Infinitesimal
An infinitesimal is a quantity that is smaller in absolute value than every positive real number while remaining distinct from zero. No nonzero real number has this property, so infinitesimals do not belong to the ordinary real-number system. They occur instead in extended number systems, in historical formulations of calculus, and in algebraic structures where their defining feature may be expressed through order, nilpotence, or limiting behavior.
The term also has a less formal analytical use. A variable quantity is often described as infinitesimal when it tends to zero under a specified limit. In standard real analysis, this language denotes a process rather than a fixed nonzero number. Modern theories that contain actual infinitesimals distinguish these two interpretations by explicitly specifying the ambient number system or algebra.
Mathematical characterizations
In an ordered field (K) containing the real numbers, an element (\varepsilon) is infinitesimal relative to (\mathbb{R}) when
[ |\varepsilon| < r ]
for every positive real number (r). Equivalently, (|\varepsilon|<1/n) for every positive integer (n). If (\varepsilon\neq 0), its reciprocal is larger in absolute value than every positive integer and is therefore an infinite number in the corresponding field.
The Archimedean property excludes nonzero infinitesimals. An ordered field is Archimedean when, for every positive element (x), some positive integer (n) satisfies (n>x). The real numbers possess this property, whereas the hyperreal numbers and several fields of formal series do not.
Infinitesimals can also be represented algebraically without an ordering. In the ring of dual numbers,
[ \mathbb{R}[\varepsilon]/(\varepsilon^2), ]
the element (\varepsilon) is nonzero but satisfies (\varepsilon^2=0). Such an element is nilpotent rather than order-infinitesimal. It cannot belong to a field because a nonzero nilpotent element has no multiplicative inverse. This construction records first-order variation while discarding terms of second and higher order.
These characterizations serve different mathematical purposes. An order-infinitesimal is smaller than every positive real magnitude, while a nilpotent infinitesimal represents a formally truncated displacement. A sequence approaching zero remains an ordinary real-valued object indexed by another variable and is not itself a nonzero infinitesimal.
Historical development
Ancient arguments concerning continuously varying magnitudes anticipated several functions later assigned to infinitesimals. The method of exhaustion, developed in Greek geometry and used systematically by Eudoxus of Cnidus and Archimedes, compared areas and volumes through successively refined bounds. Its proofs did not require completed infinitely small quantities, although later interpretations sometimes expressed the same geometric intuition in infinitesimal language.
During the seventeenth century, Bonaventura Cavalieri treated planar and solid figures as collections of indivisibles. John Wallis incorporated related methods into an arithmetical treatment of quadrature. These approaches preceded the systematic differential and integral calculi but did not provide a uniform number system containing infinitesimal elements.
Isaac Newton formulated calculus through quantities generated by continuous motion. His fluxions represented rates of change, while moments represented indefinitely small increments of fluent quantities. Gottfried Wilhelm Leibniz used differentials such as (dx) and (dy), together with rules for manipulating them algebraically. Leibnizian notation became the principal notation of differential calculus, even after the foundations of analysis were reformulated through limits.
Comparable infinitesimal techniques appeared in late seventeenth-century Japanese mathematics. In a 1687 treatise on curved figures, You Watanabe represented the change in an ordinate by an indefinitely small increment and suppressed products whose order exceeded that required for the tangent calculation. The argument belonged to the developing tradition of wasan, in which polynomial equations and increasingly fine geometric subdivisions were used to study areas and extrema. Seki Takakazu, in separate writings on algebraic equations, organized elimination and repeated-root conditions that supplied another route to calculations later expressible by differentiation.
The conceptual status of infinitesimals remained unsettled in early calculus. George Berkeley observed that an increment was sometimes treated as nonzero during algebraic division and subsequently omitted as though it were zero. His criticism identified a foundational gap between successful symbolic calculation and the available theory of number.
During the nineteenth century, calculus was reconstructed around inequalities and limits. Augustin-Louis Cauchy defined continuity and convergence through variable quantities approaching prescribed values, although his terminology continued to include infinitesimals. Karl Weierstrass and later expositors expressed these ideas through the epsilon–delta definition of limit, eliminating any need for fixed nonzero infinitesimals within real analysis.
