# What is Infinitesimal: Definition and 142 Discussions

In mathematics, an infinitesimal or 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 includes both hyperreal numbers and ordinal numbers, which is the largest ordered field.
The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though 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 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 calculation of area of a circle by representing the latter as an infinite-sided polygon. Simon Stevin's work on 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 inassignable 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.

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.

View More On Wikipedia.org
1. ### I Monad in non-standard analysis

Quick two questions: (a) In the hyperreals, is 0 considered an infinitesimal? (b) Does a monad include the real number? I seem to get contradictory answers in the Internet. Thanks.
2. ### 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##...
3. ### B How to find the infinitesimal coordinate transform along a hyperbola?

I've been told that the infinitesimal change in coordinates x and y as you rotate along a hyperbola that fits the equation b(dy)^2-a(dx)^2=r takes the form δx=bwy and δy=awx, where w is a function of the angle of rotation (I'm pretty sure it's something like sinh(theta) but it wasn't clarified...
4. ### A Infinitesimal Coordinate Transformation and Lie Derivative

I need to prove that under an infinitesimal coordinate transformation ##x^{'\mu}=x^\mu-\xi^\mu(x)##, the variation of a vector ##U^\mu(x)## is $$\delta U^\mu(x)=U^{'\mu}(x)-U^\mu(x)=\mathcal{L}_\xi U^\mu$$ where ##\mathcal{L}_\xi U^\mu## is the Lie derivative of ##U^\mu## wrt the vector...
5. ### I Analyzing Infinitesimal Motion

Hello everyone! I was wondering about this physics problem. First example: If a rocket is traveling in a straight line continuously in uniform motions from position 0 to position 1000 in 10 seconds then it will move through an infinite number of points. Since it is always changing position...
6. ### I Several Questions About Smooth Infinitesimal Analysis

Hello. I read about smooth infinitesimal analysis and I have several questions: 1.What does "ε.1" and "ε.0" mean in this proof? (photo1) (https://publish.uwo.ca/~jbell/basic.pdf , page 5-6) 2. For what purpose do we use Kock-Lawvere axiom when we deal with law of excluded middle? (photo2)...
7. ### I Proving Continuous Functions in Smooth Infinitesimal Analysis

Hello. How to prove that in smooth infinitesimal analysis every function on R is continuous? (Every function whose domain is R, the real numbers, is continuous and infinitely differentiable.) Thanks.
8. ### B Will we get an infinitesimal x when we neglect ##x^2## in ##x+x^2##?

Hello. Let's assume that we have ##2x \Delta x + \Delta x^2##. When ##\Delta x## tends to zero we can neglect ##\Delta x^2## and we'll get ##2xdx##. Let's assume that we have ##x + x^2##. When ##x## tends to zero we can neglect ##x^2##. Will we get an infinitesimal ##x## as such as ##dx##? Thanks.
9. ### I Reasoning behind Infinitesimal multiplication

Hello everyone! I have quite a bit of experience with standard calculus methods of differentiation and integration, but after seeing some of Walter Lewin's lectures I noticed in his derivation of change in momentum for a rocket ejecting a mass dm, with a change in velocity of the rockey dv, he...
10. ### Computing the infinitesimal generators for the Mobius transformation

I don't know where to start. I understand that the constraint ##ad-bc=1## gives us one less parameter since ##d=1+bc/a##. So we can rewrite our original function. I know how to compute the generators of matrix groups but in this case the generators will be functions. I also know there should be...
11. ### A Take your time, and feel free to ask if something is still unclear.

chapter 4.8 of Goldstein’s classical mechanics 3rd edition that deals with infinitesimal rotations, and the following is the part I got stuck: (p.166~167) : I'm not able to understand what the author is trying to say. How does "If ##d\boldsymbol{\Omega}## is to be a vector in the same sense...
12. ### A Exploring Infinitesimal Rotations in Classical Mechanics

Can anybody please help me to understand that why under infinitesimal rotation ##x1## transforms in the way as shown in equation 4-100? This is from Goldstein's Classical Mechanics page chapter 5 and page 168 on the Kinematics of Rigid body motion.
13. ### B The infinite and the infinitesimal

Over the years the following has continued to be my biggest question in Cosmology. In the past couple of years I wondered if we have got any closer to understanding whether our space is infinite or infinitesimal? (By infinitesimal I mean that there is no lower limit to the minimum separation of...
14. ### B Mass Reduction in Combustion: Real or Myth?

I have studied in high school that all chemical reactions obey conservation of mass, as the atoms are merely re-arranged, but when I read through special relativity, I was reading that you can show an infinitesimal change in mass (based on E=mc2) in combustion that's not noticeable that's being...
15. ### I Infinitesimal Movement Along 3-d Geodesics

I would like to determine how a point (xo,yo,zo) moves along a geodesic on a three dimensional graph when it initially starts moving in a direction according to a unit vector <vxo,vyo,vzo>. So, if I start at that point, after a very small amount of time, what is its new coordinate (x1,y1,z1) and...
16. ### I I don't understand about infinitesimal translation

and But The infinitesimal translation denoted by equ 1.6.12 and 1.6.32 And then i understand about equation 1.6.35 but equation 1.6.36 Why they take limit N go to inf ? , multiply 1/ N ? and power N ? So is the relationship below still true? ## F(Δ x'\hat{x}) = 1 - \frac{i p_x \dot{} Δ...

