5 Examples of problems where computations with newnumerals can be useful

1.5.1 The work with infinite sequences

We start by reminding traditional definitions of the infinite sequences and subse-quences. An infinite sequence an,an ∈ A,n ∈ N, is a function having as the do-main the set of natural numbers, N, and as the codomain a set A. A subsequenceis a sequence from which some of its elements have been removed. In a sequencea1,a2, . . . ,an the number n is the number of elements of the sequence. Then, the IUAallows us to consider sequences having n that can assume different finite or infinitevalues and to prove the following result.

Theorem 2. The number of elements of any infinite sequence is less or equal to ¬.


Proof. The IUA states that the set N has ¬ elements. Thus, due to the sequencedefinition given above, any sequence having N as the domain has ¬ elements.

The notion of subsequence is introduced as a sequence from which some of itselements have been removed. Thus, this definition gives infinite sequences havingthe number of members less than grossone. ⊓⊔

One of the immediate consequences of the understanding of this result is that anysequential process can have at maximum ¬ elements. Due to Postulate 1, it dependson the chosen numeral system which numbers among ¬ members of the process wecan observe.

Example 10. For example, if we consider the set, N, of extended natural numbersthen starting from the number 1, it is possible to arrive at maximum to ¬

1,2,3,4, . . . ¬−2, ¬−1,¬︸︷︷︸¬

,¬+1,¬+2,¬+3, . . . (1.23)

Starting from 2 it is possible to arrive at maximum to ¬+1

1,2,3,4, . . . ¬−2, ¬−1,¬,¬+1︸︷︷︸¬

,¬+2,¬+3, . . . (1.24)

Starting from 3 it is possible to to arrive at maximum to ¬+2

1,2,3,4, . . . ¬−2, ¬−1,¬,¬+1,¬+2︸︷︷︸¬

,¬+3, . . . (1.25)

Of course, since we have postulated that our possibilities to express numerals arefinite, it depends on the chosen numeral system which numbers among ¬ membersof these processes we can observe. ⊓⊔

It is also very important to notice a deep relation of this observation to the Axiomof Choice. The Infinite Unit Axiom postulates that any process can have at maxi-mum ¬ elements, thus the process of choice too and, as a consequence, it is notpossible to choose more than ¬ elements from a set. This observation also empha-sizes the fact that the parallel computational paradigm is significantly different withrespect to the sequential one because p parallel processes can choose p¬ elementsfrom a set. Note also that the new more precise definition of sequences allows us toobtain a new vision of Turing machines (see [47]).

It becomes appropriate now to define the complete sequence as an infinite se-quence containing ¬ elements. For example, the sequence of natural numbers iscomplete, the sequences of even and odd natural numbers are not complete. Thus,the IUA imposes a more precise description of infinite sequences. To define a se-quence an it is not sufficient just to give a formula for an, we should determine(as it happens for sequences having a finite number of elements) the first and the lastelements of the sequence. If the number of the first element is equal to one, we can


use the record an : k where an is, as usual, the general element of the sequenceand k is the number (that can be finite or infinite) of members of the sequence.

Example 11. Let us consider the following two sequences, an and cn:

an= 5, 10, . . . 5(¬−1), 5¬,

bn= 5, 10, . . . 5(2¬

5−1), 5 · 2¬

5, (1.26)

cn= 5, 10, . . . 5(4¬

5−1), 5 · 4¬

5. (1.27)

They have the same general element an = bn = cn = 5n but they are different becausethey have different numbers of members. The first sequence has ¬ elements and isthus complete, the other two sequences are not complete: bn has 2¬

5 elements and

cn has 4¬5 members. ⊓⊔

In connection with this definition the following natural question arises inevitably.Suppose that we have two sequences, for example, bn : 2¬

5 and cn : 4¬5 from

(1.26) and (1.27). Can we create a new sequence, dn : k, composed from both ofthem, for instance, as it is shown below

b1, b2, . . . b 2¬5 −2, b 2¬

5 −1, b 2¬5, c1, c2, . . . c 4¬

5 −2, c 4¬5 −1, c 4¬

5

and which will be the value of the number of its elements k?The answer is ‘no’ because due to the definition of the infinite sequence, a se-

quence can be at maximum complete, i.e., it cannot have more than ¬ elements.Starting from the element b1 we can arrive at maximum to the element c 3¬

5being

the element number ¬ in the sequence dn : k which we try to construct. Therefore,k = ¬ and

b1, . . . b 2¬5, c1, . . .c 3¬

5︸︷︷︸¬ elements

, c 3¬5 +1, . . . c 4¬

5︸︷︷︸¬5 elements

.

The remaining members of the sequence cn : 4¬5 will form the second sequence,

gn : l having l = 4¬5 − 3¬

5 = ¬5 elements. Thus, we have formed two sequences,

the first of them is complete and the second is not.To conclude this subsection, let us return to Hilbert’s paradox of the Grand Hotel

presented in Section 1.1. In the paradox, the number of the rooms in the Hotel iscountable. In our terminology this means that it has ¬ rooms. When a new guestarrives, it is proposed to move the guest occupying room 1 to room 2, the guest oc-cupying room 2 to room 3, etc. Under the IUA this procedure does not help becausethe guest from room ¬ should be moved to room ¬+1 and the Hotel has only ¬rooms. Thus, when the Hotel is full, no more new guests can be accommodated –the result corresponding perfectly to Postulate 3 and the situation taking place innormal hotels with a finite number of rooms.


1.5.2 From divergent series to expressions evaluated at differentpoints in infinity

Let us show how the new approach can be applied in such an important area astheory of divergent series. We consider two infinite series S1 = 7+ 7+ 7+ . . . andS2 = 3+ 3+ 3+ . . . The traditional analysis gives us a very poor answer that bothof them diverge to infinity. Such operations as, e.g., S2

S1and S2 −S1 are not defined.

Now, when we are able to express not only different finite numbers but alsodifferent infinite numbers such records as S1 = a1 +a2 + . . . or ∑∞

i=1 ai become un-precise (by continuation the analogy with Piraha the record ∑∞

i=1 ai becomes a kindof ∑many

i=1 ai). It is therefore necessary to indicate explicitly the number of items inthe sums S1 and S2 and it is not important if it is finite or infinite.

We emphasize again that in order to be able to calculate a sum it is necessarythat the number of items and the result are expressible in the numeral system usedfor calculations. It is important to notice that even though a sequence cannot havemore than ¬ elements, the number of items in a series can be greater than grossonebecause the process of summing up is not necessary executed by a sequential addingitems.

Example 12. Let us consider the infinite series S1 and S2 mentioned above. In orderto use our approach, it is necessary to indicate explicitly the number of their items.

Suppose that the sum S1 has k items and S2 has n items:

S1(k) = 7+7+7+ . . .+7︸︷︷︸k

, S2(n) = 3+3+3+ . . .+3︸︷︷︸n

.

Then S1(k) = 7k and S2(n) = 3n and by giving different numerical values (finite orinfinite) to k and n we obtain different numerical values for the sums. For chosenk and n it becomes possible to calculate S2(n)−S1(k) (analogously, the expressionS1(k)S2(n)

can be calculated). If, for instance, k= 5¬ and n=¬ we obtain S1(5¬)= 35¬,S2(¬) = 3¬ and it follows

S2(¬)−S1(5¬) = 3¬−35¬ =−32¬ < 0.

If k = 3¬ and n = 7¬+2 we obtain S1(3¬) = 21¬, S2(¬) = 21¬+6 and it follows

S2(7¬+2)−S1(3¬) = 21¬+6−21¬ = 6.

It is also possible to sum up sums having an infinite number of infinite or infinitesi-mal items

S3(l) = 2¬+2¬+ . . .+2¬︸︷︷︸l

, S4(m) = 4¬−1 +4¬−1 + . . .+4¬−1︸︷︷︸m

.

For l = m = 0.5¬ it follows S3(0.5¬) = ¬2 and S4(0.5¬) = 2 (remind that ¬ ·¬−1 = ¬0 = 1 (see (1.16)). It can be seen from this example that it is possible to


obtain finite numbers as the result of summing up infinitesimals. This is a directconsequence of Postulate 3. ⊓⊔

The infinite and infinitesimal numbers allow us to calculate also arithmetic andgeometric sums with an infinite number of items. Traditional approaches tell us thatif an = a1 +(n−1)d then for a finite n it is possible to use the formula

n

∑i=1

ai =n2(a1 +an).

Due to Postulate 3, we can use it also for infinite n.

Example 13. The sum of all natural numbers from 1 to ¬ can be calculated as fol-lows

1+2+3+ . . .+(¬−1)+¬ =¬

∑i=1

i =¬

2(1+¬) = 0.5¬20.5¬. (1.28)

Let us calculate now the following sum of infinitesimals where each item is ¬ timesless than the corresponding item of (1.28)

¬−1+2¬−1+ . . .+(¬−1) ·¬−1+¬ ·¬−1 =¬

∑i=1

i¬−1 =¬

2(¬−1+1) = 0.5¬10.5.

Obviously, the obtained number, 0.5¬10.5 is ¬ times less than the sum in (1.28).This example shows, particularly, that infinite numbers can also be obtained as theresult of summing up infinitesimals. ⊓⊔

Let us consider now the geometric series ∑∞i=0 qi. Traditional analysis proves that it

converges to 11−q for q such that −1 < q < 1. We are able to give a more precise

answer for all values of q. To do this we should fix the number of items in the sum.If we suppose that it contains n items, then

Qn =n

∑i=0

qi = 1+q+q2 + . . .+qn. (1.29)

By multiplying the left hand and the right hand parts of this equality by q and bysubtracting the result from (1.29) we obtain

Qn −qQn = 1−qn+1

and, as a consequence, for all q = 1 the formula

Qn = (1−qn+1)(1−q)−1 (1.30)

holds for finite and infinite n. Thus, the possibility to express infinite and infinites-imal numbers allows us to take into account infinite n and the value qn+1 being


infinitesimal for a finite q. Moreover, we can calculate Qn for infinite and finitevalues of n and q = 1, because in this case we have just

Qn = 1+1+1+ . . .+1︸︷︷︸n+1

= n+1.

Example 14. As the first example we consider the divergent series

1+3+9+ . . .=∞

∑i=0

3i.

