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In mathematics, an infinitesimal number is a quantity that is closer to
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
than any standard
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
, but that is not zero. The word ''infinitesimal'' comes from a 17th-century
Modern Latin New Latin (also called Neo-Latin or Modern Latin) is the revival of Literary Latin used in original, scholarly, and scientific works since about 1500. Modern scholarly and technical nomenclature, such as in zoological and botanical taxonomy a ...
coinage ''infinitesimus'', which originally referred to the " infinity- th" item in a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
. Infinitesimals do not exist in the standard real number system, but they do exist in other number systems, such as the surreal number system and the hyperreal number system, which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities; the augmentations are the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
s of one another. Infinitesimal numbers were introduced in the development of calculus, in which the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
was first conceived as a ratio of two infinitesimal quantities. This definition was not rigorously formalized. As calculus developed further, infinitesimals were replaced by limits, which can be calculated using the standard real numbers. Infinitesimals regained popularity in the 20th century with
Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorp ...
's development of
nonstandard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...
and the
hyperreal numbers In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains 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 hyperreal cardinal and ordinal numbers, which is the largest
ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered fiel ...
.
Vladimir Arnold Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov– ...
wrote in 1990: The crucial insight for making infinitesimals feasible mathematical entities was that they could still retain certain properties such as
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
or
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
, even if these entities were infinitely small. Infinitesimals are a basic ingredient in
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
as developed by
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
, including the
law of continuity The law of continuity is a heuristic principle introduced by Gottfried Leibniz based on earlier work by Nicholas of Cusa and Johannes Kepler. It is the principle that "whatever succeeds for the finite, also succeeds for the infinite". Kepler used ...
and the
transcendental law of homogeneity In mathematics, the transcendental law of homogeneity (TLH) is a heuristic principle enunciated by Gottfried Wilhelm Leibniz most clearly in a 1710 text entitled ''Symbolismus memorabilis calculi algebraici et infinitesimalis in comparatione potent ...
. 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 an
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
. The concept of infinitesimals was originally introduced around 1670 by either Nicolaus Mercator or
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of math ...
. Archimedes used what eventually came to be known as the
method of indivisibles In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that p ...
in his work ''
The Method of Mechanical Theorems ''The Method of Mechanical Theorems'' ( el, Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος), also referred to as ''The Method'', is one of the major surviving works of the ancient Greek polymath Ar ...
'' to find areas of regions and volumes of solids. In his formal published treatises, Archimedes solved the same problem using the
method of exhaustion The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in are ...
. The 15th century saw the work of
Nicholas of Cusa Nicholas of Cusa (1401 – 11 August 1464), also referred to as Nicholas of Kues and Nicolaus Cusanus (), was a German Catholic cardinal, philosopher, theologian, jurist, mathematician, and astronomer. One of the first German proponents of Re ...
, further developed in the 17th century by Johannes Kepler, in particular, the calculation of the area of a circle by representing the latter as an infinite-sided polygon.
Simon Stevin Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated vario ...
's work on the decimal representation of all numbers in the 16th century prepared the ground for the real continuum.
Bonaventura Cavalieri Bonaventura Francesco Cavalieri ( la, Bonaventura Cavalerius; 1598 – 30 November 1647) was an Italian mathematician and a Jesuate. He is known for his work on the problems of optics and motion, work on indivisibles, the precursors of in ...
'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 of codimension 1. 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 denoted 1/∞ 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 as
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
and Joseph-Louis Lagrange. Augustin-Louis Cauchy exploited infinitesimals both in defining continuity in his ''
Cours d'Analyse ''Cours d'Analyse de l’École Royale Polytechnique; I.re Partie. Analyse algébrique'' is a seminal textbook in infinitesimal calculus published by Augustin-Louis Cauchy in 1821. The article follows the translation by Bradley and Sandifer in de ...
'', and in defining an early form of a Dirac delta function. As Cantor and
Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
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 both
Émile Borel Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability. Biography Borel was ...
and
Thoralf Skolem Thoralf Albert Skolem (; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory. Life Although Skolem's father was a primary school teacher, most of his extended family were farmers. 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 by
Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorp ...
in 1961, who developed
nonstandard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...
based on earlier work by
