nLab infinitesimal number

Infinitesimal numbers

Infinitesimal numbers


The term ‘infinitesimal’ means the same as ‘infinitely small in absolute value’; in Latin, it literally means ‘infinity-eth’ and should be interpreted in the sense of a fraction. In the ordinary analysis of real numbers, the only infinitesimal number is zero. However, the basic intuitions of calculus since its beginnings have dealt with infinitely small (and sometimes also infinitely large) numbers. There are several different ways to develop a rigorous theory that includes infinitesimal numbers.


Briefly we recall the definition of what makes a number infinitesimal, which we give in some generality.

Let PP be a cancellative commutative monoid equipped with a strict weak order such that x<0x \lt 0 does not hold for any element xx of PP; that is, the additive identity is a bottom element with respect to the partial order induced by the negation of the strict weak order, and an element 11 such that 0<10 \lt 1. Let MM be a cancellative commutative monoid equipped with a function ||:MP{|\cdot|}\colon M \to P, thought of as a measure of absolute value, such that |0|=0{|0|} = 0. For example, MM could consist of the real numbers with PP the nonnegative real numbers; similarly, we may consider the natural numbers, the complex numbers, the cardinal numbers, and many other familiar examples of numbers. (In these examples, <\lt and ||{|\cdot|} satisfy additional compatibility properties with respect to the cancellative commutative monoids MM and PP, but these do not seem to enter into the definition. Conversely, we do not use the full cancellative commutative monoid structure of MM and PP, so the definition should work in even greater generality.)

An element xx of MM is infinitesimal if the sum of nn elements of xx has absolute value less than 11: | i=1 nx|<1{|\sum_{i = 1}^n x|} \lt 1 for every natural number nn. The sum of nn elements of an element in a cancellative commutative monoid MM always exists, since every monoid is power-associative.

According to this definition, 00 is always infinitesimal. Traditionally, one adds to the definition the requirement that i0i \neq 0, but this leads to a less useful notion of the space of all infinitesimals. We allow 00 to be infinitesimal for some of the same reasons that we allow it to be a natural number. (Although the argument is even stronger here, since it’s always decidable whether a given natural number is zero, but it may not be decidable whether a given infinitesimal is zero. This is especially important in synthetic differential geometry, where the logic is unavoidably constructive.)

In the usual systems of numbers (real, natural, complex, cardinal, etc), 00 is the only infinitesimal number. So the interesting question is how to get other infinitesimals.

Free infinitesimals

The simplest way to add an infinitesimal to an ordered ring (or rig) is freely. For example, if we start with the field \mathbb{R} of real numbers, then form the polynomial ring [x]\mathbb{R}[x], we may make xx (and every nonconstant polynomial) infinitesimal by defining x<cx \lt c for every positive real number cc and generating the rest of <\lt by requiring that [x]\mathbb{R}[x] be an ordered ring. (This amounts to defining f<gf \lt g to mean that f(x)<g(x)f(x) \lt g(x) for sufficiently small positive values of xx.)

If you want a field, then form the field of fractions (x)\mathbb{R}(x) of [x]\mathbb{R}[x], which is the field of rational functions. This is a very commonly known example, although it is more usual to make xx infinitely large (so that 1/x1/x is infinitesimal).

Similarly, one can use polynomials to define infinitesimal versions of complex numbers, cardinal numbers, etc.

The downside of this approach is that the resulting infinitesimal numbers will be subject only to those operations that apply to the category in which one forms the free construction. For example, there is no way to apply transcendental functions to infinitesimals with this approach. Presumably that could be fixed by working with C C^\infty-rings, but no algebraic theory covers everything that can be done with real numbers (or whatever numbers you start with).

Alternatively, one could formally complete the polynomial ring R[x]R[x] to get the ring of formal power series R[[x]]R[[x]]. This results in a local integral domain if RR is a field. In the case that RR is an Archimedean ordered field, then R[[x]]R[[x]] is an Archimedean ordered local integral domain; and if RR is sequentially Cauchy complete, this allows for any analytic function to be defined on R[[x]]R[[x]]. The infinitesimals in R[[x]]R[[x]] are the formal power series whose leading 0th-power term is equal to zero, or equivalently, the product of xx and a formal power series.

Nilpotent infinitesimals

Instead of working with free infinitesimals, we could also work with nilpotent infintiesimals. This requires working in an Archimedean ordered Artinian local \mathbb{R} -algebras. Since these are not ordered fields, the strict weak order is not a strict total order, the preorder is not a partial order, and the apartness relation is not tight. There is an equivalence relation aba \approx b which says that aa is approximately equal to bb, and is defined by aba \approx b if and only if the difference aba - b is a zero divisor.

Given an Archimedean ordered Artinian local \mathbb{R}-algebra AA, since AA is a local ring, the quotient of AA by its ideal DD of non-invertible elements is the real numbers \mathbb{R}, and the canonical function used in defining the quotient ring is the function :A\Re:A \to \mathbb{R} which takes a number aAa \in A to its purely real component (a)\Re(a) \in \mathbb{R}. Since AA is an ordered \mathbb{R}-algebra, there is a strictly monotone ring homomorphism h:Ah:\mathbb{R} \to A. A number aAa \in A is purely real if h((a))=ah(\Re(a)) = a, and a number aAa \in A is purely infinitesimal if it is in the ideal of zero divisors DD, the fiber of \Re at the real number 00. Zero is the only number in AA which is both purely real and purely infinitesimal.

