What is infinitesimals: Definition and 35 Discussions

In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinity-th" item in a sequence.
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 reciprocals of one another.
Infinitesimal numbers were introduced in the development of calculus, in which the derivative 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's development of nonstandard analysis and the hyperreal 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.
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.
The crucial insight for making infinitesimals feasible mathematical entities was that they could still retain certain properties such as angle or slope, even if these entities were infinitely small.Infinitesimals are a basic ingredient in calculus as developed by Leibniz, including the law of continuity and the transcendental 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 an integral.
The concept of infinitesimals was originally introduced around 1670 by either Nicolaus Mercator or Gottfried Wilhelm Leibniz. Archimedes used what eventually came to be known as the method of indivisibles in his work The Method of Mechanical Theorems 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 15th century saw the work of Nicholas of Cusa, 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'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 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 and Joseph-Louis Lagrange. Augustin-Louis Cauchy exploited infinitesimals both in defining continuity in his Cours d'Analyse, and in defining an early form of a Dirac delta function. As Cantor and Dedekind 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 and Thoralf 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 in 1961, who developed nonstandard analysis based on earlier work by Edwin Hewitt in 1948 and Jerzy Łoś in 1955. The hyperreals implement an infinitesimal-enriched continuum and the transfer principle implements Leibniz's law of continuity. The standard part function implements Fermat's adequality.

View More On Wikipedia.org
  1. J

    I Integration with different infinitesimal intervals

    Some sources state a similar format of the following $$\int_a^{a+da}f(x)dx=f(a)da$$ Which had me thinking whether the following integration can exist $$\int_a^{a+dx}f(x)dx=f(a)dx$$ I have difficulty grasping some aspects about these integrations 1. Regarding the 1st integration, shouldn't ##a##...
  2. bhobba

    Insights Beginner's Guide to Precalculus, Calculus and Infinitesimals

    [url="https://www.physicsforums.com/insights/precalculus-calculus-and-infinitesimals/"]Continue reading...
  3. bhobba

    Insights Precalculus, Calculus and Infinitesimals

    See my insights article for those interested in an unconventional approach to doing Precalculus at an accelerated pace and beginning Calculus. It is different from the usual way that a precalculus is done text in that it covers in the US what is called Algebra 1, Geometry, Algebra 2, and...
  4. bhobba

    Insights What Are Numbers? - Insights for Beginners

    Hi Everyone I have been doing further investigation into infinitesimals since I wrote my insight article. I had an issue with the original article; the link to the foundations of natural numbers, integers, and rational numbers was somewhat advanced. I did need to write an insights article at...
  5. bhobba

    B How can hyperreal numbers make infinitesimals logically sound in calculus?

    When I learned calculus, the intuitive idea of infinitesimal was used. These are numbers so small that, for all practical purposes (say 1/trillion to the power of a trillion) can be taken as zero but are not. That way, when defining the derivative, you do not run into 0/0, but when required...
  6. M

    B Representation of infinitesimals in different ways

    Hello. There are 4 types of infinitesimals: 1) dx=1/N, N is the number of elemets of the set of the natural numbers (letter N is used to indicate the cardinality of the set of natural numbers) 2) Hyperreal numbers: ε=1/ω, ω is number greater than any real number. 3) Surreal numbers: { 0, 1...
  7. F

    I Infinitesimals' rates of approaching 0

    When comparing 2 infinitesimals, does the higher order one approach 0 faster or slower?
  8. LCSphysicist

    Is dlπ/2 equivalent to dl/(dlπ/2)?

    dl = Infinitesimal length of the segment. dlπ/2 = the semicircle length lim dl-> zero dl/(dlπ/2) = 2/π, no zero, so the answer would be yes. But second the book, the answer is no, where am i wrong?
  9. Aleoa