Edwin Hewitt Edwin Hewitt (January 20, 1920, Everett, Washington – June 21, 1999) was an American mathematician known for his work in abstract harmonic analysis and for his discovery, in collaboration with Leonard Jimmie Savage, of the Hewitt–Savage z ...
in 1948 and
Jerzy Łoś Jerzy Łoś (born 22 March 1920 in Lwów, Poland (now Lviv, Ukraine) – 1 June 1998 in Warsaw) () was a Polish mathematician, logician, economist, and philosopher. He is especially known for his work in model theory, in particular for " Łoś's th ...
in 1955. The hyperreals implement an infinitesimal-enriched continuum and the
transfer principle In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the first- ...
implements Leibniz's law of continuity. The standard part function implements Fermat's
adequality Adequality is a technique developed by Pierre de Fermat in his treatise ''Methodus ad disquirendam maximam et minimam''Eleatic School. The
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
mathematician Archimedes (c. 287 BC – c. 212 BC), in ''
The Method of Mechanical Theorems ''The Method of Mechanical Theorems'' ( el, Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος), also referred to as ''The Method'', is one of the major surviving works of the ancient Greek polymath Ar ...
'', was the first to propose a logically rigorous definition of infinitesimals. His
Archimedean property In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, typica ...
defines a number ''x'' as infinite if it satisfies the conditions , ''x'', >1, , ''x'', >1+1, , ''x'', >1+1+1, ..., and infinitesimal if ''x''≠0 and a similar set of conditions holds for ''x'' 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 mathematician John Wallis introduced the expression 1/∞ in his 1655 book ''Treatise on the Conic Sections''. The symbol, which denotes the reciprocal, or inverse, of 
The infinity symbol (\infty) is a List of mathematical symbols, mathematical symbol representing the concept of infinity. This symbol is also called a lemniscate, after the lemniscate curves of a similar shape studied in algebraic geometry, or ...
, is the symbolic representation of the mathematical concept of an infinitesimal. In his ''Treatise 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 a
thought experiment A thought experiment is a hypothetical situation in which a hypothesis, theory, or principle is laid out for the purpose of thinking through its consequences. History The ancient Greek ''deiknymi'' (), or thought experiment, "was the most anc ...
of adding an infinite number of parallelograms of infinitesimal width to form a finite area. This concept was the predecessor to the modern method of integration used in
integral calculus In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
. The conceptual origins of the concept of the infinitesimal 1/∞ can be traced as far back as the Greek philosopher
Zeno of Elea Zeno of Elea (; grc, Ζήνων ὁ Ἐλεᾱ́της; ) was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known ...
, whose Zeno'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. Prior to the invention of calculus mathematicians were able to calculate tangent lines using
Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
's method of
adequality Adequality is a technique developed by Pierre de Fermat in his treatise ''Methodus ad disquirendam maximam et minimam''René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...
'
method of normals In calculus, the method of normals was a technique invented by Descartes for finding normal and tangent lines to curves. It represented one of the earliest methods for constructing tangents to curves. The method hinges on the observation that the ...
. There is debate among scholars as to whether the method was infinitesimal or algebraic in nature. When Newton and
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
invented the
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, they made use of infinitesimals, Newton's '' fluxions'' and Leibniz' '' differential''. The use of infinitesimals was attacked as incorrect by Bishop Berkeley in his work '' The Analyst''. Mathematicians, scientists, and engineers continued to use infinitesimals to produce correct results. In the second half of the nineteenth century, the calculus was reformulated by Augustin-Louis Cauchy,
Bernard Bolzano Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Gonzal Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his lib ...
,
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
,
Cantor A cantor or chanter is a person who leads people in singing or sometimes in prayer. In formal Jewish worship, a cantor is a person who sings solo verses or passages to which the choir or congregation responds. In Judaism, a cantor sings and lead ...
,
Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
, and others using the
(ε, δ)-definition of limit Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches z ...
and
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
. While the followers of Cantor, Dedekind, and Weierstrass sought to rid analysis of infinitesimals, and their philosophical allies like
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
and Rudolf Carnap declared that infinitesimals are ''pseudoconcepts'',
Hermann Cohen Hermann Cohen (4 July 1842 – 4 April 1918) was a German Jewish philosopher, one of the founders of the Marburg school of neo-Kantianism, and he is often held to be "probably the most important Jewish philosopher of the nineteenth century ...
and his
Marburg school In late modern philosophy, late modern continental philosophy, neo-Kantianism (german: Neukantianismus) was a revival of the 18th-century philosophy of Immanuel Kant. The Neo-Kantians sought to develop and clarify Kant's theories, particularly hi ...
of
neo-Kantianism In late modern continental philosophy, neo-Kantianism (german: Neukantianismus) was a revival of the 18th-century philosophy of Immanuel Kant. The Neo-Kantians sought to develop and clarify Kant's theories, particularly his concept of the "thin ...
sought to develop a working logic of infinitesimals. The mathematical study of systems containing infinitesimals continued through the work of
Levi-Civita Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made signific ...
, 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 (see
hyperreal number In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
s).