To fix it, we should decide the number of items, n, at the sum and, for example, forn = ¬2 we obtain

¬2

∑i=0

3i = 1+3+9+ . . .+3¬2=

1−3¬2+1

1−3= 0.5(3¬2

+1 −1).

Analogously, for n = ¬2 +1 we obtain

1+3+9+ . . .+3¬2+3¬2

+1 = 0.5(3¬2+2 −1).

If we now find the difference between the two sums

0.5(3¬2+2 −1)− (0.5(3¬2

+1 −1)) = 3¬2+1(0.5 ·3−0.5) = 3¬2

+1

we obtain the newly added item 3¬2+1. ⊓⊔

Example 15. In this example, we consider the series ∑∞i=1

12i . It is well-known that it

converges to one. However, we are able to give a more precise answer. In fact, dueto Postulate 3, the formula

n

∑i=1

12i =

12(1+

12+

122 + . . .+

12n−1 ) =

12·

1− 12

n

1− 12

= 1− 12n

can be used directly for infinite n, too. For example, if n = ¬ then

¬

∑i=1

12i = 1− 1

2¬,

where 12¬ is infinitesimal. Thus, the traditional answer ∑∞

i=112i = 1 was just a finite

approximation to our more precise result using infinitesimals. ⊓⊔

Example 16. In this example, we consider divergent series with alternate signs. Letus start from the famous series

S5 = 1−1+1−1+1−1+ . . .


In literature there exist many approaches giving different answers regarding thevalue of this series (see [18]). All of them use various notions of average. How-ever, the notions of sum and average are different. In our approach we do not appealto average and calculate the required sum directly. To do this we should indicateexplicitly the number of items, k, in the sum. Then

S5(k) = 1−1+1−1+1−1+1− . . .︸︷︷︸k

=

0, if k = 2n,1, if k = 2n+1,

and it is not important is k finite or infinite. For example, S5(¬) = 0 because thenumber ¬

2 being the result of division of ¬ by 2 has been introduced as the numberof elements of a set and, therefore, it is integer. As a consequence, ¬ is even number.Analogously, S5(¬−1) = 1 because ¬−1 is odd. ⊓⊔

It is important to emphasize that, as it happens in the case of the finite numberof items in a sum, the obtained answers do not depend on the way the items in theentire sum are re-arranged. In fact, if we know the exact infinite number of items inthe sum and the order of alternating the signs is clearly defined, we know also theexact number of positive and negative items in the sum.

Let us illustrate this point by supposing, for instance, that we want to re-arrangethe items in the sum S1(2¬) in the following way

S1(2¬) = 1+1−1+1+1−1+1+1−1+ . . .

However, we know that the sum has 2¬ items and the number 2¬ is even. Thismeans that in the sum there are ¬ positive and ¬ negative items. As a result, the re-arrangement considered above can continue only until the positive items present inthe sum will not finish and then it will be necessary to continue to add only negativenumbers. More precisely, we have

S1(2¬) = 1+1−1+1+1−1+ . . .+1+1−1︸︷︷︸¬ positive and ¬

2 negative items

−1−1− . . .−1−1−1︸︷︷︸¬2 negative items

= 0,

where the result of the first part in this re-arrangement is calculated as (1+1−1) ·¬

2 = ¬

2 and the result of the second part is equal to − ¬

2 .

Example 17. Let us consider now the following divergent series

S6 = 1−2+3−4+ . . .

It can be easily considered as the difference of two arithmetic progressions afterwe have fixed the number of items, k, in the sum S6(k). Suppose that it containsgrossone items. Then it follows

S6(¬) = 1−2+3−4+ . . .− (¬−2)+(¬−1)−¬ =


(1+3+5+ . . .+(¬−3)+(¬−1))− (2+4+6+ . . .+(¬−2)+¬) =

(1+¬−1)¬4

− (2+¬)¬

4=

¬2 −2¬−¬2

4=−¬

2. ⊓⊔

1.5.3 Calculating limits and expressing irrational numbers

Let us now discuss the problem of calculation of limits from the point of view of ourapproach. In traditional analysis, if a limit limx→a f (x) exists, then it gives us a verypoor – just one value – information about the behavior of f (x) when x tends to a.Now we can obtain significantly richer information because we are able to calculatef (x) directly at any finite, infinite, or infinitesimal point that can be expressed bythe new positional system even if the limit does not exist.

Thus, limits equal to infinity can be substituted by precise infinite numerals andlimits equal to zero can be substituted by precise infinitesimal numerals8. This isvery important for practical computations because these substitutions eliminate in-determinate forms.

Example 18. Let us consider the following two limits

limx→+∞

(5x3 − x2 +1061) = +∞, limx→+∞

(5x3 − x2) = +∞.

Both give us the same result, +∞, and it is not possible to execute the operation

limx→+∞

(5x3 − x2 +1061)− limx→+∞

(5x3 − x2).

that is an indeterminate form of the type ∞−∞ in spite of the fact that for any finitex it follows

5x3 − x2 +1061 − (5x3 − x2) = 1061. (1.31)

The new approach allows us to calculate exact values of both expressions, 5x3 −x2 + 1061 and 5x3 − x2 + 10, at any infinite (and infinitesimal) x expressible in thechosen numeral system. For instance, the choice x = 3¬2 gives the value

5(3¬2)3 − (3¬2)2 +1061 = 135¬6-9¬41061

for the first expression and 135¬6-9¬4 for the second one. We can easily calculatethe difference of these two infinite numbers, thus obtaining the same result as wehad for finite values of x in (1.31):

135¬6-9¬41061 − (135¬6-9¬4) = 1061. ⊓⊔

8 Naturally, if we speak about limits of sequences, limn→∞ a(n), then n ∈N and, as a consequence,it follows that n should be less than or equal to grossone.


An additional advantage of the usage of the Infinity Computer for calculatinglimits arises in the following situations. Suppose that we have a computer procedurecalculating f (x), we do not know the corresponding analytic formulae for f (x), for acertain argument a the value f (a) is not defined (or a traditional computer producesan overflow or underflow message), and it is necessary to calculate the limx→a f (x).Traditionally, this situation requires a human intervention and an additional theoret-ical investigation whereas the Infinity Computer is able to process it automaticallyworking numerically with the expressions involved in the procedure. It is sufficientto calculate f (x), for example, at a point x = a+¬−1 in cases of finite a or a = 0and x = ¬ in the case when we are interested in the behavior of f (x) at infinity.Obviously, if the limit does not exist but there exist limits from the right and fromthe left, it is sufficient to calculate x = a+¬−1 and x = a−¬−1, respectively.

Example 19. Suppose that we have two procedures evaluating f (x) = x2+2xx and

g(x)= 34x . Obviously, f (0) and g(0) are not defined and it is not possible to calculate

limx→0 f (x), limx→∞ f (x) and limx→0 g(x), limx→∞ g(x) using traditional computers.Then, suppose that we are interested in evaluating the expression

h(x) = ( f (x)−2) ·g(x).

It is easy to see that h(x) = 34 for any finite value of x. On the other hand, thefollowing limits

limx→0

h(x) = (limx→0

f (x)−2) · limx→0

g(x),

limx→∞

h(x) = ( limx→∞

f (x)−2) · limx→∞

g(x)

cannot be evaluated. The Infinity Computer can calculate h(x) numerically for dif-ferent infinitesimal and infinite values of x obtaining the same result that takes placefor finite x. For example, it follows

h(¬−1) =

((¬−1)2 +2¬−1

¬−1 −2

)· 34¬−1 = (¬−1 +2−2) ·34¬ = 34,

h(¬)=

(¬2 +2¬

¬−2

)· 34

¬=(¬+2−2)·34¬−1 = 34. ⊓⊔

It is necessary to emphasize the fact that expressions can be calculated even whentheir limits do not exist. Thus, we obtain a very powerful tool for studying divergentprocesses.

Example 20. The limit limn→+∞ f (n), f (n) = (−1)nn3, does not exist. However, wecan easily calculate expression (−1)nn3 at different infinite points n. For instance,for n = ¬ it follows f (¬) = ¬3 because grossone is even and for the odd n =0.5¬−1 it follows

f (0.5¬−1)=−(0.5¬−1)3 =−0.125¬30.75¬2-1.5¬11. ⊓⊔


Limits with the argument tending to zero can be considered analogously. In thiscase, we can calculate the corresponding expression at any infinitesimal point usingthe new positional system and obtain a significantly more reach information.

Example 21. If x is a fixed finite number then

limh→0

(x+h)2 − x2

h= 2x. (1.32)

In the new positional system we obtain

(x+h)2 − x2

h= 2x+h. (1.33)

If, for instance, h=¬−1, the answer is 2x¬0¬−1, if h= 4.2¬−2 we obtain the value2x¬04.2¬−2, etc. Thus, the value of the limit (1.32), for a finite x, is just the finiteapproximation of the number (1.33) having finite and infinitesimal parts. ⊓⊔

Let us make a remark regarding irrational numbers. Among their properties, theyare characterized by the fact that we do not know any numeral system that wouldallow us to express them by a finite number of symbols used to express other num-bers. Thus, special numerals (e,π,

√2,√

3, etc.) are introduced by describing theirproperties in a way (similarly, all other numerals, e.g., symbols ‘0’ or ‘1’, are intro-duced also by describing their properties). These special symbols are then used inanalytical transformations together with ordinary numerals.

For example, it is possible to work directly with the symbol e in analytical trans-formations by applying suitable rules defining this number together with numeralstaking part in a chosen numeral system S . At the end of transformations, the ob-tained result will be be expressed in numerals from S and, probably, in terms of e.If it is then required to execute some numerical computations, this means that it isnecessary to substitute e by a numeral (or numerals) from S that will allow us toapproximate e in some way.

The same situation takes place when one uses the new numeral system, i.e., whilewe work analytically we use just the symbol e in our expressions and then, if wewish to work numerically we should pass to approximations. The new numeral sys-tem opens a new perspective on the problem of the expression of irrational numbers.Let us consider one of the possible ways to obtain an approximation of e, i.e., byusing the limit

e = limn→+∞

(1+1n)n = 2.71828182845904 . . . (1.34)

In our numeral system the expression (1 + 1n )

n can be written directly for finiteand/or infinite values of n. For n = ¬ we obtain the number e0 designated so inorder to distinguish it from the record (1.34)

e0 = (1+1¬)¬ = (¬0¬−1)¬. (1.35)


It becomes clear from this record why the number e cannot be expressed in a posi-tional numeral system with a finite base. Due to the definition of a sequence underthe IUA, such a system can have at maximum ¬ numerals – digits – to express frac-tional part of a number (see section 1.5.5 for details) and, as it can be seen from(1.35), this quantity is not sufficient for e because the item 1

¬¬ is present in it.Naturally, it is also possible to construct more exotic e-type numbers by substi-

tuting ¬ in (1.35) by any infinite number written in the new positional system withinfinite base. For example, if we substitute ¬ in (1.35) by ¬2 we obtain the number

e1 = (1+1

¬2 )¬

2= (¬0¬−2)¬

2.

