INFINITESIMAL NUMBERS

"The most important equation in mathematics is x² = 0" (Michael Atiyah).

"Infinitesimals are the way to explain the continuum" (Charles Sanders Peirce).

"There is no smallest among the small, for there is always something still smaller" (Anaxagoras).

---

Concept and Definition of Infinitesimal Number
An infinitesimal (or infinitesimal) number −usually denoted as ε− is defined as "a number greater than zero and less than any positive real number". As with the definition of infinity, the infinitesimal is not a real number because if it were, the definition would be contradictory, since the infinitesimal would have to be less than itself.

There are several ways to define an infinitely small number. The two simplest ways might be, for example:

1. As the inverse of an increasingly larger number tending to infinity.
2. By a recursive process: dividing 1 successively by half.

In both cases we are dealing with a dynamic number, which never stabilizes. It is unreachable, as is the case with irrational numbers, such as √2. Apparently, there is no way to define it exactly, to concretize it, because if we do so, we arrive, as we have said, at a contradiction. However, surprisingly, there is a simple way to do it, which we explain below.

The modern definition of infinitesimal is an entity ε such that its square is zero: ε² = 0. This definition of infinitesimal has many advantages:

• It is simple, the simplest possible definition. But this simplicity gives it maximum power and creativity.
  
  
• It is a way of reifying it, in this case at the operational level by means of an imaginary expression.
  
  
• It is not a concrete number. It can be considered a qualitative number, the quality is that of being an infinitely small number.
  
  
• A parallelism is established with the imaginary unit, also defined by an imaginary expression: (i² = −1). This parallelism is such that, just as there are complex numbers of the form a+bi we also define the so-called "dual numbers", which are of the form a+bε, where a and b are real numbers.
  
  
• We must distinguish between infinitesimal and infinitesimal. The former is a noun. The second is a qualifier of an infinitely small entity.
  
  
• As in the case of the imaginary unit i, which is not √(−1), ε is not √(0), for it is a substitution, not an equality of an equation, for if it were an equation, ε would be zero. No doubt Michael Atiyah, in the chapter header quote, is referring to a substitution and not an equation.
  
  The enormous importance Atiyah attaches to this expression is justified because this definition of infinitesimal constitutes a bridge between the discrete and the continuous. In fact, its most important aspect is that it makes it possible to formalize the concept of the derivative of a function in an operational way.
  
  
• It allows to be included in expressions with real numbers.
  
  
• It is constant and static. It does not depend on any variable or process.
  
  
• It allows to formalize somewhat fuzzy aspects such as: infinitely close real numbers, continuous functions, etc.
  
  
• It allows to define infinite infinitesimal numbers. Indeed, if ε is an infinitesimal, then rε is also so for every real number r because (rε)² = 0.
  
  Therefore, all rε expressions are infinitesimal, they all have the same operational structure, and there are as many infinitesimals as there are real numbers, i.e., infinities.
  
  
• The fact that ε is non-zero, allows us to imagine it as a segment of the infinitely small real line, thus establishing a close relationship between continuum and infinitesimals. Thus, to the continuum we can associate two classes of elements:
  
  
  1. Real numbers, which correspond to geometric points, without extension.
  2. Infinitesimals, which correspond to infinitely small segments, but with extension.

In both cases we are dealing with imaginary entities because a geometric point has no real existence and neither does an infinitely small segment.


Specification in MENTAL
Definition

The infinitesimal is defined by the imaginary substitution expression (nilpotent)

(ε*ε = 0)

Properties

• ⟨( (ε^n = ε) ← n>1 )⟩ // follows from definition
  
  
• ⟨( (r*ε)*(r*ε) = 0 )⟩ // id.
  
  
• (ε*0 = 0) // ε behaves like a real number
  
  
• (ε−ε = 0) // id.
  
