Inmathematics, aninfinitesimal number is a non-zero quantity that is closer to0 than any non-zeroreal number is. The wordinfinitesimal comes from a 17th-centuryModern Latin coinageinfinitesimus, which originally referred to the "infinitieth" item in asequence.
Infinitesimals do not exist in the standard real number system, but they do exist in other number systems, such as thesurreal number system and thehyperreal number system, which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities; the augmentations are thereciprocals of one another.
Infinitesimal numbers were introduced in thedevelopment of calculus, in which thederivative was first conceived as a ratio of two infinitesimal quantities. This definition was notrigorously formalized. As calculus developed further, infinitesimals were replaced bylimits, which can be calculated using the standard real numbers.
In the 3rd century BCArchimedes used what eventually came to be known as themethod of indivisibles in his workThe Method of Mechanical Theorems to find areas of regions and volumes of solids.[1] In his formal published treatises, Archimedes solved the same problem using themethod of exhaustion.
Infinitesimals regained popularity in the 20th century withAbraham Robinson's development ofnonstandard analysis and thehyperreal numbers, which, after centuries of controversy, showed that a formal treatment of infinitesimal calculus was possible. Following this, mathematicians developed surreal numbers, a related formalization of infinite and infinitesimal numbers that include both hyperrealcardinal andordinal numbers, which is the largestordered field.
Vladimir Arnold wrote in 1990:
Nowadays, when teaching analysis, it is not very popular to talk about infinitesimal quantities. Consequently, present-day students are not fully in command of this language. Nevertheless, it is still necessary to have command of it.[2]
An insight[whose?] for making infinitesimals feasible mathematical entities was that they could still retain certain properties such asangle orslope, even if these entities were infinitely small.[3]
Infinitesimals are a basic ingredient incalculus as developed byLeibniz, including thelaw of continuity and thetranscendental law of homogeneity. In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective in mathematics,infinitesimal means infinitely small, smaller than any standard real number. Infinitesimals are often compared to other infinitesimals of similar size, as in examining the derivative of a function. An infinite number of infinitesimals are summed to calculate anintegral.
The modern concept of infinitesimals was introduced around 1670 by eitherNicolaus Mercator orGottfried Wilhelm Leibniz.[4] The 15th century saw the work ofNicholas of Cusa, further developed in the 17th century byJohannes Kepler, in particular, the calculation of the area of a circle by representing the latter as an infinite-sided polygon.Simon Stevin's work on the decimal representation of all numbers in the 16th century prepared the ground for the real continuum.Bonaventura Cavalieri's method of indivisibles led to an extension of the results of the classical authors. The method of indivisibles related to geometrical figures as being composed of entities ofcodimension 1.[clarification needed]John Wallis's infinitesimals differed from indivisibles in that he would decompose geometrical figures into infinitely thin building blocks of the same dimension as the figure, preparing the ground for general methods of the integral calculus. He exploited an infinitesimal denoted1/∞ in area calculations.
The use of infinitesimals by Leibniz relied upon heuristic principles, such as the law of continuity: what succeeds for the finite numbers succeeds also for the infinite numbers and vice versa; and the transcendental law of homogeneity that specifies procedures for replacing expressions involving unassignable quantities, by expressions involving only assignable ones. The 18th century saw routine use of infinitesimals by mathematicians such asLeonhard Euler andJoseph-Louis Lagrange.Augustin-Louis Cauchy exploited infinitesimals both in definingcontinuity in hisCours d'Analyse, and in defining an early form of aDirac delta function. As Cantor andDedekind were developing more abstract versions of Stevin's continuum,Paul du Bois-Reymond wrote a series of papers on infinitesimal-enriched continua based on growth rates of functions. Du Bois-Reymond's work inspired bothÉmile Borel andThoralf Skolem. Borel explicitly linked du Bois-Reymond's work to Cauchy's work on rates of growth of infinitesimals. Skolem developed the first non-standard models of arithmetic in 1934. A mathematical implementation of both the law of continuity and infinitesimals was achieved byAbraham Robinson in 1961, who developednonstandard analysis based on earlier work byEdwin Hewitt in 1948 andJerzy Łoś in 1955. Thehyperreals implement an infinitesimal-enriched continuum and thetransfer principle implements Leibniz's law of continuity. Thestandard part function implements Fermat'sadequality.