The numbers considered above take their origins in the limit (1.34). Similarly, otherformulae leading to approximations of e expressed in traditional numeral systemsgive us other new numbers that can be expressed in the new numeral system. Thesame way of reasoning can be used with respect to other irrational numbers, too.

1.5.4 Measuring infinite sets with elements defined by formulae

We have already discussed in Section 1.3 how we calculate the number of elementsfor sets being results of the usual operations (intersection, union, etc.) with finite setsand infinite sets of the type Nk,n. In order to have a possibility to work with infinitesets having a more general structure than the sets Nk,n, we need to develop morepowerful instruments. Suppose that we have an integer function g(i) > 0 strictlyincreasing on indexes i = 1,2,3, . . . and we wish to know how many elements arethere in the set

G = g(1),g(2),g(3), . . ..

In our terminology this question has no any sense because of the following reason.In the finite case, to define a set it is not sufficient to say that it is finite. It is

necessary to indicate its number of elements explicitly as, e.g., in this example

G1 = g(i) : 1 ≤ i ≤ 5,

or implicitly, as it is made here:

G2 = g(i) : i ≥ 1, 0 < f (i)≤ b, (1.36)

where b is finite.Now we have mathematical tools to indicate the number of elements for infinite

sets, too. Thus, analogously to the finite case and due to Postulate 3, it is not suf-ficient to say that a set has infinitely many elements. It is necessary to indicate itsnumber of elements explicitly or implicitly. For instance, the number of elements ofthe set

G3 = g(i) : 1 ≤ i ≤ ¬10


is indicated explicitly: the set G3 has ¬10 elements.If a set is given in the form (1.36) where b is infinite, then its number of elements,

J, can be determined asJ = maxi : g(i)≤ b (1.37)

if we are able to determine the inverse function g−1(x) for g(x). Then, J = [g−1(b)],where [u] is integer part of u. Note that if b = ¬, then the set G2 ⊆ N since all itselements are integer, positive, and g(i)≤ ¬ due to (1.37).

Example 22. Let us consider the following set, A1(k,n), having g(i) = k+n(i−1),

A1(k,n) = g(i) : i ≥ 1, g(i)≤ ¬, 1 ≤ k ≤ n, n ∈ N.

It follows from the IUA that A1(k,n) = Nk,n from (1.3). By applying (1.37) we findfor A1(k,n) its number of elements

J1(k,n) = [¬−kn +1] = [¬−k

n ]+1 = ¬n −1+1 = ¬

n . ⊓⊔

Example 23. Analogously, the set

A2(k,n, j) = k+ni j : i ≥ 0, 0 < k+ni j ≤ ¬, 0 ≤ k < n, n ∈ N, j ∈ N,

has J2(k,n, j) = [j√

¬−kn ] elements. ⊓⊔

1.5.5 Measuring infinite sets of numerals and their comparison

Let us calculate the number of elements in some well-known infinite sets of numer-als using the designation |A| to indicate the number of elements of a set A.

Theorem 3. The number of elements of the set, Z, of integers is |Z|= 2¬1.

Proof. The set Z contains ¬ positive numbers, ¬ negative numbers, and zero. Thus,

|Z|= ¬+¬+1 = 2¬1. ⊓⊔

Traditionally, rational numbers are defined as ratio of two integer numbers. Thenew approach allows us to calculate the number of numerals in a fixed numeralsystem. Let us consider a numeral system Q1 containing numerals of the form

pq, p ∈ Z, q ∈ Z, q = 0. (1.38)

Theorem 4. The number of elements of the set, Q1, of rational numerals of the type(1.38) is |Q1|= 4¬22¬1.

Proof. It follows from Theorem 3 that the numerator of (1.38) can be filled in by2¬1 and the denominator by 2¬ numbers. Thus, number of all possible combina-tions is


|Q1|= 2¬1 ·2¬ = 4¬22¬1. ⊓⊔

It is necessary to notice that in Theorem 4 we have calculated different numeralsand not different numbers. For example, in the numeral system Q1 the number 0 canbe expressed by 2¬ different numerals

0−¬

,0

−¬+1,

0−¬+2

, . . .0−2

,0−1

,01,

02, . . .

0¬-2

,0

¬-1,

0¬

and numerals such as −1−2 and 1

2 have been calculated as two different numerals.The following theorem determines the number of elements of the set Q2 containingnumerals of the form

− pq,

pq, p ∈ N, q ∈ N, (1.39)

and zero is represented by one symbol 0.

Theorem 5. The number of elements of the set, Q2, of rational numerals of the type(1.39) is |Q2|= 2¬21.

Proof. Let us consider positive rational numerals. The form of the rational numeralpq , the fact that p, q ∈ N, and the IUA impose that both p and q can assume values

from 1 to ¬. Thus, the number of all possible combinations is ¬2. The same numberof combinations we obtain for negative rational numbers and one is added becausewe count zero as well. ⊓⊔

Let us now calculate the number of elements of the set, Rb, of real numbersexpressed by numerals in the positional system by the record

(an−1an−2 . . .a1a0.a−1a−2 . . .a−(q−1)a−q)b (1.40)

where the symbol b indicates the radix of the record and n, q ∈ N.

Theorem 6. The number of elements of the set, Rb, of numerals (1.40) is |Rb| =b2¬.

Proof. In formula (1.40) defining the type of numerals we deal with there are twosequences of digits: the first one, an−1an−2 . . .a1a0, is used to express the integer partof the number and the second, a−1a−2 . . .a−(q−1)a−q, for its fractional part. Due todefinition of sequence and the IUA, each of them can have at maximum ¬ elements.Thus, it can be at maximum ¬ positions on the left of the dot and, analogously, ¬positions on the right of the dot. Every position can be filled in by one of the b digitsfrom the alphabet 0,1, . . . ,b− 1. Thus, we have b¬ combinations to express theinteger part of the number and the same quantity to express its fractional part. Asa result, the positional numeral system using the numerals of the form (1.40) canexpress b2¬ numbers. ⊓⊔

Note that the result of theorem 6 does not consider the practical situation ofwriting down concrete numerals. Obviously, the number of numerals of the type


(1.40) that can be written in practice is finite and depends on the chosen numeralsystem for writing digits.

It is worthwhile to notice also that all the numerals numerals of the type (1.40)represent different numbers. In addition, minimal and maximal numbers expressiblein Rb can be explicitly indicated.

Example 24. For instance, in the decimal positional system R10 the numerals

1.999 . . .99︸︷︷︸¬ digits

, 2.000 . . .00︸︷︷︸¬ digits

represent different numbers and their difference is equal to

2.000 . . .00︸︷︷︸¬ digits

−1.999 . . .9︸︷︷︸¬ digits

= 0.000 . . .01︸︷︷︸¬ digits

.

Analogously the smallest and the largest numbers expressible in R10 can be easilyindicated. They are, respectively,

−999 . . .9︸︷︷︸¬ digits

.999 . . .9︸︷︷︸¬ digits

, 999 . . .9︸︷︷︸¬ digits

.999 . . .9︸︷︷︸¬ digits

. ⊓⊔

On the other hand, the traditional point of view on real numbers tells that thereexist real numbers that can be represented in positional systems by two different in-finite sequences of digits, for instance, in the decimal positional system the records2.000000 . . . and 1.99999 . . . represent the same number. Note that there is no anycontradiction between the traditional and the new points of view. They just use dif-ferent lens in their mathematical microscopes to observe numbers. The instrumentsused in the traditional point of view for this purpose was just too weak to distinguishtwo different numbers in the records 2.000000 . . . and 1.99999 . . ..

Note that traditionally it was accepted that any positional numeral system is ableto represent all real numbers (‘the whole real line’). In this section, we have shownthat any numeral system is just an instrument that can be used to observe certainreal numbers. This instrument can be more or less powerful, e.g., the positionalsystem (1.40) with the radix 10 is more powerful than the positional system (1.40)with the radix 2 but neither of the two is able to represent irrational numbers. Twonumeral systems can allow us to observe either the same sets of numbers, or sets ofnumbers having an intersection, or two disjoint sets of numbers. Due to Postulate 2,we are not able to answer the question ‘What is the whole real line?’ because thisis the question asking ‘What is the object of the observation?’, we are able just toinvent more and more powerful numeral systems that will allow us to improve ourobservations of numbers by using newly introduced numerals.

Theorem 7. The sets Z,Q1,Q2, and Rb are not monoids under addition.

Proof. The proof is obvious and is so omitted. ⊓⊔


1.6 Relations to results of Georg Cantor

We start this subsection by calculating the number of points at the interval [0,1). Todo this we need a definition of the term ‘point’ and mathematical tools to indicate apoint. Since this concept is one of the most fundamental, it is very difficult to findan adequate definition for it. If we accept (as is usually done in modern mathemat-ics) that a point in [0,1) is determined by a numeral x called the coordinate of thepoint where x ∈ S and S is a set of numerals, then we can indicate the point by itscoordinate x and are able to execute required calculations.

It is important to emphasize that we have not postulated that x belongs to the set,R, of real numbers as it is usually done. Since we can express coordinates only bynumerals, then different choices of numeral systems lead to various sets of numeralsand, as a consequence, to different sets of points we can refer to. The choice of anumeral system will define what is the point for us and we shall not be able towork with those points which coordinates are not expressible in the chosen numeralsystem (remind Postulate 2). Thus, we are able to calculate the number of points ifwe have already decided which numerals will be used to express the coordinates ofpoints.

Different numeral systems can be chosen to express coordinates of the pointsin dependence on the precision level we want to obtain. For example, Piraha arenot able to express any point. If the numbers 0 ≤ x < 1 are expressed in the formp−1¬

, p ∈ N, then the smallest positive number we can distinguish is 1¬

and theinterval [0,1) contains the following points

0,1¬,

2¬, . . .