First-order properties

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 no quantification over sets, 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'' and ''y'', ''xy'' = ''yx''." However, statements of the form "for any ''set'' ''S'' of numbers ..." may not carry over. Logic with this limitation on quantification is referred to as
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
. 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, the completeness property cannot be expected to carry over, because the reals are the unique complete ordered field up to isomorphism. We can distinguish three levels at which a non-Archimedean number system could have first-order properties compatible with those of the reals: # An
ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered fiel ...
obeys all the usual axioms of the real number system that can be stated in first-order logic. For example, the commutativity axiom ''x'' + ''y'' = ''y'' + ''x'' holds. # A
real closed field In mathematics, a real closed field is a field ''F'' that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. D ...
has all the first-order properties of the real number system, regardless of whether they are usually taken as axiomatic, for statements involving the basic ordered-field relations +, ×, and ≤. This is a stronger condition than obeying the ordered-field axioms. More specifically, one includes additional first-order properties, such as the existence of a root for every odd-degree polynomial. For example, every number must have a cube root. # The system could have all the first-order properties of the real number system for statements involving ''any'' relations (regardless of whether those relations can be expressed using +, ×, and ≤). For example, there would have to be a sine function that is well defined for infinite inputs; the same is true for every real function. 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, the transcendental 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.


Number systems that include infinitesimals


Formal series


Laurent series

An example from category 1 above is the field of Laurent 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 refers to this system as the super-reals, not to be confused with the
superreal number In abstract algebra, the superreal numbers are a class of extensions of the real numbers, introduced by H. Garth Dales and W. Hugh Woodin as a generalization of the hyperreal numbers and primarily of interest in non-standard analysis, model theo ...
system of Dales and Woodin. Since a Taylor 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.


The Levi-Civita field

The
Levi-Civita field In mathematics, the Levi-Civita field, named after Tullio Levi-Civita, is a non-Archimedean ordered field; i.e., a system of numbers containing infinite and infinitesimal quantities. Each member a can be constructed as a formal series of the form ...
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.


Transseries

The field of
transseries In mathematics, the field \mathbb^ of logarithmic-exponential transseries is a non-Archimedean ordered differential field which extends comparability of asymptotic growth rates of elementary nontrigonometric functions to a much broader class of ob ...
is larger than the Levi-Civita field. An example of a transseries is: :e^\sqrt+\ln\ln x+\sum_^\infty e^x x^, where for purposes of ordering ''x'' is considered infinite.


Surreal numbers

Conway's
surreal number In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals ...
s fall into category 2, except that the surreal numbers form a proper class and not a set, 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. There is a natural extension of the exponential function to the surreal numbers.


Hyperreals

The most widespread technique for handling infinitesimals is the hyperreals, developed by
Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorp ...
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 the
transfer principle In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the first- ...
, proved by
Jerzy Łoś Jerzy Łoś (born 22 March 1920 in Lwów, Poland (now Lviv, Ukraine) – 1 June 1998 in Warsaw) () was a Polish mathematician, logician, economist, and philosopher. He is especially known for his work in model theory, in particular for " Łoś's th ...
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 \mathbb has a natural counterpart ^*\mathbb, which contains both finite and infinite integers. A proposition such as \forall n \in \mathbb, \sin n\pi=0 carries over to the hyperreals as \forall n \in ^*\mathbb, ^*\!\!\sin n\pi=0 .


Superreals

The
superreal number In abstract algebra, the superreal numbers are a class of extensions of the real numbers, introduced by H. Garth Dales and W. Hugh Woodin as a generalization of the hyperreal numbers and primarily of interest in non-standard analysis, model theo ...
system of Dales and Woodin is a generalization of the hyperreals. It is different from the super-real system defined by David Tall.