The notion of infinitely small quantities was discussed by theEleatic School. TheGreek mathematicianArchimedes (c. 287 BC – c. 212 BC), inThe Method of Mechanical Theorems, was the first to propose a logically rigorous definition of infinitesimals.[5] HisArchimedean property defines a numberx as infinite if it satisfies the conditions|x| > 1,|x| > 1 + 1,|x| > 1 + 1 + 1, ..., and infinitesimal ifx ≠ 0 and a similar set of conditions holds forx and the reciprocals of the positive integers. A number system is said to be Archimedean if it contains no infinite or infinitesimal members.
The English mathematicianJohn Wallis introduced the expression 1/∞ in his 1655 bookTreatise on the Conic Sections. The symbol, which denotes the reciprocal, or inverse, of ∞, is the symbolic representation of the mathematical concept of an infinitesimal. In hisTreatise on the Conic Sections, Wallis also discusses the concept of a relationship between the symbolic representation of infinitesimal 1/∞ that he introduced and the concept of infinity for which he introduced the symbol ∞. The concept suggests athought experiment of adding an infinite number ofparallelograms of infinitesimal width to form a finite area. This concept was the predecessor to the modern method of integration used inintegral calculus. The conceptual origins of the concept of the infinitesimal 1/∞ can be traced as far back as the Greek philosopherZeno of Elea, whoseZeno's dichotomy paradox was the first mathematical concept to consider the relationship between a finite interval and an interval approaching that of an infinitesimal-sized interval.
Infinitesimals were the subject of political and religious controversies in 17th century Europe, including a ban on infinitesimals issued by clerics in Rome in 1632.[6]
Prior to the invention of calculus mathematicians were able to calculate tangent lines usingPierre de Fermat's method ofadequality andRené Descartes'method of normals. There is debate among scholars as to whether the method was infinitesimal or algebraic in nature. WhenNewton andLeibniz invented thecalculus, they made use of infinitesimals, Newton'sfluxions and Leibniz'differential. The use of infinitesimals was attacked as incorrect byBishop Berkeley in his workThe Analyst.[7] Mathematicians, scientists, and engineers continued to use infinitesimals to produce correct results. In the second half of the nineteenth century, the calculus was reformulated byAugustin-Louis Cauchy,Bernard Bolzano,Karl Weierstrass,Cantor,Dedekind, and others using the(ε, δ)-definition of limit andset theory.While the followers of Cantor, Dedekind, and Weierstrass sought to rid analysis of infinitesimals, and their philosophical allies likeBertrand Russell andRudolf Carnap declared that infinitesimals arepseudoconcepts,Hermann Cohen and hisMarburg school ofneo-Kantianism sought to develop a working logic of infinitesimals.[8] The mathematical study of systems containing infinitesimals continued through the work ofLevi-Civita,Giuseppe Veronese,Paul du Bois-Reymond, and others, throughout the late nineteenth and the twentieth centuries, as documented by Philip Ehrlich (2006). In the 20th century, it was found that infinitesimals could serve as a basis for calculus and analysis (seehyperreal numbers).
In extending the real numbers to include infinite and infinitesimal quantities, one typically wishes to be as conservative as possible by not changing any of their elementary properties. This guarantees that as many familiar results as possible are still available. Typically,elementary means that there is noquantification oversets, but only over elements. This limitation allows statements of the form "for any number x..." For example, the axiom that states "for any number x,x + 0 = x" would still apply. The same is true for quantification over several numbers, e.g., "for any numbers x andy,xy = yx." However, statements of the form "for anyset S of numbers ..." may not carry over. Logic with this limitation on quantification is referred to asfirst-order logic.
The resulting extended number system cannot agree with the reals on all properties that can be expressed by quantification over sets, because the goal is to construct a non-Archimedean system, and the Archimedean principle can be expressed by quantification over sets. One can conservatively extend any theory including reals, including set theory, to include infinitesimals, just by adding a countably infinite list of axioms that assert that a number is smaller than 1/2, 1/3, 1/4, and so on. Similarly, thecompleteness property cannot be expected to carry over, because the reals are the unique complete ordered field up to isomorphism.
There are three categories in which a non-Archimedean number system could have first-order properties compatible with those of the reals:
Systems in category 1, at the weak end of the spectrum, are relatively easy to construct but do not allow a full treatment of classical analysis using infinitesimals in the spirit of Newton and Leibniz. For example, thetranscendental functions are defined in terms of infinite limiting processes, and therefore there is typically no way to define them in first-order logic. Increasing the analytic strength of the system by passing to categories 2 and 3, we find that the flavor of the treatment tends to become less constructive, and it becomes more difficult to say anything concrete about the hierarchical structure of infinities and infinitesimals.