¬−2¬

,¬−1

¬. (1.41)

It is easy to see that they are ¬. If we want to count the number of intervals of theform [a−1,a),a ∈N, on the ray x ≥ 0, then, due to Postulate 3, the definition of se-quence, and Theorem 2, not more than ¬ intervals of this type can be distinguishedon the ray x ≥ 0. They are

[0,1), [1,2), [2,3), . . . [¬−3,¬−2), [¬−2,¬−1), [¬−1,¬).

Within each of them we are able to distinguish ¬ points and, therefore, at the entireray ¬2 points can be observed. Analogously, the ray x < 0 is represented by theintervals

[−¬,−¬+1), [−¬+1,−¬+2), . . . [−2,−1), [−1,0).

Hence, this ray also contains ¬2 such points and on the whole line 2¬2 points ofthis type can be represented and observed.

Note that the point −¬ is included in this representation and the point ¬ is ex-cluded from it. Let us slightly modify our numeral system in order to have ¬ rep-resentable. For this purpose, intervals of the type (a− 1,a],a ∈ N, should be con-sidered to represent the ray x > 0 and the separate symbol, 0, should be used to


represent zero. Then, on the ray x > 0 we are able to observe ¬2 points and, anal-ogously, on the ray x < 0 we also are able to observe ¬2 points. Finally, by addingthe symbol used to represent zero we obtain that on the entire line 2¬2 + 1 pointscan be observed.

It is important to stress that the situation with counting points is a direct conse-quence of Postulate 2 and is typical for natural sciences where it is well-known thatinstruments influence results of observations. It is similar to the work with micro-scope or fractals (see [26]): we decide the level of the precision we need and obtaina result dependent on the chosen level of accuracy. If we need a more precise or amore rough answer, we change the lens of our microscope.

In our terms this means to change one numeral system with another. For instance,instead of the numerals considered above, let us choose a positional numeral systemwith the radix b

(.a1a2 . . .aq−1aq)b, q ∈ N, (1.42)

to calculate the number of points within the interval [0,1).

Theorem 8. The number of elements of the set of numerals of the type (1.42) isequal to b¬.

Proof. Formula (1.42) defining the type of numerals we deal with contains a se-quence of digits a1a2 . . .aq−1aq. Due to the definition of the sequence and Theo-rem 2, this sequence can have at maximum ¬ elements, i.e., q ≤ ¬. Thus, it can beat maximum ¬ positions on the the right of the dot. Every position can be filled inby one of the b digits from the alphabet 0,1, . . . ,b−1. Thus, we have b¬ combi-nations. As a result, the positional numeral system using the numerals of the form(1.42) can express b¬ numbers. ⊓⊔

Corollary 1. The entire line contains 2¬b¬ points of the type (1.42).

Proof. We have already seen above that it is possible to distinguish 2¬ unit intervalswithin the line. Thus, the whole number of points of the type (1.42) on the line isequal to 2¬b¬. ⊓⊔

In this example of counting, we have changed the tool to calculate the numberof points within each unit interval from (1.41) to (1.42), but used the old way tocalculate the number of intervals, i.e., by natural numbers. If we are not interestedin subdividing the line at intervals and want to obtain the number of the points onthe line directly by using positional numerals of the type (1.40). Then, as it hasalready has been established in Theorem 6, the number of points expressible by thenumerals (1.40) is |Rb|= b2¬.

It is obligatory to say in this occasion that the results presented above should beconsidered as a more precise analysis of the situation discovered by the genius ofCantor. He has proved, by using his famous diagonal argument, that the number ofelements of the set N is less than the number of real numbers at the interval [0,1)without calculating the latter. To do this he expressed real numbers in a positional


Fig. 1.2 Due to Cantor, the interval (0,1) and the entire real number line have the same number ofpoints

numeral system. We have shown that this number will be different depending on theradix b used in the positional system (1.42) to express real numbers. However, all ofthe obtained numbers, b¬, are more than the number of elements of the set of naturalnumbers, ¬, and, therefore, the diagonal argument maintains its force.

Let us now return to the problem of comparison of infinite sets and considerCantor’s famous result showing that the number of points over the interval (0,1) isequal to the number of points over the whole real line, i.e.,

|R|= |(0,1)|. (1.43)

The proof of this counterintuitive fact is given by establishing a one-to-one corre-spondence between the elements of the two sets. Such a mapping can be done byusing for example the function

y = tan(0.5π(2x−1)), x ∈ (0,1), (1.44)

illustrated in Fig. 1.2. Cantor shows by using Fig. 1.2 that to any point x ∈ (0,1)a point y ∈ (−∞,∞) can be associated and vice versa. Thus, he concludes that therequested one-to-one correspondence between the sets R and (0,1) has been estab-lished and, therefore, this proves (1.43).

Our point of view is different: the number of elements is an intrinsic characteris-tic of each set (for both finite and infinite cases) that does not depend on any objectoutside the set. Thus, in Cantor’s example from Fig. 1.2 we have (see Fig. 1.3) threemathematical objects: (i) a set, XS1 , of points over the interval (0,1) which we areable to distinguish using a numeral system S1; (ii) a set, YS2 , of points over the ver-tical real line which we are able to distinguish using a numeral system S2; (iii) thefunction (1.44) described using a numeral system S3. All these three mathemati-


Fig. 1.3 Three independent mathematical objects: the set XS1 represented by dots, the set YS2 rep-resented by stars, and function (1.44)

cal objects are independent each other. The sets XS1 and YS2 can have the same ordifferent number of elements.

Thus, we are not able to evaluate f (x) at any point x. We are able to do thisonly at points from XS1 . Of course, in order to be able to execute these evaluationsit is necessary to conciliate the numeral systems S1,S2, and S3. The fact that wehave made evaluations of f (x) and have obtained the corresponding values does notinfluence minimally the numbers of elements of the sets XS1 and YS2 . Moreover, itcan happen that the number y = f (x) cannot be expressed in the numeral system S2and it is necessary to approximate it by a number y ∈ S2. This situation, very wellknown to computer scientists, is represented in Fig. 1.3.

Let us remind one more famous example related to the one-to-one correspon-dence and taking its origins in studies of Galileo Galilei: even numbers can be putin a one-to-one correspondence with all natural numbers in spite of the fact that theyare a part of them:

even numbers: 2, 4, 6, 8, 10, 12, . . .

natural numbers: 1, 2, 3, 4 5, 6, . . .(1.45)

Again, our view on this situation is different since we cannot establish a one-to-one correspondence between the sets because they are infinite and we, due toPostulate 1, are able to execute only a finite number of operations. We cannot usethe one-to-one correspondence as an executable operation when it is necessary towork with infinite sets.

However, we already know that the number of elements of the set of naturalnumbers is equal to ¬ and ¬ is even. Since the number of elements of the set of


even numbers is equal to ¬2 , we can write down not only initial (as it is usually

done traditionally) but also the final part of (1.45)

2, 4, 6, 8, 10, 12, . . . ¬−4, ¬−2, ¬ 1, 2, 3, 4 5, 6, . . . ¬

2 −2, ¬2 −1, ¬

2

(1.46)

concluding so (1.45) in a complete accordance with Postulate 3. Note that record(1.46) does not affirms that we have established the one-to-one correspondenceamong all even numbers and a half of natural ones. We cannot do this due to Postu-late 1. The symbols ‘. . .’ indicate an infinite number of numbers and we can executeonly a finite number of operations. However, record (1.46) affirms that for any evennumber expressible in the chosen numeral system it is possible to indicate the cor-responding natural number in the lower row of (1.46).

We conclude this section by the following remark. With respect to our method-ology, the mathematical results obtained by Piraha, Cantor, and those presented inthis chapter do not contradict to each other. They all are correct with respect tomathematical languages used to express them. This relativity is very important andit has been emphasized in Postulate 2. For instance, the result of Piraha 1+2=‘many’is correct in their language in the same way as the result 1+2=3 is correct in themodern mathematical languages. Analogously, the result (1.45) is correct in Can-tor’s language and the more powerful language developed in this chapter allows usto obtain a more precise result (1.46) that is correct in the new language.

The choice of the mathematical language depends on the practical problem thatare to be solved and on the accuracy required for such a solution. Again, the result ofPiraha ‘many’+1=‘many’ is correct. If one is satisfied with its accuracy, the answer‘many’ can be used (and is used by Piraha) in practice. However, if one needs a moreprecise result, it is necessary to introduce a more powerful mathematical language(a numeral system in this case) allowing one to express the required answer in amore accurate way.

1.7 New computational possibilities for mathematical modelling

The computational capabilities of the Infinity Computer allow one to construct newand more powerful mathematical models able to take into account infinite and in-finitesimal changes of parameters. In this section, the main attention is given toinfinitesimals that can increase the accuracy of models and computations, in gen-eral. It is shown that the introduced infinitesimal numerals and the formalization ofthe concept ‘point’ given in the previous sections can be successfully used in prac-tical calculations. Examples related to computations of probabilities and areas (andvolumes) of objects having several parts of different dimensions are given.

It becomes also possible in several occasions to automatize the process of thesolving of computational problems avoiding an interruption of the work of computer


procedures and the necessity of a human intervention required when one workswith traditional computers. It is necessary to emphasize that the examples describedin this section are related to numerical computations at the Infinity Computer. Nosymbolic computations are required to work with infinite and infinitesimal numberswhen one uses the Infinity Computer.

1.7.1 Usage of infinitesimals for solving systems of linearequations

Very often in computations, an algorithm performing calculations encounters a sit-uation where the problem to divide by zero occurs. Then, obviously, this operationcannot be executed. If it is known that the problem under consideration has a solu-tion, then a number of additional computational steps trying to avoid this divisionis performed. A typical example of this kind is the operation of pivoting used whenone solves systems of linear equations by an algorithm such as Gauss-Jordan elim-ination. Pivoting is the interchanging of rows (or both rows and columns) in orderto avoid division by zero and to place a particularly ‘good’ element in the diagonalposition prior to a particular operation.

The following two simple examples give just an idea of a numerical usage ofinfinitesimals and show that the usage of infinitesimals can help to avoid pivoting incases when the pivotal element is equal to zero. We emphasize again that the InfinityComputer (see [41]) works with infinite and infinitesimal numbers expressed in thepositional numeral system (1.14), (1.15) numerically, not symbolically.