  
• ⟨( (r1<r2) → (r1*ε)<(r2*ε) )⟩
  
  
• ⟨( (r1 + ε*r2)*(r1 − ε*r2) = r1^2 )⟩
  
  
• ⟨( r^n = (1 + n*ε) )⟩

Examples of expressions with infinitesimal

1. (x = 1+ε) // is self-evaluating
x*x // ev. (1 + 2*ε)
x^3 // ev. (1 + 3*ε)
x^4 // ev. (1 + 4*ε)
   
   In general,
   ⟨( (x^n = 1 + n*ε )⟩
   
   
2. (x = 1 + 2*ε) // is self-evaluating
x*x // ev. (1 + 4*ε)
x+x // ev. (2 + 4*ε)
   
   
3. (x = 1 + 2*ε) // self-evaluate
(y = 1 - 2*ε) // self-evaluates
x*y // ev. 1

Infinitesimals of higher order

An infinitesimal of order 2, symbolized as ε/2, is defined as

( (ε/2)*(ε/2) = ε/1 )

where ((ε/1) = ε).

Analogously, an infinitesimal of order 3 is:

( (ε/3)*(ε/3) = ε/2 )

In general, an infinitesimal of order n:

⟨( (ε/n)*(ε/n) = ε/(n−1) )⟩


Infinitely close real numbers

If r is a real number, r+ε is a real number infinitely close to r.

We could define two infinitely close numbers, r1 and r2 by the expression:

⟨( (r1∼r2) ↔ (abs(r1−r2) = ε) )⟩

where abs is the absolute value:

⟨( abs(r) = (r ← r>0 →' −r) )⟩


Continuity of a function

The continuity expression (on the right) of a function f(r) at a point r = r0 is:

⟨( (r = r0+ε) → (f(r) = f(r0) + ε) )⟩

Expressed in another, even simpler and more general way:

⟨( (r1 ∼ r2) ↔ (f(r1) ∼ f(r2)) )⟩

that is, if r1 is infinitely close to r2, then so are their corresponding functional values f(r1) and f(r2).


Dual Numbers
Analogy with complex numbers

Dual numbers are an extension of the real numbers. They have the form (x + εy), with x and y being real numbers and ε an entity such that its square is zero (ε² = 0). The analogy with complex numbers covers the following aspects:

• The symbol ε is an imaginary number representing an infinitely small number.
  
  
• In a dual number (x + εy), x is the real part and y is the imaginary part. When y is zero, we have the real numbers.
  
  
• The conjugate number of (x + εy) is (x − εy). The product of a dual number by its conjugate is x². In the case of conjugate complex numbers, the result is x² + y².
  
  
• Dual numbers, like complex numbers, can be represented graphically, on the Cartesian axes x and y.
  
  
• Every circle of radius rε has as circumference length 2πrε, and as area π(rε)² = 0. And every circle of radius ri has as circumference length 2πri, and as area π(ri)2 = −πr². So circles for radius values (r) i, ε and 1 have surface −π, 0 and π, respectively. Then, at the circle level, ε is between the values i and 1.

Operations with dual numbers



Functions with dual numbers

Dual expresión	Equivalent expression
√(x + εy)	√x + εy/(2√(x)
ex+εy	ex(1 + εy)
ln(x + εy)	ln(x) + εy/x
sen(x + εy)	sen(x) + εy*cos(x)
cos(x + εy)	cos(x) − εy*sen(x)
tan(x + εy)	tan(x) + εy/cos2(x)
asen(x + εy)	asen(x) + εy/√(1 + x2)
acos(x + εy)	acos(x) − εy/√(1 + x2)
atan(x + εy)	atan(x) + εy/√(1 + x2)

We can check the usefulness of these expressions. For example, by means of the expression

e^x+εy = e^x (1 + εy) we can calculate the derivative of e^x: (e^{(x + ε)} − e^x)/ε = e^x

---

Addenda
A brief history of infinitesimals

The concept of infinitesimals has been surrounded, since its origins, by a great philosophical controversy regarding its existence. They were historically rejected as metaphysical entities that could not be formally defined. And, in the case of their existence, of their true nature: constant, variable, imaginary number, qualitative number (without magnitude), relative zero (instead of absolute zero), dynamic number, generic number, etc. It has always been a challenge to logic and traditional concepts. Some of the criticisms made of it were the following:

• The infinitesimal is something ambiguous, not exactly defined.
  