    Deriving the lever law using infinitesimals

    I'm trying to derive the lever law by myself, however, I'm stuck. Please follow the logic of my calculations. Every object in the picture has the same mass. I want to prove that, under the effect of the gravitational force, I can replace the objects in A and C with the two objects in B, and...
  10. S

    I Precise intuition about limits and infinitesimals

    I've understood the formal definition of limits and its various applications. However, I'm trying to dive more into the history of how the concept of limits were conceived (more than what Wikipedia tends to cover), and how to formally understand and visualise infinitesimals. For example, I know...
  11. E

    B Using infinitesimals to find the volume of a sphere/surface

    I've always thought of dxat the end of an integral as a "full stop" or something to tell me what variable I'm integrating with respect to. I looked up the derivation of the formula for volume of a sphere, and here, dx is taken as an infinitesimally small change which is multiplied by the area of...
  12. bcrowell

    Teaching about infinitesimals and integration

    This doesn't seem to me like an accurate characterization of NSA. In a synthetic treatment of the reals, we posit some axioms about the reals, and we simply assume that there exists a systems that obeys those axioms. In a constructive description of the reals, we build them up using Dedekind...
  13. TheDemx27

    How do you Treat Infinitesimals?

    All throughout calculus texts, the authors have always put conditions on the manipulation of differentials. They say that for the chain rule, the cancellation of the differentials is simply a way to remember the formula. When doing separation of variable for ODEs, texts always say something...
  14. Vinay080

    What is the Euler's stand on infinitesimals?

    Euler was the master in analysisng anything. This can be seen in his words in the preface of his book "Mmathematica" (translated by Ian Bruce), where he speaks on the text of Hermann "Phoronomiam": Euler has given many insightful words on analysisng things in his preface of many other books...
  15. S

    Can one define higher order infinitesimals?

    Say ##A##, ##B##, ##C##,... are finite numbers; real, complex, quarternians, tensors, or what have you. "First Order" infinitesimals are finite variables prepended with the letter ##d##. Infinitesimal of any order, are prepended with ##d^n## where ##n## is the infinitesimal "order". Finite...
  16. C

    Infinitesimals as interval limits in integration

    Ok so what I want to know is, is this valid? If so what does it mean?
  17. koustav

    Infinitesimals of order higher

    in a certain problem it was written Δθ=∂θ/∂x*Δx + ∂θ/∂y*Δy + ∂θ/∂z*Δz + infinitesimals of order higher than Δx,Δy and Δz.can anyone tell me what is "infinitesimals of order higher than Δx,Δy and Δz?"
  18. dexterdev

    A doubt related to infinitesimals in continuous fourier transform.

    Hi all, Only few days back I got the idea of probability density function. (Till that day , I believed that pdf plot shows the probability. Now I know why it is density function.) Now I have a doubt on CTFT (continuous time Fourier transform). This is a concept I got from my...
  19. J

    Can You Calculate Electric Fields Without Infinitesimals?

    I'm trying to find a way to use calculus without infinitesimals and I'm stuck on this physics problem. It's a uniform charge distribution question. Basically a half circle with radius r and you have to find the electric field at a point that is along its x-axis. The E_y component will be 0...
  20. bcrowell

    Calculus Elementary Calculus: An Approach Using Infinitesimals by Jerome H. Keisler

    Author: Jerome H. Keisler Title: Elementary Calculus: An Approach Using Infinitesimals Download Link: http://www.math.wisc.edu/~keisler/calc.html Prerequisities: High School Mathematics Table of Contents: Introduction Real and Hyperreal Numbers The Real Line Functions of Real...
  21. M

    Infinitesimals in integration vs delta x in summations

    Hi, I first had a question regarding infinitesimals. What does it mean when the infinitesimal is at the beginning of the integral? For example: ∫dxf(x) is this the same as ∫f(x)dx ? My second question was how to convert a summation to an integral and a summation into an integral...
  22. N

    Separable Equation - Notation Question w/ Infinitesimals

    Please see below link for the two different styles of solving a separable equation. http://en.wikipedia.org/wiki/Separation_of_variables#Ordinary_differential_equations_.28ODE.29 Which one is more proper? Why? My DE teacher told me that strictly speaking it's wrong to use the first method...
  23. W

    Can Infinitesimals Make Calculus Easier to Understand?