Example 25. Solution to the system[0 12 2

] [x1x2

]=

[22

]is obviously given by x∗1 = −1, x∗2 = 2. It cannot be found by the method of Gausswithout pivoting since the first pivotal element a11 = 0.

Since all the elements of the matrix are finite numbers, let us substitute the ele-ment a11 = 0 by ¬−1 and perform exact Gauss transformations without pivoting:[

¬−1 1 22 2 2

]→[

1 ¬ 2¬0 −2¬+2 −4¬+2

]→

[1 ¬ 2¬

0 1 −4¬+2−2¬+2

][

1 0 2¬−¬ · −4¬+2−2¬+2

0 1 −4¬+2−2¬+2

]→

[1 0 2¬

−2¬+2

0 1 −4¬+2−2¬+2

]→

[1 0 −1+ 1

1−¬0 1 2− 1

1−¬

].

It follows immediately that the solution to the initial system is given by the finiteparts of numbers −1+ 1

1−¬and 2− 1

1−¬.


We have introduced the number ¬−1 once and, as a result, we have obtainedexpressions where the maximal power of grossone is one and there are rational ex-pressions depending on grossone, as well. It is possible to manage these rationalexpressions in two ways: (i) to execute division in order to obtain its result in theform (1.14), (1.15); (ii) without executing division. In the latter case, we just con-tinue to work with rational expressions. In the case (i), since we need finite numbersas final results, in the result of division it is not necessary to store the parts cp¬p

with p < −1. These parts can be forgotten because in any way the result of theirsuccessive multiplication with the numbers of the type c1¬1 (remind that 1 is themaximal exponent present in the matrix under consideration) will give exponentsless than zero, i.e., numbers with these exponents will be infinitesimals that are notinteresting for us in this computational context.

Thus, by using the positional numeral system (1.14), (1.15) with the radixgrossone we obtain [

1 ¬ 2¬

0 1 −4¬+2−2¬+2

]→[

1 ¬ 2¬

0 1 2¬0+1¬−1

][

1 0 2¬−¬ · (2¬01¬−1)

0 1 2¬01¬−1

]→[

1 0 −1¬0

0 1 2¬01¬−1

].

The finite parts of numbers −1¬0 and 2¬01¬−1, i.e., −1 and 2 respectively, thenprovide the required solution. 2

Example 26. Solution to the system0 0 12 0 −11 2 3

x1x2x3

=

131

is the following: x∗1 = 2, x∗2 =−2, and x∗3 = 1. The coefficient matrix of this systemhas the first two leading principal minors equal to zero. Consequently, the first twopivots, in the Gauss transformations, are zero. We solve the system without pivotingby substituting the zero pivot by ¬−1, when necessary.

Let us show how the exact computations are executed:0 0 1 12 0 −1 31 2 3 1

→

1 0 ¬ ¬0 0 −2¬−1 −2¬+30 2 −¬+3 −¬+1

1 0 ¬ ¬

0 1 −2¬2 −¬ −2¬2 +3¬0 2 −¬+3 −¬+1

→

1 0 ¬ ¬

0 1 −2¬2 −¬ −2¬2 +3¬

0 0 4¬2 +¬+3 4¬2 −7¬+1


1 0 ¬ ¬

0 1 −2¬2 −¬ −2¬2 +3¬

0 0 1 4¬2−7¬+14¬2

+¬+3

→

1 0 0 8¬2

+2¬4¬2

+¬+3

0 1 0 −8¬2+10¬

4¬2+¬+3

0 0 1 4¬2−7¬+14¬2

+¬+3

It is easy to see that the finite parts of the numbers

x∗1 =8¬2 +2¬

4¬2 +¬+3= 2− 6

4¬2 +¬+3,

x∗2 =−8¬2 +10¬

4¬2 +¬+3=−2+

12¬+64¬2 +¬+3

,

x∗3 =4¬2 −7¬+14¬2 +¬+3

= 1− 8¬+24¬2 +¬+3

,

coincide with the corresponding solution x∗1 = 2, x∗2 =−2, and x∗3 = 1.In this procedure we have introduced the number ¬−1 two times. As a result,

we have obtained expressions where the maximal power of grossone is equal to 2and there are rational expressions depending on grossone, as well. By reasoninganalogously to Example 26, when we execute divisions, in the obtained results itis not necessary to store the parts of the type cp¬p, p < −2, because in any waythe result of their successive multiplication with the numbers of the type c2¬2 willgive finite exponents less than zero. That is, numbers with these exponents will beinfinitesimals that are not interesting for us in this computational context. Thus, byusing the positional numeral system (1.14), (1.15), we obtain1 0 ¬ ¬

0 1 −2¬2 −¬ −2¬2 +3¬

0 0 1 4¬2−7¬+14¬2

+¬+3

→

1 0 ¬ ¬

0 1 −2¬2 −¬ −2¬2 +3¬

0 0 1 1¬0-2¬−1

.Note that the number 1¬0-2¬−1 does not contain the part of the type c−2¬−2 be-cause the coefficient c−2 obtained after the executed division is such that c−2 = 0.Then we proceed as follows1 0 ¬ ¬

0 1 0 −20 0 1 1¬0-2¬−1

→

1 0 0 20 1 0 −20 0 1 1¬0-2¬−1

.The obtained solutions x∗1 = 2 and x∗2 = −2 have been obtained exactly withoutinfinitesimal parts and x∗3 = 1 is derived from the finite part of 1¬0-2¬−1. 2

We conclude this section by emphasizing that zero pivots in the matrix are sub-stituted dynamically by ¬−1. Thus, the number of the introduced infinitesimals ¬−1

depends on the number of zero pivots.


Fig. 1.4 What is the probability that the rotating disk stops in such a way that the point A will beexactly in front of the arrow?

1.7.2 Applications in probability theory and calculating volumes

A formalization of the concept ‘point’ introduced above allows us to execute moreaccurately computations having relations with this concept. Very often in scien-tific computing and engineering it is required to construct mathematical models formulti-dimensional objects. Usually this is done by partitioning the modelled objectin several parts having the same dimension and each of the parts is modelled sepa-rately. Then additional efforts are made in order to provide a correct functioning ofa model unifying the obtained sub-models and describing the entire object.

Another interesting applied area is linked to stochastic models dealing withevents having probability equal to zero. In this subsection, we first show that thenew approach allows us to distinguish the impossible event having the probabilityequal to zero (i.e., P(∅) = 0) and events having an infinitesimal probability. Thenwe show how infinitesimals can be used in calculating volumes of objects consistingof parts having different dimensions.

Let us consider the problem presented in Fig. 1.4 from the traditional point ofview of probability theory. We start to rotate a disk having radius r with the point Amarked at its border and we would like to know the probability P(E) of the followingevent E: the disk stops in such a way that the point A will be exactly in front of thearrow fixed at the wall. Since the point A is an entity that has no extent it is calculatedby considering the following limit

P(E) = limh→0

h2πr

= 0.

where h is an arc of the circumference containing A and 2πr is its length.However, the point A can stop in front of the arrow, i.e., this event is not impos-

sible and its probability should be strictly greater than zero, i.e., P(E)> 0. The newapproach allows us to calculate this probability numerically.

First of all, in order to state the experiment more rigorously, it is necessary tochoose a numeral system to express the points on the circumference. This choice


will fix the number of points, K, that we are able to distinguish on the circumfer-ence. Definition of the notion point allows us to define elementary events in ourexperiment as follows: the disk has stopped and the arrow indicates a point. As aconsequence, we obtain that the number, N(Ω), of all possible elementary events,ei, in our experiment is equal to K where Ω =∪N(Ω)

i=1 ei is the sample space of our ex-periment. If our disk is well balanced, all elementary events are equiprobable and,therefore, have the same probability equal to 1

N(Ω) . Thus, we can calculate P(E)directly by subdividing the number, N(E), of favorable elementary events by thenumber, K = N(Ω), of all possible events.

For example, if we use numerals of the type i¬, i ∈ N, then K = ¬. The number

N(E) depends on our decision about how many numerals we want to use to representthe point A. If we decide that the point A on the circumference is represented by mnumerals we obtain

P(E) =N(E)N(Ω)

=mK

=m¬

> 0.

where the number m¬

is infinitesimal if m is finite. Note that this representation isvery interesting also from the point of view of distinguishing the notions ‘point’ and‘arc’. When m is finite than we deal with a point, when m is infinite we deal with anarc.

In the case we need a higher accuracy, we can choose, for instance, numerals ofthe type i¬−2,1 ≤ i ≤ ¬2, for expressing points at the disk. Then it follows K = ¬2

and, as a result, we obtain P(E) = m¬−2 > 0.This example with the rotating disk, of course, is a particular instance of the

general situation taking place in the traditional probability theory where the prob-ability that a continuous random variable X attains a given value a is zero, i.e.,P(X = a) = 0. While for a discrete random variable one could say that an eventwith probability zero is impossible, this can not be said in the case of a continuousrandom variable. As we have shown by the example above, in our approach thissituation does not take place because this probability can be expressed by infinites-imals. As a consequence, probabilities of such events can be computed and used innumerical models describing the real world (see [36] for a detailed discussion onthe modelling continuity by infinitesimals in the framework of the approach usinggrossone).

Moreover, the obtained probabilities are not absolute, they depend on the accu-racy chosen for the mathematical model describing the experiment. There is againa straight analogy with Physics where it is not possible to obtain results that havea precision higher than the accuracy of the measurement of the data. We also can-not obtained a precision that is higher than the precision of numerals used in themathematical model.

Let us now consider two examples showing that the new approach allows us tocalculate areas and volumes of a more general class of objects than the traditionalone. In Fig. 1.5 two figures are shown. The traditional approach tells us that both ofthem have area equal to one. In the new approach, if we use numerals of the typei¬−1, i ∈N, to express points within a unit interval, then the unit interval consists of


Fig. 1.5 It is possible to calculate and to distinguish areas of these two objects

Fig. 1.6 New possibilities for calculating volumes of objects

¬ points and in the plane each point has the infinitesimal area ¬−1 ·¬−1 = ¬−2. Asa consequence, this value will be our accuracy in calculating areas in this example.Suppose now that the vertical line added to the square at the right figure in Fig. 1.5has the width equal to one point. Then we are able to calculate the area, S2, of theright figure and it will be possible to distinguish it from the area, S1, of the squareon the left

S1 = 1 ·1 = 1, S2 = 1 ·1+1 ·¬−1 = 1¬01¬−1.