An example from category 1 above is the field ofLaurent series with a finite number of negative-power terms. For example, the Laurent series consisting only of the constant term 1 is identified with the real number 1, and the series with only the linear term x is thought of as the simplest infinitesimal, from which the other infinitesimals are constructed. Dictionary ordering is used, which is equivalent to considering higher powers of x as negligible compared to lower powers.David O. Tall[9] refers to this system as the super-reals, not to be confused with thesuperreal number system of Dales and Woodin. Since aTaylor series evaluated with a Laurent series as its argument is still a Laurent series, the system can be used to do calculus on transcendental functions if they are analytic. These infinitesimals have different first-order properties than the reals because, for example, the basic infinitesimal x does not have a square root.
TheLevi-Civita field is similar to the Laurent series, but is algebraically closed. For example, the basic infinitesimal x has a square root. This field is rich enough to allow a significant amount of analysis to be done, but its elements can still be represented on a computer in the same sense that real numbers can be represented in floating-point.[10]
The field oftransseries is larger than the Levi-Civita field.[11] An example of a transseries is:
where for purposes of orderingx is considered infinite.
Conway'ssurreal numbers fall into category 2, except that the surreal numbers form aproper class and not a set.[12] They are a system designed to be as rich as possible in different sizes of numbers, but not necessarily for convenience in doing analysis, in the sense that every ordered field is a subfield of the surreal numbers.[13] There is a natural extension of the exponential function to the surreal numbers.[14]: ch. 10
The most widespread technique for handling infinitesimals is the hyperreals, developed byAbraham Robinson in the 1960s. They fall into category 3 above, having been designed that way so all of classical analysis can be carried over from the reals. This property of being able to carry over all relations in a natural way is known as thetransfer principle, proved byJerzy Łoś in 1955. For example, the transcendental function sin has a natural counterpart *sin that takes a hyperreal input and gives a hyperreal output, and similarly the set of natural numbers has a natural counterpart, which contains both finite and infinite integers. A proposition such as carries over to the hyperreals as .
Thesuperreal number system of Dales and Woodin is a generalization of the hyperreals. It is different from the super-real system defined byDavid Tall.
Inlinear algebra, thedual numbers extend the reals by adjoining one infinitesimal, the new element ε with the property ε2 = 0 (that is, ε isnilpotent). Every dual number has the formz =a +bε witha andb being uniquely determined real numbers.
One application of dual numbers isautomatic differentiation. This application can be generalized to polynomials in n variables, using theExterior algebra of an n-dimensional vector space.
Synthetic differential geometry orsmooth infinitesimal analysis have roots incategory theory. This approach departs from the classical logic used in conventional mathematics by denying the general applicability of thelaw of excluded middle – i.e.,not (a ≠b) does not have to meana =b. Anilsquare ornilpotent infinitesimal can then be defined. This is a numberx wherex2 = 0 is true, butx = 0 need not be true at the same time. Since the background logic isintuitionistic logic, it is not immediately clear how to classify this system with regard to classes 1, 2, and 3. Intuitionistic analogues of these classes would have to be developed first.
Cauchy used an infinitesimal to write down a unit impulse, infinitely tall and narrow Dirac-type delta function satisfying in a number of articles in 1827, see Laugwitz (1989). Cauchy defined an infinitesimal in 1821 (Cours d'Analyse) in terms of a sequence tending to zero. Namely, such a null sequence becomes an infinitesimal in Cauchy's andLazare Carnot's terminology.
Modern set-theoretic approaches allow one to define infinitesimals via theultrapower construction, where a null sequence becomes an infinitesimal in the sense of an equivalence class modulo a relation defined in terms of a suitableultrafilter. The article by Yamashita (2007) contains bibliography on modernDirac delta functions in the context of an infinitesimal-enriched continuum provided by thehyperreals.
The method of constructing infinitesimals of the kind used in nonstandard analysis depends on themodel and which collection ofaxioms are used. We consider here systems where infinitesimals can be shown to exist.
In 1936Maltsev proved thecompactness theorem. This theorem is fundamental for the existence of infinitesimals as it proves that it is possible to formalise them. A consequence of this theorem is that if there is a number system in which it is true that for any positive integern there is a positive numberx such that 0 < x < 1/n, then there exists an extension of that number system in which it is true that there exists a positive numberx such that for any positive integern we have 0 < x < 1/n. The possibility to switch "for any" and "there exists" is crucial. The first statement is true in the real numbers as given inZFCset theory : for any positive integern it is possible to find a real number between 1/n and zero, but this real number depends onn. Here, one choosesn first, then one finds the correspondingx. In the second expression, the statement says that there is anx (at least one), chosen first, which is between 0 and 1/n for anyn. In this casex is infinitesimal. This is not true in the real numbers (R) given by ZFC. Nonetheless, the theorem proves that there is a model (a number system) in which this is true. The question is: what is this model? What are its properties? Is there only one such model?