  
• Exact calculations cannot be made with inexact quantities.
  
  
• How the sum of infinitesimals can produce a finite result.
  
  
• How can non-zero terms be disregarded and produce exact results.
  
  
• "What are these evanescent increments? They are neither finite quantities nor infinitely small quantities nor anything. Can we not call them phantoms of defunct quantities?" (George Berkeley, in his 1734 essay The Analyst).

The most important historical milestones were:

• The roots of the concept of infinitesimal go back to the ancient Greeks (Archimedes and Eudoxus) who used infinitesimals in a completely intuitive way for the calculation of surfaces and volumes of geometric objects (circle, sphere, cone, etc.).
  
  
• In the 17th century, Newton and Leibniz introduced the concept of differential (infinitesimal increment of a variable) and the so-called "differential calculus", a very powerful technique that allowed unifying many problems and reducing their complexity, despite lacking solid mathematical foundations.
  
  
• Newton used the symbol "o" as a variable that could approach zero as much as he wanted, without coinciding. Leibniz used the symbol dx (small portion of x) and also considered it a variable quantity. He also claimed that it was not a simple and absolute zero, but a relative zero and a form without magnitude. And that, although they did not really exist, they could be reasoned with as if they did. For Leibniz, the ultimate goal was to achieve a theory of infinitesimals that would be part of the very structure of the real numbers.
  
  
• Euler believed that the differentials were quantities quantitatively equal to zero, and qualitatively different from zero.
  
  
• In the 19th century, Cauchy introduced the generic concept of limit, showing that the concepts of infinitesimal, derivative and integral could be derived from this concept. For Cauchy, an infinitesimal (as a limit) is a variable, whose value changes actively, decreasing indefinitely, converging to the value zero.
  
  
• Weierstrass formalized the concept of Cauchy limit. He does not consider infinitesimals as variables, unlike Cauchy, but as a generic member of a possible set of values. Around 1872 infinitesimals were formally eliminated from mathematics and replaced by formalized limits to define derivatives, integrals, convergent sequences, etc.
  
  
• During the 20th century, the concept of infinitesimal was revived, and there was also interest in higher-order infinitesimals. Physicists used differentials, since they were more intuitive and easier to use than limits.
  
  
• In the 1960s Abraham Robinson introduced "Non-standard analysis", with the concept of "hyperreal numbers" [see Applications - Mathematics - Hyperreal Numbers].
  
  
• William Kingdon Clifford introduced dual numbers in 1873 to study non-Euclidean geometry.
  
  
• Eduard Study, in 1903, defined dual angles to specify the relationship between two lines in Euclidean space. A dual angle is a dual number of the form φ+εs, where φ is the angle between two lines in 3D space and s is the distance between the two lines.
  
  
• Finally, the infinitesimal ε is introduced as an imaginary number, defined by the simple substitution formula ε² = 0, a formalization similar to that of the imaginary unit defined as i²  = −1. With the expression ε² = 0, infinitesimals have been formalized in an incredibly simple way and empowered as true mathematical entities.

Bibliography

• Bell, John L. A Primer of Infinitesimal Analysis. Cambridge University press, 2008.
  
  
• Cheng, Harry H. Programming with Dual Numbers and its Applications in Mechanisms Design. Engineering with Computers, vol. 10, no. 4, 1994, pp. 212-229. Disponible en Internet.
  
  
• Dovan, W.G. Los infinitésimos. Editorial Andina, 1981.
  
  
• Guénon, René. Los principios del cálculo infinitesimal. Sanz y Torres, 2007.
  
  
• Ian Stewart. Fantasmas de cantidades difuntas. Capítulo 6 de De aquí al infinito. Crítica. Grijalbo Mondadori, S.A., Barcelona, 1996.
  
  
• Kauffman, Louis H. Virtual Logic. Infinitesimals and Zero Numbers. Cybernetics And Human Knowing, vol. 7, no. 1, 2000, pp. 83-90. Disponible en Internet.
  
  
• Stroyan, K.D. y Luxemburg, W.A.U. Introduction to the Theory of Infinitesimals. Academic Press, New York, 1976.