If the added vertical line has the width equal to three points then it follows

S2 = 1 ·1+3 ·¬−1 = 1¬03¬−1.

The volume of the figure shown in Fig. 1.6 is calculated analogously:

V = 1 ·1 ·1+1 ·1 ·¬−1 +1 ·¬−1 ·¬−1 = 1¬01¬−11¬−2.

If the accuracy of the considered numerals of the type i¬−1, i ∈ N, is not sufficient,we can increase it by using, for instance, numerals of the type i¬−2,1 ≤ i ≤ ¬2.Then the unit interval consists of ¬2 points and at the plane each point has the


infinitesimal area ¬−2 ·¬−2 = ¬−4. As a result, by a complete analogy to the pre-vious case we obtain for lines having the width, for instance, equal to five points inall three dimensions that

S2 = 1 ·1+5 ·¬−2 = 1¬05¬−2,

V = 1 ·1 ·1+1 ·1 ·5 ·¬−2 +1 ·5 ·¬−2 ·5 ·¬−2 = 1¬05¬−225¬−4.

1.8 Traditional and blinking fractals and their quantitativeanalysis using infinite and infinitesimal numbers

Fractal objects have been very well studied during the last few decades (see, e.g.,[10, 26] and references given therein) and have been applied in various fields (seenumerous applications given in [9, 10, 15, 26, 49]). However, mathematical analysisof fractals (except, of course, a very well developed theory of fractal dimensions)very often continues to have mainly a qualitative character and tools for a quantita-tive analysis of fractals at infinity are not very rich yet.

In this section, we propose to apply the methodology developed above for a quan-titative analysis of traditional and newly introduced blinking fractals.

1.8.1 Blinking fractals

Let us start by introducing the new class of objects – blinking fractals – that are notcovered by traditional theories studying self-similarity processes. Traditional frac-tals are constructed using the principle of self-similarity that infinitely many timesrepeats a basic object (some times slightly modified in time). However, there existprocesses and objects that evidently are very similar to classical fractals but cannotbe covered by the traditional approaches because several self-similarity mechanismsparticipate in the process of their construction. Before going to a general definitionof blinking fractals let us give just three examples of them.

The first example is derived from one of the famous fractal constructions – thecoast of Britain – as follows. Suppose that we have made a picture of the coast twotimes using the same scale of the map: at the moment of the early sunrise and at themoment of late sunset. Then, due to the long shadows present at these moments anddirected to the opposite sides we shall have two different pictures. If we suppose,for example, that sunset corresponds to shadows on the left and sunrise to shadowson the right, then we can indicate them as L and R, correspondingly. If now we startto make pictures (starting from sunrise) alternating moments of the photographingfrom sunrise to sunset and decreasing the scale each time, we shall obtain a se-ries of pictures being very similar to traditional fractals but different because leftshadows will alternate right shadows at this sequence as follows: R,L,R,L,R,L, . . .


Thus, there are two fractal mechanisms working in our process. Each of them canbe represented by one of subsequences R,R,R, . . . and L,L,L, . . . and the traditionalanalysis does not allow us to say what will be the limit fractal object and will it haveL or R type of shadow.

Fig. 1.7 The rotating prism having the triangular face red and the rectangular face blue.

The second example is constructed as follows. Let us take a prism (see Fig. 1.7)that is rotating around its vertical axis and observe it at two different moments. Thefirst is the moment when we see its face being the blue rectangular with sides 1 and√

2. Since we look exactly at the front of the prism we see the rectangular as thesquare with the length one on side. The second moment is when we look at the facebeing the red right isosceles triangle with the legs equal to one. Then we apply tothis three-dimensional object the two following self-similarity rules: we substituteeach prism by four smaller prisms during the time passing between each even andodd observation and by two smaller prisms during the time passing between eachodd and even observation. Thus, at the odd iterations we observe application of thefirst mechanism shown in the top of Fig. 1.8. The second mechanism shown in thebottom of this figure is applied during the even iterations. As a result, starting fromthe blue square one on side at iteration 0 we observe the pictures (see Fig. 1.9)with alternating blue squares and red triangles. Again, as it was with the aboveexample related to the coast of Britain, we can extract two fractal subsequencesbeing traditional fractals. The mechanism of the first one dealing with blue squaresis shown in Fig. 1.10. The second mechanism dealing with red triangles is presentedin Fig. 1.11. Traditional approaches are not able to say anything about the behaviorof this process at infinity. Does there exist a limit object of this process? If it exists,what can we say about its structure? Does it consist of red triangles or blue squares?What is the area of this (again, if it exists) limit object? All these questions remainwithout answers.

Before we discuss the last example linked, as it was with our first example, toanother famous fractal construction – Cantor’s set (see Fig. 1.12) – let us make a


Fig. 1.8 We observe that each blue square is transformed in four red triangles and each red triangleis transformed in two blue squares.

Fig. 1.9 The first four iterations of the process that has started from one blue square and uses twoself-similarity mechanisms.

few comments reminding that very often we can give certain numerical answersto questions regarding fractals only if a finite number of steps in the procedureof their construction has been considered. The same questions very often remainwithout any answer if we consider an infinite number of steps. If a finite number of


Fig. 1.10 The first traditional fractal mechanism regarding blue squares that can be separated fromthe process shown in Fig. 1.9.

Fig. 1.11 The second traditional fractal mechanism regarding red triangles that can be separatedfrom the process shown in Fig. 1.9.

Step 0

Step 1

Step 2

Step 3

Step 4

Fig. 1.12 Cantor’s set.

steps, n, has been done in construction of Cantor’s set, then we are able to describenumerically the set being the result of this operation. It will have 2n intervals havingthe length 1

3n each. Obviously, the set obtained after n+1 iterations will be differentand we also are able to measure the lengths of the intervals forming the secondset. It will have 2n+1 intervals having the length 1

3n+1 each. The situation changesdrastically in the limit because we are not able to distinguish results of n and n+1steps of the construction if n is infinite.

We also are not able to distinguish at infinity the results of the following twoprocesses that both use Cantor’s construction but start from different positions. Thefirst one is the usual Cantor’s set and it starts from the interval [0,1], the secondstarts from the couple of intervals [0, 1

3 ] and [ 23 ,1]. In spite of the fact that for any

given finite number of steps, n, the results of the constructions will be different forthese two processes we have no tools to distinguish them at infinity.


Step 0

Step 1

Step 2

Step 3

Step 4

Fig. 1.13 At each odd iteration we remove the open interval being the middle third part from eachof the intervals present in the construction and at each even iteration from each interval present inthe set we remove open intervals being the second and the last fourth parts.

Let us now slightly change the process of construction used in Cantor’s set tocreate a new example of a blinking fractal. At each odd iteration we shall maintainCantor’s rule, i.e., we remove the open interval being the middle third part from eachof 2n intervals present in the construction at the n-th iteration, where n = 2k−1. Incontrast, if n = 2k from each interval present in the set corresponding to the n-thiteration we remove open intervals being the second and the last fourth parts (seeFig. 1.13). Again, as it was in the two previous examples, we have two differentmechanisms working in this process and we are not able to say anything with respectto the structure of the resulting object at infinity. All the examples considered abovehave two different fractal mechanisms participating in their construction. Naturally,examples with more than two such mechanisms can be easily given.

To conclude this subsection we give the following general definition of objectsthat will be studied in this chapter together with traditional fractals. Objects con-structed using the principle of self-similarity with an infinite cyclic application ofseveral fractal rules are called blinking fractals.

1.8.2 Quantitative analysis of traditional and blinking fractals

Starting from Cantor’s set we show how lengths of traditional fractals can be calcu-lated at infinity.

We remind that if a finite number of steps, n, has been executed in Cantor’sconstruction starting from the interval [0,1] then we are able to describe numericallythe set being the result of this operation. It will have 2n intervals having the length13n each. Obviously, the set obtained after n+ 1 iterations will be different and wealso are able to measure the lengths of the intervals forming the second set. It willhave 2n+1 intervals having the length 1

3n+1 each. The situation changes drastically inthe limit because traditional approaches are not able to distinguish results of n andn+1 steps of the construction if n is infinite. Now, we can do it using the introducedinfinite and infinitesimal numbers.


Since the construction of Cantor’s set is a process, it cannot contain more then¬ steps (see discussion related to the example (1.23)-(1.25)). Thus, if we start theprocess from the interval [0,1], after ¬ steps Cantor’s set consists of 2¬ intervals andtheir total length, L(n), is expressed in infinitesimals: L(¬) = ( 2

3 )¬, i.e., the set has

a well defined infinite number of intervals and each of them has the infinitesimallength equal to 3−¬. Analogously, after ¬− 1 steps Cantor’s set consists of 2¬−1

intervals and their total length is expressed in infinitesimals: L(¬) = ( 23 )

¬−1. Thus,the length L(n) for any (finite or infinite) number of steps, n, where 1 ≤ n ≤ ¬ andis expressible in the chosen numeral system can be calculated.

It is important to notice here that (again due to the limitation illustrated by theexample (1.23)-(1.25)) it is not possible to count one by one all the intervals atCantor’s set if their number is superior to ¬. For instance, after ¬ steps it has 2¬

intervals and they cannot be counted one by one because 2¬ > ¬ and any process(including that of the sequential counting) cannot have more that ¬ steps.

It becomes possible to study by a complete analogy other classical fractals. Forinstance, we immediately obtain that the length of the Koch Curve starting from theinterval [0,1] after ¬ steps has the infinite length equal to ( 4

3 )¬ because it consists of

4¬ segments having the length ( 13 )

¬ each. In the same way we can calculate the areaof the Sierpinski Carpet. If its construction starts from the unit square then after ¬steps we obtain the set of squares having the total infinitesimal area equal to ( 8

9 )¬

because it consists of 8¬ squares and each of them has area equal to ( 19 )

¬.Consider now two processes that both use Cantor’s construction but start from

different initial conditions. Traditional approaches do not allow us to distinguishthem at infinity in spite of the fact that for any given finite number of steps, n, theresults of the constructions are different and can be calculated. Using the new ap-proach we are able to study the processes numerically also at infinity. For example,if the first process is the usual Cantor’s set and it starts from the interval [0,1] andthe second one starts from the couple of intervals [0, 1

3 ] and [ 23 ,1] then after ¬

2 stepsthe result of the first process will be the set consisting of 2

¬2 intervals and its length

L(¬2 ) = ( 2

3 )¬2 . The second set after ¬

2 steps will consists of 2¬2 +1 intervals and its

length L(¬2 +1) = ( 2

3 )¬2 +1.