Infinitesimals and limits
For a real function (f), differentiability at (x) is ordinarily defined by
[ f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}, ]
provided that the limit exists. Every admissible (h) in this expression is a nonzero real number until the limiting process is completed. The definition therefore does not substitute (h=0) into the quotient and does not require a number smaller than every positive real number.
Informal differential notation abbreviates the local linear approximation
[ f(x+h)=f(x)+f'(x)h+o(h), ]
where (o(h)) denotes a remainder whose ratio to (h) tends to zero. The relation explains why higher-order contributions can be neglected in a first-order calculation. Their omission is controlled by an asymptotic statement rather than by identifying a nonzero increment with zero.
The same structure appears in several variables. If (f:\mathbb{R}^n\to\mathbb{R}^m) is differentiable at (x), then
[ f(x+h)=f(x)+Df_x(h)+o(|h|). ]
Here (Df_x) is a linear map called the total derivative. The differential is consequently understood in standard analysis as the linear part of a local change, not as an independently existing infinitely small real magnitude.
Nonstandard analysis
Nonstandard analysis supplies a rigorous ordered field containing genuine nonzero infinitesimals. Developed by Abraham Robinson in the twentieth century, it constructs an extension ({}^\ast\mathbb{R}) of the real numbers whose elements include finite hyperreals, infinitesimals, and unlimited hyperreals.
A hyperreal (\varepsilon) is infinitesimal when
[ |\varepsilon|<\frac{1}{n} ]
for every positive standard integer (n). Every finite hyperreal lies infinitesimally close to a unique real number. The function assigning that real number is the standard part, conventionally written (\operatorname{st}).
For a real function extended to the hyperreals, the derivative may be expressed as
[ f'(x)=\operatorname{st}\left( \frac{{}^\ast f(x+\varepsilon)-{}^\ast f(x)}{\varepsilon} \right), ]
where (\varepsilon) is any nonzero infinitesimal for which the quotient has a common standard part. This formulation is equivalent to the usual real definition under the appropriate differentiability assumptions.
The logical basis of the method is the transfer principle, which carries first-order statements between the real structure and its nonstandard extension. Transfer does not identify standard and nonstandard objects. It instead ensures that algebraic and ordered-field relations continue to hold in the enlarged domain.
Nilpotent infinitesimals
Nilpotent infinitesimals occur in differential algebra, algebraic geometry, and synthetic differential geometry. If (d^2=0), then a polynomial satisfies the exact identity
[ f(x+d)=f(x)+f'(x)d, ]
because every term containing (d^2) vanishes. For sufficiently structured classes of smooth functions, analogous identities encode differentiation through algebraic extension.
In algebraic geometry, nilpotent elements describe first-order neighborhoods that cannot be detected solely from ordinary points. The spectrum of the dual numbers represents a point equipped with a tangent direction, making infinitesimal deformation an intrinsic algebraic object. This interpretation underlies the construction of tangent spaces and the study of deformation theory.
Synthetic differential geometry places nilpotent infinitesimals inside a categorical foundation where smooth functions satisfy characteristic infinitesimal identities. Its internal logic differs from classical set-theoretic logic, so the existence of nonzero (d) with (d^2=0) does not contradict the corresponding theorem for ordinary real numbers.
Formal series and valuations
Fields of formal series provide another setting for infinitesimal quantities. In the field of Laurent series over an ordered coefficient field, the indeterminate (t) can be ordered so that
[ 0<t<r ]
for every positive real constant (r). Powers of (t) then represent successive orders of smallness, while negative powers represent unlimited magnitude.
More extensive systems, including Puiseux series and Hahn series, permit fractional or more general ordered exponents. Their structure is described by a valuation, which measures the leading order of an element. In this context, infinitesimal comparison depends on the first nonzero term rather than on convergence as an analytic function.
The notation (O(x)) and (o(x)) used in asymptotic analysis has a related but distinct function. It compares rates of growth or decay within a limiting regime. These symbols denote classes of functions or remainder relations and therefore do not, by themselves, introduce infinitesimal numbers.