The advantage of working with nilpotent infinitesimals is that the ring homomorphism h:Ah:\mathbb{R} \to A preserves smooth functions: given a natural number nn \in \mathbb{N} and a purely infinitesimal number ϵD\epsilon \in D such that ϵ n+1=0\epsilon^{n + 1} = 0, then for every smooth function fC ω()f \in C^\omega(\mathbb{R}), there is a function f A:AAf_A:A \to A such that for all real numbers xx \in \mathbb{R}, f A(h(x))=h(f(x))f_A(h(x)) = h(f(x)) and

f A(h(x)+ϵ)= i=0 n1i!h(d ifdx i(x))ϵ if_A(h(x) + \epsilon) = \sum_{i = 0}^{n} \frac{1}{i!} h\left(\frac{d^i f}{d x^i}(x)\right) \epsilon^i

If we restrict to Archimedean ordered local Artinian \mathbb{R}-algebras AA where every element of the nilradical DD is a nilsquare element, where for all ϵD\epsilon \in D, ϵ 2=0\epsilon^2 = 0, then the ring homomorphism h:Ah:\mathbb{R} \to A preserves differentiable functions; for every differentiable function f:f:\mathbb{R} \to \mathbb{R}, there is a function f A:AAf_A:A \to A such that for all real numbers xx \in \mathbb{R} and nilpotent elements ϵD\epsilon \in D, f A(h(x))=h(f(x))f_A(h(x)) = h(f(x)) and

f A(h(x)+ϵ)=h(f(x))+h(dfdx(x))ϵf_A(h(x) + \epsilon) = h(f(x)) + h\left(\frac{d f}{d x}(x)\right) \epsilon

This allows us to define certain differentiable functions on the real numbers, such as the exponential function, the sine function, and the cosine function, through its property as solutions to systems of first-order ordinary differential equations, without having to define the notion of limit of a sequence, series, integral, and prove the fundamental theorem of calculus and/or convergence of Taylor series.

One could also work with partial functions instead of functions. Given a predicate PP on the real numbers \mathbb{R}, let II denote the set of all elements in \mathbb{R} for which PP holds. A partial function f:f:\mathbb{R} \to \mathbb{R} is equivalently a function f:If:I \to \mathbb{R} for any such predicate PP and set II.

A function f:If:I \to \mathbb{R} is smooth at a subset SIS \subseteq I with injection j:Sj:S \hookrightarrow \mathbb{R} if it has a function (D ()j)():×S(D^{(-)} j)(-):\mathbb{N} \times S \to \mathbb{R} with (D 0j)(a)=a(D^0 j)(a) = a for all aSa \in S, such that for all Archimedean ordered Artinian local \mathbb{R} -algebras AA with ring homomorphism h A:Ah_A:\mathbb{R} \to A, natural numbers nn \in \mathbb{N}, and purely infinitesimal elements ϵD\epsilon \in D such that ϵ n+1=0\epsilon^{n + 1} = 0

f A(h A(j(a))+ϵ)= i=0 n1i!h A((D ij)(a))ϵ if_A(h_A(j(a)) + \epsilon) = \sum_{i = 0}^{n} \frac{1}{i!} h_A((D^i j)(a)) \epsilon^i

Special cases include being smooth at an element aIa \in I, which is the same as being smooth at the singleton subset {a}\{a\}, and being smooth which is the same as being smooth at the improper subset of II.

This approach is used in synthetic differential geometry, where one adds additional axioms to the foundations stating that the only functions on AA are the smooth functions.

Nonstandard analysis

If the point is to do with infinitesimal numbers everything that we can do with ordinary numbers, then why not use high-powered logic to do this for us? That is the approach taken by Abraham Robinson in nonstandard analysis.

Nonstandard analysis can be applied to mathematics as a whole, so it treats complex numbers, cardinal numbers, and the rest all together.

Smooth toposes

Another approach is to focus only on what we want to do with infinitesimals in a particular field. Since infinitesimals were used to do calculus, then let's just do calculus. We can take as axiomatic the familiar properties of smooth functions used in calculus, including the ways these were applied to infinitesimals before analysis was made rigorous (by modern standards) in the 19th century. This is the approach taken by Bill Lawvere and others in synthetic differential geometry.

Other approaches

The surreal numbers have infinitesimals, and include both the real numbers and the ordinal numbers. But they are not much good for ordinary calculus, although there are some indications that they might have applications in asymptotics.

There is also an interpretation of pre-Cauchy calculus in which infinitesimals are interpreted as infinite sequences that converge to 00. I don't know much about this.


Invertible infinitesimals in NSA and SDG

All the infinitesimals appearing in nonstandard analysis are invertible, since the hyperreal numbers form a field. (This is also true for all the surreal infinitesimals.) By contrast, most of the infinitesimals appearing in synthetic differential geometry are nilpotent, and hence not invertible. However, there are some models that do contain invertible infinitesimals, and hence also ‘infinite’ numbers as their inverses (see smooth natural number). Two such models are the smooth toposes called 𝒵\mathcal{Z} and \mathcal{B} in

These toposes are related to the smooth topos called 𝒢\mathcal{G} there, which has nilpotent but no invertible infinitesimals, by a transfer theorem (chapter VII, section 4) valid for a certain class of coherent formulas. Additionally, the ‘object of nonstandard smooth natural numbers’ in these toposes is defined by an ‘algebra of unbounded sequences’, similar in spirit to the unbounded sequences which represent infinitely large numbers in nonstandard analysis. However, it is not clear whether any more precise comparison can be made.

ring with infinitesimalsfunction
dual numbersdifferentiable function
Weil ringsmooth function
power series ringanalytic function


Last revised on December 25, 2023 at 22:40:29. See the history of this page for a list of all contributions to it.