Let us answer now to the following traditional problem: How many points arethere at Cantor’s set? From our new point of view this formulation is not sufficientlyprecise. Now, when it becomes possible to distinguish different sets at different iter-ations we should say: How many points there are at Cantor’s set being the result ofn steps of Cantor’s procedure started from the initial set consisting of k intervals? Inthe following without loss of generality we consider the case k = 1 and calculate thenumber of the points in the set Cn being the result of n steps of Cantor’s procedurestarting from the interval [0,1].

Then, as it has been shown in Section 1.6, we are able to calculate the number ofpoints if we have decided which numerals will be used to express the coordinatesof the points within the interval [0,1]. Moreover, we shall be able to do such calcu-lations only if the numeral system chosen to express coordinates of the points willbe powerful enough to distinguish the points within the intervals generated during


this process. Obviously, we shall be able to distinguish within Cn no more pointsthan our chosen numeral system will allow us. For instance, if we give to a personfrom our primitive Piraha tribe the set C2 consisting of four intervals, this personoperating with his poor numeral system consisting of the numerals I, II, and ‘many’will not be able to say us how many intervals there are in this set and which arecoordinates of, for example, their end points. This happens because his system istoo poor both for counting the intervals and for expressing coordinates of their endpoints. His answer will be just ‘many’ for the number of intervals and he will beable to indicate the coordinate of only one point – 1. However, if we give him theset C0 his answer will be correct for the intervals – there is one interval – and he willbe able to indicate the coordinate of the same point – 1.

Thus, the situation with counting points in Cantor’s set again is similar to thework with a microscope: we decide the level of the precision we need and obtain aresult dependent on the chosen level. If we need a more precise or a more rough an-swer, we change the level of accuracy of our microscope. If we need a high precisionand need to distinguish many points, we should take a powerful numeral system toexpress the coordinates. In the case when we need a low precision, a weak numeralsystem can be taken.

The introduced mathematical tools allow us to give answers to similar questionsnot only for traditional but for blinking fractals, too. We start by considering theblinking fractal described in Figs. 1.7– 1.11. Since the answers depend on the initialconditions, we suppose without loss of generality that the process starts from theblue square one unit of length on side. This means that during any (finite or infinite)even iteration we observe blue squares and during any odd iteration we see redtriangles. We shall indicate the set obtained after n iterations by Pn. The area An ofthe set Pn is calculated as follows. For any (finite or infinite) n= 2k,k ≥ 0, it consistsof 23k squares with the side equal to 2−2k. Thus, the area of Pn is

A2k = (2−2k)2 ·23k = 2−k.

For n = 2k−1,k ≥ 1, the set Pn consists of 23k−1 right isosceles triangles with thelegs equal to 2−2k+1. In this case the area of Pn is calculated as follows

A2k−1 = 0.5(2−2k+1)2 ·23k−1 = 2−k. (1.47)

For example, for the infinite n = 0.5¬ the set P0.5¬ consists of 20.75¬ blue squares(because the number 0.5¬ is even), their total area is infinitesimal and is equal toA0.5¬ = 2−0.25¬. Analogously, if the number of iterations is n = 0.5¬+ 1 then theset P0.5¬+1 consists of red triangles and k from (1.47) is equal to −0.25¬+ 1. Thenumber of triangles is 20.75¬+2 and their total area is infinitesimal and is equal toA0.5¬+1 = 2−0.25¬+1.

Finally, let us consider the blinking fractal from Fig. 1.13. We shall indicate theset obtained after n iterations by Fn. The length Ln of the set Fn is calculated asfollows. For any (finite or infinite) n = 2k,k ≥ 0, it consists of 22k intervals and eachof them has the length 3−k ·4−k. Thus,


L2k = 22k ·3−k ·4−k = 3−k.

Analogously, for n = 2k−1,k ≥ 1, we obtain that Fn consists of 22k−1 intervals andeach of them has the length 3−k ·4−k+1. Thus,

L2k−1 = 22k−1 ·3−k ·4−k+1 = 2 ·3−k.

For example, for the infinite odd n = 0.5¬− 1 the set F0.5¬−1 consists of 20.5¬−1

intervals and their total length is infinitesimal and is equal to L0.5¬−1 = 2 ·3−0.25¬.

1.9 Concepts of continuity in Physics and Mathematics

The goal of this section is to discuss mathematical and physical definitions of con-tinuity and to develop a new, more physical point of view on this notion using theinfinite and infinitesimal numbers introduced above. The new point of view is il-lustrated by a detailed consideration of one of the most fundamental mathematicaldefinitions – function.

In Physics, the ‘continuity’ of an object is relative. For example, if we observe atable by eyes, then we see it continuous. If we use a microscope for our observation,we see that the table is discrete. This means that we decide how to see the object,as a continuous or as a discrete, by the choice of the instrument for observation. Aweak instrument – our eyes – is not able to distinguish its internal small separateparts (e.g., molecules) and we see the table as a continuous object. A sufficientlystrong microscope allows us to see the separate parts and the table becomes discretebut each small part now is viewed as continuous.

In this connection, fractals become a very useful tool for describing physical ob-jects. Let us return to Figs. 1.12 and 1.13 and suppose that we observe two beamsconsisting of two different materials at Step 0 by eye and we see both of themcontinuous. Then we take a microscope with a weak lens number 1, look at the mi-croscope and see the pictures corresponding to Step 1 in Figs. 1.12 and 1.13, i.e.,that the beams are not continuous but consist of two smaller parts that, in their turn,now seem to us to be continuous. Then we proceed by taking a stronger lens num-ber 2, look again at the microscope and see the pictures corresponding to Step 2in Figs. 1.12 and 1.13. First, we see now that the beams consist of four smallerparts and each of them seems to be continuous. Second, we see that their locationsare different (remind, that we have supposed that the beams have been made us-ing different materials). By increasing the force of lenses we can observe picturesviewed at Steps 3, 4, etc. obtaining higher levels of discretization. Thus, continu-ity in Physics is resolution dependent and fractal ideas can serve as a good tool formodeling the physical relative continuity.

In contrast, in the traditional mathematics any mathematical object is either con-tinuous or discrete. For example, the same function cannot be both continuous anddiscrete. Thus, this contraposition of discrete and continuous in the traditional math-


ematics does not reflect properly the physical situation that we observe in practice.For fortune, the infinite and infinitesimal numbers introduced in the previous sec-tions give us a possibility to develop a new theory of continuity that is closer to thephysical world and better reflects the new discoveries made by physicists (remind,that the foundations of the mathematical analysis have been established centuriesago and, therefore, do not take into account the subsequent revolutionary results inPhysics, e.g., appearance of quantum Physics). We start by introducing a definitionof the one-dimensional continuous set of points based on the above considerationand Postulate 2 and establish relations to such a fundamental notion as functionusing the infinite and infinitesimal numbers.

We remind that traditionally a function f (x) is defined as a binary relation amongtwo sets X and Y (called the domain and the codomain of the relation) with the addi-tional property that to each element x∈X corresponds exactly one element f (x)∈Y .We consider now a function f (x) defined over a one-dimensional interval [a,b]. Itfollows immediately from the previous sections that to define a function f (x) overan interval [a,b] it is not sufficient to give a rule for evaluating f (x) and the valuesa and b because we are not able to evaluate f (x) at any point x ∈ [a,b] (for exam-ple, traditional numeral systems do not allow us to express any irrational number ζand, therefore, we are not able to evaluate f (ζ)). However, the traditional definitionof a function includes in its domain points at which f (x) cannot be evaluated, thusintroducing ambiguity.

In order to be precise in the definition of a function, it is necessary to indicateexplicitly a numeral system, S , we intend to use to express points from the interval[a,b]. Thus, a function f (x) is defined when we know a rule allowing us to obtainf (x) given x and its domain, i.e., the set [a,b]S of points x ∈ [a,b] expressible inthe chosen numeral system S . We suppose hereinafter that the system S is used towrite down f (x) (of course, the choice of S determines a class of formulae and/orprocedures we are able to express using S ) and it allows us to express any number

y = f (x), x ∈ [a,b]S .

The number of points of the domain [a,b]S can be finite or infinite but the set[a,b]S is always discrete. This means that for any point x ∈ [a,b]S it is possible todetermine its closest right and left neighbors, x+ and x−, respectively, as follows

x+ = minz : z ∈ [a,b]S , z > x, x− = maxz : z ∈ [a,b]S , z < x. (1.48)

Apparently, the obtained discrete construction leads us to the necessity to aban-don the nice idea of continuity, which is a very useful notion used in different fieldsof mathematics. But this is not the case. In contrast, the new approach allows us tointroduce a new definition of continuity very well reflecting the physical world.

Let us consider n+1 points at a line

a = x0 < x1 < x2 < .. . < xn−1 < xn = b (1.49)


Fig. 1.14 It is not possible to say whether this function is continuous or discrete until we have notintroduced a unit of measure and a numeral system to express distances between the points

and suppose that we have a numeral system S allowing us to calculate their coordi-nates using a unit of measure µ (for example, meter, inch, etc.) and to construct sothe set X = [a,b]S expressing these points.

The set X is called continuous in the unit of measure µ if for any x ∈ (a,b)S it fol-lows that the differences x+−x and x−x− from (1.48) expressed in units µ are equalto infinitesimal numbers. In our numeral system with radix grossone this means thatall the differences x+− x and x− x− contain only negative grosspowers. Note thatit becomes possible to differentiate types of continuity by taking into account val-ues of grosspowers of infinitesimal numbers (continuity of order ¬−1, continuity oforder ¬−2, etc.).

This definition emphasizes the physical principle that there does not exist anabsolute continuity: it is relative (see discussion in page 56) with respect to thechosen instrument of observation which in our case is represented by the unit ofmeasure µ. Thus, the same set can be viewed as a continuous or not in dependenceof the chosen unit of measure.