19. ### Finite rotations and infinitesimal rotations

Hi I am using Kleppner and it states that finite rotations do not commute but infinitesimal rotations do commute. I follow the logic in the book but i don't understand the concept. Surely a finite rotation consists of many , many infinitesimal rotations and if they commute why doesn't the finite...
20. ### Infinitesimal coordinate transformation of the metric

I kinda know how to do this problem, it is just that I hit a sign problem. If I take the partial derivative of the coordinate transformation with respect to ##x'^\mu##, I get writing it first in the inverse form, ##x^\alpha = x'^\alpha - \epsilon^\alpha## ##\frac{\partial x^\alpha}{\partial...
21. ### I Finding an infinitesimal limit function

I have the function: ##\sqrt{\left(\frac{x}{h}+1\right)^{2}+\left(\frac{y}{h}\right)^{2}}-\sqrt{\left(\frac{x}{h}\right)^{2}+\left(\frac{y}{h}\right)^{2}}## I would like to find an analytical solution, the equivalent function, in the limit of h approaching zero.Additional info which might be...
22. ### Infinitesimal Perturbation in a potential well

If I calculate ## <\psi^0|\epsilon|\psi^0>## and ## <\psi^0|-\epsilon|\psi^0>## separately and then add, the correction seems to be 0 since ##\epsilon## is a constant perturbation term. SO how should I approach this? And how the Δ is relevant in this calculation?
23. ### I A random question comes to mind, about the infinitesimal area of rings

I know the area of a thin ring of radius ##r## can be expressed as ##2\pi rdr##, however, I wonder if I use the usual way of calculating area of a ring, can I reach the same conclusion? I got this: $$4\pi(r+dr)^2-4\pi r^2=4\pi r^2+8\pi rdr+4\pi (dr)^2-4\pi r^2=8\pi rdr+4\pi (dr)^2$$And now I'm...
24. ### I Variation of geometrical quantities under infinitesimal deformation

This question is about 2-d surfaces embedded inR3It's easy to find information on how the metric tensor changes when $$x_{\mu}\rightarrow x_{\mu}+\varepsilon\xi(x)$$ So, what about the variation of the second fundamental form, the Gauss and the mean curvature? how they change? I found some...
25. ### The Effects of Incredibly Small Inertia on Acceleration: A Scientific Inquiry

I have a question about inertia (as in mass and Newton's first law) being extremely small. Now, say the inertia of an object is, say, 0.00000000000000000000005 kilograms, or something like that. Would a light, weak force exerted on the object accelerate the object to high speeds, or would it...

40. ### I Extending an infinitesimal operator

I notice that in Quantum Mechanics when extending an infinitesimal operation to a finite one, we should end with the exponential. For example: (rf. Sakurai, Modern Quantum Mechanics) $$D(\boldsymbol{\hat n}, d\phi) = 1 - \frac{i}{\hbar} ( \boldsymbol {J \cdot \hat n})d\phi$$ This is the...
41. ### I Epsilon-delta vs. infinitesimal

Background: mechanical engineer with a flawed math education (and trying to make up for it). I have recently read this statement (and others like it): "We shall also informally use terminology such as "infinitesimal" in order to avoid having to discuss the (routine) "epsilon-delta" analytical...
42. ### I Infinitesimal cube and the stress tensor

The Cauchy stress tensor at a material point is usually visualized using an infinitesimal cube. The stress vectors (traction vectors) on opposite sides of the cube are equal in magnitude and opposite in direction. As a result, the infinitesimal cube is in equilibrium. However, when we derive...
43. ### A Does an infinitesimal generator of acceleration exist?

I am trying to determine what types of field theories have a Lagrangian that is symmetric under an Infinitesimal acceleration coordinate transformations. Does an infinitesimal generator of acceleration exist? How could I go about constructing this matrix?
44. ### B What are surreal numbers and how do they work?

Hey guys! I have heard of this concept in various places and sort of understands what it attempts to do. Can anybody please explain it to me in more detail like how it works, how to notate it, and how to expand it to infinities and infinitesimals. Thanks in advance! Aakash Lakshmanan xphysx.com...
45. ### Gaussian integration in infinitesimal limit

Homework Statement Given the wave function of a particle \Psi(x,0) = \left(\frac{2b}{\pi}\right)^{1/4}e^{-bx^2} , what is the probability of finding the particle between 0 and \Delta x , where \Delta x can be assumed to be infinitesimal. Homework EquationsThe Attempt at a Solution I proceed...
46. ### I Is space infinitesimal? Can it be divided unlimited times?

Is space infinitesimal? By this I mean can it be divided an unlimited number of times? If we take a 1m ruler and divide it in 2 and we get a 50cm ruler How many times can we keep doing this? (disregard that the ruler is made of atoms) So for many years I have been told that the limit is the...
47. ### 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...
48. ### Infinitesimal Lorentz transformations

Homework Statement Show that an infinitesimal boost by v^j along the x^j-axis is given by the Lorentz transformation \Lambda^{\mu}_{\nu} = \begin{pmatrix} 1 & v^1 & v^2 & v^3\\ v^1 & 1 & 0 & 0\\ v^2 & 0 & 1 & 0\\ v^3 & 0 & 0 & 1 \end{pmatrix} Show that an infinitesimal rotation by theta^j...
49. ### I Infinitesimal area element in polar coordinate

We know, that the infinitesimal area element in Cartesian coordinate system is ##dy~dx## and in Polar coordinate system, it is ##r~dr~d\theta##. This inifinitesimal area element is calculated by measuring the area of the region bounded by the lines ##x,~x+dx, ~y,~y+dy## (for polar coordinate...
50. ### Investigations into the infinitesimal Lorentz transformation

Homework Statement [/B] A Lorentz transformation ##x^{\mu} \rightarrow x'^{\mu} = {\Lambda^{\mu}}_{\nu}x^{\nu}## is such that it preserves the Minkowski metric ##\eta_{\mu\nu}##, meaning that ##\eta_{\mu\nu}x^{\mu}x^{\nu}=\eta_{\mu\nu}x'^{\mu}x'^{\nu}## for all ##x##. Show that this implies...