Dual numbers

In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...
, the
dual number In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0. Du ...
s extend the reals by adjoining one infinitesimal, the new element ε with the property ε2 = 0 (that is, ε is
nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cla ...
). Every dual number has the form ''z'' = ''a'' + ''b''ε with ''a'' and ''b'' being uniquely determined real numbers. One application of dual numbers is
automatic differentiation In mathematics and computer algebra, automatic differentiation (AD), also called algorithmic differentiation, computational differentiation, auto-differentiation, or simply autodiff, is a set of techniques to evaluate the derivative of a function s ...
. This application can be generalized to polynomials in n variables, using the
Exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
of an n-dimensional vector space.


Smooth infinitesimal analysis

Synthetic differential geometry or smooth infinitesimal analysis have roots in category theory. This approach departs from the classical logic used in conventional mathematics by denying the general applicability of the
law of excluded middle In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradi ...
– i.e., ''not'' (''a'' ≠ ''b'') does not have to mean ''a'' = ''b''. A ''nilsquare'' or ''
nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cla ...
'' infinitesimal can then be defined. This is a number ''x'' where ''x''2 = 0 is true, but ''x'' = 0 need not be true at the same time. Since the background logic is
intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
, 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.


Infinitesimal delta functions

Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
used an infinitesimal \alpha to write down a unit impulse, infinitely tall and narrow Dirac-type delta function \delta_\alpha satisfying \int F(x)\delta_\alpha(x) = F(0) 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 and Lazare Carnot's terminology. Modern set-theoretic approaches allow one to define infinitesimals via the
ultrapower The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factor ...
construction, where a null sequence becomes an infinitesimal in the sense of an equivalence class modulo a relation defined in terms of a suitable
ultrafilter In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter o ...
. The article by Yamashita (2007) contains bibliography on modern Dirac delta functions in the context of an infinitesimal-enriched continuum provided by the hyperreals.


Logical properties

The method of constructing infinitesimals of the kind used in nonstandard analysis depends on the
model A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
and which collection of axioms are used. We consider here systems where infinitesimals can be shown to exist. In 1936 Maltsev proved the
compactness theorem In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generally ...
. 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 integer ''n'' there is a positive number ''x'' 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 number ''x'' such that for any positive integer ''n'' 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 in ZFC
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
: for any positive integer ''n'' it is possible to find a real number between 1/''n'' and zero, but this real number depends on ''n''. Here, one chooses ''n'' first, then one finds the corresponding ''x''. In the second expression, the statement says that there is an ''x'' (at least one), chosen first, which is between 0 and 1/''n'' for any ''n''. In this case ''x'' 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 a
one-dimensional In physics and mathematics, a sequence of ''n'' numbers can specify a location in ''n''-dimensional space. When , the set of all such locations is called a one-dimensional space. An example of a one-dimensional space is the number line, where the ...
linearly ordered In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive) ...
set of numbers, but fundamentally, there are two different approaches: : 1) Extend the number system so that it contains more numbers than the real numbers. : 2) Extend the axioms (or extend the language) so that the distinction between the infinitesimals and non-infinitesimals can be made in the real numbers themselves. In 1960,
Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorp ...
provided an answer following the first approach. The extended set is called the hyperreals 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 called nonstandard. In 1977 Edward Nelson provided an answer following the second approach. The extended axioms are IST, which stands either for Internal 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.


Infinitesimals in teaching

Calculus textbooks based on infinitesimals include the classic ''
Calculus Made Easy ''Calculus Made Easy'' is a book on infinitesimal calculus originally published in 1910 by Silvanus P. Thompson, considered a classic and elegant introduction to the subject. The original text continues to be available as of 2008 from Macmilla ...
'' by Silvanus P. Thompson (bearing the motto "What one fool can do another can") and the German text ''Mathematik fur Mittlere Technische Fachschulen der Maschinenindustrie'' by R. Neuendorff. Pioneering works based on
Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorp ...
's infinitesimals include texts by Stroyan (dating from 1972) and
Howard Jerome Keisler Howard Jerome Keisler (born 3 December 1936) is an American mathematician, currently professor emeritus at University of Wisconsin–Madison. His research has included model theory and non-standard analysis. His Ph.D. advisor was Alfred Tarski a ...
( 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. Another elementary calculus text that uses the theory of infinitesimals as developed by Robinson is ''Infinitesimal Calculus'' by Henle and Kleinberg, originally published in 1979. 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 the ''hyperhyper''reals, and demonstrate some applications for the extended model. An elementary calculus text based on smooth infinitesial 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.