There are in fact many ways to construct such aone-dimensionallinearly ordered set of numbers, but fundamentally, there are two different approaches:
In 1960,Abraham Robinson provided an answer following the first approach. The extended set is called thehyperreals and contains numbers less in absolute value than any positive real number. The method may be considered relatively complex but it does prove that infinitesimals exist in the universe of ZFC set theory. The real numbers are called standard numbers and the new non-real hyperreals are callednonstandard.
In 1977Edward Nelson provided an answer following the second approach. The extended axioms are IST, which stands either forInternal set theory or for the initials of the three extra axioms: Idealization, Standardization, Transfer. In this system, we consider that the language is extended in such a way that we can express facts about infinitesimals. The real numbers are either standard or nonstandard. An infinitesimal is a nonstandard real number that is less, in absolute value, than any positive standard real number.
In 2006 Karel Hrbacek developed an extension of Nelson's approach in which the real numbers are stratified in (infinitely) many levels; i.e., in the coarsest level, there are no infinitesimals nor unlimited numbers. Infinitesimals are at a finer level and there are also infinitesimals with respect to this new level and so on.
Calculus textbooks based on infinitesimals include the classicCalculus Made Easy bySilvanus P. Thompson (bearing the motto "What one fool can do another can"[15]) and the German textMathematik fur Mittlere Technische Fachschulen der Maschinenindustrie by R. Neuendorff.[16]
Pioneering works based onAbraham Robinson's infinitesimals include texts byStroyan (dating from 1972) andHoward Jerome Keisler (Elementary Calculus: An Infinitesimal Approach). Students easily relate to the intuitive notion of an infinitesimal difference 1-"0.999...", where "0.999..." differs from its standard meaning as the real number 1, and is reinterpreted as an infinite terminating extended decimal that is strictly less than 1.[17][18]
Another elementary calculus text that uses the theory of infinitesimals as developed by Robinson isInfinitesimal Calculus by Henle and Kleinberg, originally published in 1979.[19] The authors introduce the language of first-order logic, and demonstrate the construction of a first order model of the hyperreal numbers. The text provides an introduction to the basics of integral and differential calculus in one dimension, including sequences and series of functions. In an Appendix, they also treat the extension of their model to thehyperhyperreals, and demonstrate some applications for the extended model.
An elementary calculus text based on smooth infinitesimal analysis is Bell, John L. (2008). A Primer of Infinitesimal Analysis, 2nd Edition. Cambridge University Press. ISBN 9780521887182.
A more recent calculus text utilizing infinitesimals is Dawson, C. Bryan (2022), Calculus Set Free: Infinitesimals to the Rescue, Oxford University Press. ISBN 9780192895608.
In a related but somewhat different sense, which evolved from the original definition of "infinitesimal" as an infinitely small quantity, the term has also been used to refer to a function tending to zero. More precisely, Loomis and Sternberg'sAdvanced Calculus defines the function class of infinitesimals,, as a subset of functions between normed vector spaces by
,
as well as two related classes (seeBig-O notation) by
, and
.[20]
The set inclusionsgenerally hold. That the inclusions are proper is demonstrated by the real-valued functions of a real variable,, and:
but and.
As an application of these definitions, a mapping between normed vector spaces is defined to be differentiable at if there is a [i.e, a bounded linear map] such that
in a neighborhood of. If such a map exists, it is unique; this map is called thedifferential and is denoted by,[21] coinciding with the traditional notation for the classical (though logically flawed) notion of a differential as an infinitely small "piece" ofF. This definition represents a generalization of the usual definition of differentiability for vector-valued functions of (open subsets of) Euclidean spaces.
Let be aprobability space and let. An array ofrandom variables is called infinitesimal if for every, we have:[22]
The notion of infinitesimal array is essential in some central limit theorems and it is easily seen by monotonicity of the expectation operator that any array satisfyingLindeberg's condition is infinitesimal, thus playing an important role inLindeberg's Central Limit Theorem (a generalization of thecentral limit theorem).
Theorem 24.29. The surreal number system is the largest ordered field
Relates to Infinitesimal | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| |||||||||||||||||||||||||||||||||