Example 27. The set of six equidistant points

X1 = a,x1,x2,x3,x4,x5 (1.50)

from Fig. 1.14 can have the distance d between the points equal to ¬−1 in a unitof measure µ and to be, therefore, continuous in µ. Usage of a new unit of measureν = ¬−3µ implies that d = ¬2 in ν and the set X1 is not continuous in ν. ⊓⊔

Note that the introduced definition does not require that all the points from X areequidistant. For instance, if in Fig. 1.14 for a unit measure µ the largest over the set[a,b]S distance x6 − x5 is infinitesimal then the whole set is continuous in µ.

The set X is called discrete in the unit of measure µ if for all points x ∈ (a,b)S itfollows that the differences x+− x and x− x− from (1.48) expressed in units µ arenot infinitesimal numbers. In our numeral system with radix grossone this meansthat in all the differences x+ − x and x− x− negative grosspowers cannot be thelargest ones. For instance, the set X1 from (1.50) is discrete in the unit of measure ν


from Example 27. Of course, it is also possible to consider intermediate cases wheresets have continuous and discrete parts (see again discussion in page 56 related tobeams from Figs. 1.12 and 1.13).

The introduced notions allow us to give the following very simple definition of afunction continuous at a point. A function f (x) defined over a set [a,b]S continuousin a unit of measure µ is called continuous in the unit of measure µ at a point x ∈(a,b)S if both differences f (x)− f (x+) and f (x)− f (x−) are infinitesimal numbersin µ, where x+ and x− are from (1.48). For the continuity at points a, b it is sufficientthat one of these differences is infinitesimal. The notions of continuity from theleft and from the right in a unit of measure µ at a point are introduced naturally.Similarly, the notions of a function discrete, discrete from the right, and discretefrom the left can be defined.

The function f (x) is continuous in the unit of measure µ over the set [a,b]S if itis continuous in µ at all points of [a,b]S . Again, it becomes possible to differentiatetypes of continuity by taking into account values of grosspowers of infinitesimalnumbers (continuity of order ¬−1, continuity of order ¬−2, etc.) and to considerfunctions in such units of measure that they become continuous or discrete overcertain subintervals of [a,b]. In the further consideration we shall often fix the unitof measure µ and write just ‘continuous function’ instead of ‘continuous function inthe unit of measure µ’. Let us give three simple examples illustrating the introduceddefinitions.

Example 28. We start by showing that the function f (x) = x2 is continuous overthe set X2 defined as the interval [0,1] where numerals i

¬,0 ≤ i ≤ ¬, are used to

express its points in units µ. First of all, note that the set X2 is continuous in µ becauseits points are equidistant with the distance d = ¬−1. Since this function is strictlyincreasing, to show its continuity it is sufficient to check the difference f (x)− f (x−)at the point x = 1. In this case, x− = 1−¬−1 and we have

f (1)− f (1−¬−1) = 1− (1−¬−1)2 = 2¬−1(−1)¬−2.

This number is infinitesimal, thus f (x) = x2 is continuous over the set X2. ⊓⊔

Example 29. Consider the same function f (x) = x2 over the set X3 defined as theinterval [¬− 1,¬] where numerals ¬− 1+ i

¬,0 ≤ i ≤ ¬, are used to express its

points in units µ. Analogously, the set X3 is continuous and it is sufficient to checkthe difference f (x)− f (x−) at the point x = ¬ to show continuity of f (x) over thisset. In this case,

x− = ¬−1+¬−1

¬= ¬−¬−1,

f (x)− f (x−) = f (¬)− f (¬−¬−1) = ¬2 − (¬−¬−1)2 = 2¬0(−1)¬−2.

This number is not infinitesimal because it contains the finite part 2¬0 and, as aconsequence, f (x) = x2 is not continuous over the set X3. ⊓⊔

Example 30. Consider f (x) = x2 defined over the set X4 being the interval [¬−1,¬]where numerals ¬−1+ i

¬2 ,0≤ i≤¬2, are used to express its points in units µ. The


set X4 is continuous and we check the difference f (x)− f (x−) at the point x = ¬.We have

x− = ¬−1+¬2 −1

¬2 = ¬−¬−2,

f (x)− f (x−) = f (¬)− f (¬−¬−2) = ¬2 − (¬−¬−2)2 = 2¬−1(−1)¬−4.

Since the obtained result is infinitesimal, f (x) = x2 is continuous over X4. ⊓⊔

Let us consider now a function f (x) defined by formulae over a set X = [a,b]S sothat different expressions can be used over different subintervals of [a,b]. The term‘formula’ hereinafter indicates a single expression used to evaluate f (x).

Example 31. The function g(x) = 2x2 −1,x ∈ [a,b]S , is defined by one formula andfunction

f (x) =

max−10x,5x−1, x ∈ [c,0)S ∪ (0,d]S ,4x, x = 0, c < 0, d > 0, (1.51)

is defined by three formulae, f1(x), f2(x), and f3(x) where

f1(x) =−10x, x ∈ [c,0)S ,f2(x) = 4x, x = 0,f3(x) = 5x−1, x ∈ (0,d]S . ⊓⊔

(1.52)

Consider now a function f (x) defined in a neighborhood of a point x as follows

f (ξ) =

f1(ξ), x− l ≤ ξ < x,f2(ξ), ξ = x,f3(ξ), x < ξ ≤ x+ r,

(1.53)

where the number l is any number such that the same formula f1(ξ) is used todefine f (ξ) at all points ξ such that x− l ≤ ξ < x. Analogously, the number r is anynumber such that the same formula f3(ξ) is used to define f (ξ) at all points ξ suchthat x < ξ ≤ x+r. Of course, as a particular case it is possible that the same formulais used to define f (ξ) over the interval [x− l,x+ r], i.e.,

f (ξ) = f1(ξ) = f2(ξ) = f3(ξ), ξ ∈ [x− l,x+ r]. (1.54)

It is also possible that (1.54) does not hold but formulae f1(ξ) and f3(ξ) are definedat the point x and are such that at this point they return the same value, i.e.,

f1(x) = f2(x) = f3(x). (1.55)

If condition (1.55) holds, we say that function f (x) has continuous formulae at thepoint x. Of course, in the general case, formulae f1(ξ), f2(ξ), and f3(ξ) can be orcannot be defined out of the respective intervals from (1.53). In cases where con-dition (1.55) is not satisfied we say that function f (x) has discontinuous formulae


at the point x. Definitions of functions having formulae which are continuous ordiscontinuous from the left and from the right are introduced naturally.

Example 32. Let us study the following function

f (x) =

¬2 + x2−1

x−1 , x = 1,a, x = 1,

(1.56)

at the point x = 1. By using designations (1.53) and the fact that for x = 1 it followsx2−1x−1 = x+1 we have

f (ξ) =

f1(ξ) = ¬2 +ξ+1, ξ < 1,f2(ξ) = a, ξ = 1,f3(ξ) = ¬2 +ξ+1, ξ > 1,

Sincef1(1) = f3(1) = ¬2 +2, f2(1) = a,

we obtain that if a = ¬2 + 2, then the function (1.56) has continuous formulae9 atthe point x = 1. Analogously, the function (1.51) has continuous formulae at thepoint x = 0 from the left and discontinuous from the right. ⊓⊔

Thus, functions having continuous formulae at a point can be continuous or dis-crete at this point in dependence of the chosen unit of measure. Analogously, func-tions having discontinuous formulae at a point can be continuous or discrete at thispoint again in dependence of the chosen unit of measure. The notion of continuity ofa function depends on the chosen unit of measure and numeral system S and it canbe used for functions defined by formulae, computer procedures, tables, etc. In con-trast, the notion of a function having continuous formulae works only for functionsdefined by formulae and does not depend on units of measure or numeral systemschosen to express its domain. It is related only to properties of formulae.

We conclude this section by the note that the expressed numerical point of viewon the definition of continuity has been then extended in [40] to the differentialcalculus for one-dimensional functions assuming finite, infinite, and infinitesimalvalues over finite, infinite, and infinitesimal domains.

1.10 A brief conclusion

In this chapter, a new computational methodology has been introduced. It allows usto express, by a finite number of symbols, not only finite numbers but infinite andinfinitesimals, as well, an to execute numerical computations with all of them. A

9 Note, that even if a=¬2+2+ε, where ε is an infinitesimal number (remind that all infinitesimalsare not equal to zero), we are able to establish that the function has discontinuous formulae.


number of theoretical and applied problems where the new way of counting helps alot has been discussed.

It has been emphasized that the philosophical triad – researcher, object of in-vestigation, and tools used to observe the object – existing in such natural sciencesas Physics and Chemistry, exists in Mathematics, too. In natural sciences, the in-strument used to observe the object influences the results of observations. The samehappens in Mathematics where numeral systems used to express numbers are amongthe instruments of observations used by mathematicians. The usage of powerful nu-meral systems gives the possibility to obtain more precise results in Mathematics,in the same way as the usage of a good microscope gives the possibility to obtainmore precise results in Physics.

When a mathematician chooses a mathematical language (an instrument), in thismoment he/she chooses both a set of numbers that can be observed through thenumerals available in the chosen numeral system and the accuracy of results that canbe obtained during computations. In the cases where two languages having differentaccuracies can be applied, it does not usually make sense to mix the languages, i.e.,to compose mathematical expressions using symbols from both languages, becausethe result of such a mixing either has no any sense or has the lower of the twoaccuracies.

The analysis done in the chapter shows that the traditional mathematical languageusing for computations the symbol ∞ very often does not possess a sufficiently highaccuracy when one deals with problems having their interesting properties at infin-ity. However, the new numeral system and the new way of counting described in thischapter do not contradict the traditional approaches. They just describe objects withdifferent accuracies. It has been discovered that situations that can be illustrated bythe following metaphor can take place. Suppose that we have measured two dis-tances A and B with the accuracy equal to 1 meter and we have found that both ofthem are equal to 25 meters. Suppose now that we want to measure them with theaccuracy equal to 1 centimeter. Then, very probably, we shall obtain something likeA = 2487 centimeters and B = 2538 centimeters, i.e., A = B. Both answers, A = Band A = B, are correct but with different accuracies and both of them can be usedsuccessfully in different situations. For instance, if one just wants to go for a walk,then the accuracy of the answer A = B expressed in meters is sufficient. However, ifone needs to connect some devices with a cable, then a higher accuracy is requiredand the answer expressed in centimeters should be used.

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