Functions tending to zero

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's ''Advanced Calculus'' defines the function class of infinitesimals, \mathfrak, as a subset of functions f:V\to W between normed vector spaces by
\mathfrak(V,W) = \,
as well as two related classes \mathfrak,\mathfrak (see
Big-O notation Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
) by
\mathfrak(V,W) = \, and
\mathfrak(V,W) = \.
The set inclusions \mathfrak(V,W)\subsetneq\mathfrak(V,W)\subsetneq\mathfrak(V,W)generally hold. That the inclusions are proper is demonstrated by the real-valued functions of a real variable f:x\mapsto , x, ^, g:x\mapsto x , and h:x\mapsto x^2 :
f,g,h\in\mathfrak(\mathbb,\mathbb),\ g,h\in\mathfrak(\mathbb,\mathbb),\ h\in\mathfrak(\mathbb,\mathbb) but f,g\notin\mathfrak(\mathbb,\mathbb) and f\notin\mathfrak(\mathbb,\mathbb).
As an application of these definitions, a mapping F:V\to W between normed vector spaces is defined to be differentiable at \alpha\in V if there is a T\in\mathrm(V,W) .e, a bounded linear map V\to Wsuch that
(\alpha+\xi)-F(\alpha)T(\xi)\in \mathfrak(V,W)
in a neighborhood of \alpha. If such a map exists, it is unique; this map is called the ''differential'' and is denoted by dF_\alpha, coinciding with the traditional notation for the classical (though logically flawed) notion of a differential as an infinitely small "piece" of ''F''. This definition represents a generalization of the usual definition of differentiability for vector-valued functions of (open subsets of) Euclidean spaces.


Array of random variables

Let (\Omega,\mathcal,\mathbb) be a
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
and let n\in\mathbb. An array \ of random variables is called infinitesimal if for every \epsilon>0, we have: :\max_\mathbb\\to 0\text n\to\infty 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 satisfying Lindeberg's condition is infinitesimal, thus playing an important role in Lindeberg's Central Limit Theorem (a generalization of the
central limit theorem In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themsel ...
).


See also

*
Cantor function In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. ...
*
Differential (infinitesimal) In mathematics, differential refers to several related notions derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions. The term is used in various branches of mathema ...
*
Indeterminate form In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this s ...
* Infinitesimal calculus *
Infinitesimal transformation In mathematics, an infinitesimal transformation is a limiting form of ''small'' transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space. This is conventionally represented by a 3×3 s ...
*
Instant In physics and the philosophy of science, instant refers to an infinitesimal interval in time, whose passage is instantaneous. In ordinary speech, an instant has been defined as "a point or very short space of time," a notion deriving from its ety ...
*
Nonstandard calculus In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some arguments in calculus that were previously considered m ...
* Model theory


Notes


References

* B. Crowell, "Calculus" (2003) * Dawson, C. Bryan, "Calculus Set Free: Infinitesimals to the Rescue" (2022) Oxford University Press *Ehrlich, P. (2006) The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes. Arch. Hist. Exact Sci. 60, no. 1, 1–121. * Malet, Antoni. "Barrow, Wallis, and the remaking of seventeenth-century indivisibles". ''Centaurus'' 39 (1997), no. 1, 67–92. * J. Keisler, "Elementary Calculus" (2000) University of Wisconsin * K. Stroyan "Foundations of Infinitesimal Calculus" (1993) * Stroyan, K. D.; Luxemburg, W. A. J. Introduction to the theory of infinitesimals. Pure and Applied Mathematics, No. 72. Academic Press arcourt Brace Jovanovich, Publishers New York-London, 1976. *
Robert Goldblatt __notoc__ Robert Ian Goldblatt (born 1949) is a mathematical logician who is Emeritus Professor in the School of Mathematics and Statistics at Victoria University, Wellington, New Zealand. His most popular books are ''Logics of Time and Computatio ...
(1998) "Lectures on the hyperreals" Springer. * Cutland et al. "Nonstandard Methods and Applications in Mathematics" (2007) Lecture Notes in Logic 25, Association for Symbolic Logic. * "The Strength of Nonstandard Analysis" (2007) Springer. * * Yamashita, H.: Comment on: "Pointwise analysis of scalar Fields: a nonstandard approach" . Math. Phys. 47 (2006), no. 9, 092301; 16 pp. J. Math. Phys. 48 (2007), no. 8, 084101, 1 page. {{Authority control Calculus History of calculus Infinity Nonstandard analysis History of mathematics Mathematical logic