    I've been playing around with a free PDF Calculus book lately. But, I have no way to check the logic used to get to a particular answer. I've been trying to find the standard part for: (1/ɛ)((1/sqrt(4+ɛ))-(1/2)) I've tried every way I could think of to algebraically manipulate this in...
  24. Phrak

    Infinitesimals and Infini-tesa-tesimals

    Infinitesimals and "Infini-tesa-tesimals" Positive infinitesimals are defined as greater than zero, and less than 1/n, where n is any number 1,2,3... The set of negative infinitesimals is the same, but where negative infinitesimals are less than zero and greater than 1/-n. Infinities are the...
  25. nomadreid

    Infinitesimals in the Cantor set

    I am not sure into which rubric to put this, but since there is some Model Theory here, I am putting it in this one. First, I define the Cantor set informally: A(0) = [0,1] A(n+1) = the set of closed intervals obtained by taking out the open middle third of each interval contained in A(n)...
  26. Rasalhague

    Quotient of infinitesimals indeterminate?

    In Lectures on the hyperreals: an introduction to nonstandard analysis, pp. 50-51, Goldblatt includes among his hyperreal axioms that the sum of two infinitesimals is infinitesimal, that the product of an infinitesimal and an appreciable (i.e. nonzero real) number is infinitesimal, and that the...
  27. Rasalhague

    Dedekind cuts & infinitesimals

    "A real number is a Dedekind cut in the set Q of rational numbers: a partition of Q into a pair of nonempty disjoint subsets <L,U> with every element of L less than every element of U and L having no largest member. Thus 21/2 can be identified with the cut: L = {q in Q: q2 < 2}, U = {q in Q: q2...
  28. I

    Infinitesimals - finding the E field

    The problem is to find the E field a distance z above the centre of a square sheet of charge of side length a and charge density \sigma. The solutions use the result for a square loop and integrate to get the result for the square sheet, however it's the change \lambda \to \sigma \frac{da}{2}...
  29. A

    Are there any infinitesimals in R?

    Homework Statement Prove that infinitesimals are not a subset of R. Homework Equations N/A The Attempt at a Solution Well, I had two ideas about how to prove this but I'm really not sure about either. Proof 1 was the first idea I had but I think it's probably wrong since it has...
  30. S

    Exploring Limit Laws with Infinitesimals

    Have a look! Hi all, Well, i am posting the work that i have done in proving that theorem. Like i said it is nothing important, but rather it is important only for me, so if you could have a look at what i have done i would really appreciate it. The theorem that i have tried to prove...
  31. E

    Infinitesimals and differentials

    Homework Statement I hate infinitesimals and differentials. When I learned calculus, we used Liebniz notation df/dx only as a convenience for using the chain rule. In physics, apparently, people just play around with differentials and infinitesimals and expect to get the right answer...
  32. D

    Can Calculus be Formulated with Infinitesimals Without Introducing New Numbers?

    Is there a formulation of calculus that uses infinitesimals rigorously without introducing an additional number system (non-standard analysis) and without deviating from classical logic?
  33. D

    Non-standard calculus (infinitesimals)

    Compute the standard part of this, please: \frac{ \sqrt{H+1}}{ \sqrt{2H} + \sqrt{H-1}}, where H is positive infinite. It probably should be some algebra trick I'm not familar with.
  34. J

    Definition for a infinitesimals number

    I know what all of these are, but I’ve never seen or heard a formal definition for them, could someone please provide one? 1) a number 2) a real number 3) a integer 4) a rational number 5) a irrational number 6) a transcendental number 7) a infinitesimals number 8) a hyper real number...