User Contributed Dictionary
Adverb
 In an infinitesimal manner.
 To an infinitesimal degree.
Extensive Definition
Infinitesimals have been used to express the idea
of objects so small that there is no way to see them or to measure
them. For everyday life, an infinitesimal object is an object which
is smaller than any possible measure. When used as an adjective in
the vernacular, "infinitesimal" means extremely small, but not
necessarily "infinitely small".
Before the nineteenth century none of the
mathematical concepts as we know them today were formally defined,
but many of these concepts were already there. The founders of
calculus, Leibniz, Newton, Euler, Lagrange, the Bernoullis and many
others, used infinitesimals in the way shown below and achieved
essentially correct results even though no formal definition was
available (similarly, there was no formal definition of real
numbers at the time).
History of the infinitesimal
The first mathematician to
make use of infinitesimals was Archimedes (c.
250 BC).
The Archimedean
property is the property of an ordered algebraic
structure of having no nonzero infinitesimals.
In India from the
12th
century until the 16th
century, infinitesimals were discovered for use with differential
calculus by Indian
mathematician Bhaskara and
various Keralese
mathematicians.
When Newton and
Leibniz
developed calculus,
they made use of infinitesimals. A typical argument might go:

 To find the derivative f′(x) of the function f(x) = x2, let dx be an infinitesimal. Then,

 since dx is infinitely small.
This argument, while intuitively appealing, and
producing the correct result, is not mathematically
rigorous. The use of infinitesimals was attacked as incorrect
by Bishop
Berkeley in his work The Analyst.
The fundamental problem is that dx is first treated as nonzero
(because we divide by it), but later discarded as if it were
zero.
The naive definition of an infinitesimal is this:
a number whose absolute
value is less than any nonzero positive number. From this
definition, it can be shown than there are no nonzero real
infinitesimals, using the property of the least
upper bound. Considering just the positive numbers, the only
way for a number to be less than all numbers would be to be the
least positive number. If h is such a number, then what is h/2? Or
if h is indivisible, is it still a number? Also, intuitively, one
would require that the reciprocal of an infinitesimal is infinitely
large (in modulus) or unlimited, but this would make it the
greatest number when clearly, there is no "last" biggest
number.
Despite this, the real numbers can in fact be
extended and modified to include infinitesimals, forming such
systems as the dual numbers
or the hyperreals,
but this can only be done if certain properties of the real numbers
are removed.
It was not until the second half of the nineteenth
century that the calculus was given a formal mathematical
foundation by Karl
Weierstrass and others using the notion of a limit.
In the 20th century, it was found that infinitesimals could after
all be treated rigorously. Neither formulation is wrong, and both
give the same results if used correctly.
Modern uses of infinitesimals
Infinitesimal is necessarily a relative concept. If epsilon is infinitesimal with respect to a class of numbers it means that epsilon cannot belong to that class. This is the crucial point: infinitesimal must necessarily mean infinitesimal with respect to some other type of numbers.The path to formalisation
Proving or disproving the existence of
infinitesimals of the kind used in nonstandard analysis depends on
the model and
which collection of axioms
are used. We consider here systems where infinitesimals can be
shown to exist.
In 1936 Maltsev proved the
compactness
theorem. This theorem is fundamental for the existence of
infinitesimals as it proves that it is possible to formalise them.
A consequence of this theorem is that if there is a number system
in which it is true that for any positive integer n there is a
positive number x such that
0 < x < 1/n, then there
exists an extension of that number system in which it is true that
there exists a positive number x such that for any positive integer
n we have 0 < x < 1/n.
The possibility to switch "for any" and "there exists" is crucial.
The first statement is true in the real numbers as given in
ZFC set theory :
for any positive integer n it is possible to find a real number
between 1/n and zero, only this real number will depend on n. Here,
one chooses n first, then one finds the corresponding x. In the
second expression, the statement says that there is an x (at least
one), chosen first, which is between 0 and 1/n for any n. In this
case x is infinitesimal. This is not true in the real numbers (R)
given by ZFC. Nonetheless, the theorem proves that there is a
model (a
number system) in which this will be true. The question is: what is
this model? What are its properties? Is there only one such model?
There are in fact many ways to construct such a onedimensional linearly
ordered set of numbers, but fundamentally, there are two
different approaches:
 1) Extend the number system so that it contains more numbers than the real numbers.
 2) Extend the axioms (or extend the language) so that the distinction between the infinitesimals and noninfinitesimals can be made in the real numbers.
In 1960, Abraham
Robinson provided an answer following the first approach. The
extended set is called the hyperreals
and contains numbers less in absolute value than any positive real
number. The method may be considered relatively complex but it does
prove that infinitesimals exist in the universe of ZFC set theory.
The real numbers are called standard numbers and the new nonreal
hyperreals are called nonstandard.
In 1977 Edward
Nelson provided an answer following the second approach. The
extended axioms are IST, which stands either for Internal
Set Theory or for the initials of the three extra axioms:
Idealization, Standardization, Transfer. In this system we consider
that the language is extended in such a way that we can express
facts about infinitesimals. The real numbers are either standard or
nonstandard. An infinitesimal is a nonstandard real number which is
less, in absolute value, than any positive standard real
number.
In 2006 Karel
Hrbacek developed an extension of Nelson's approach in which
the real numbers are stratified in (infinitely) many levels i.e, in
the coarsest level there are no infinitesimals nor unlimited
numbers. Infinitesimals are in a finer level and there are also
infinitesimals with respect to this new level and so on.
All of these approaches are mathematically
rigorous.
This allows for a definition of infinitesimals
which refers to these approaches:
A definition
 An infinitesimal number is a nonstandard number whose modulus is less than any nonzero positive standard number.
What standard and nonstandard refer to depends on
the chosen context.
Alternatively, we can have
synthetic differential geometry or
smooth infinitesimal analysis with its roots in category
theory. This approach departs dramatically from the classical
logic used in conventional mathematics by denying the law
of excluded middlei.e., not (a ≠ b) does not have to mean a =
b. A nilsquare or nilpotent infinitesimal can
then be defined. This is a number x where x2 = 0 is true, but x = 0
need not be true at the same time. With an infinitesimal such as
this, algebraic proofs using infinitesimals are quite rigorous,
including the one given above.
Notes
References
 J. Keisler "Elementary Calculus" (2000) University of Wisconsin http://www.math.wisc.edu/~keisler/calc.html
 K. Stroyan "Foundations of Infinitesimal Calculus" (1993) http://www.math.uiowa.edu/%7Estroyan/InfsmlCalculus/InfsmlCalc.htm
 Robert Goldblatt (1998) "Lectures on the hyperreals" Springer. http://www.springer.com/west/home/generic/order?SGWID=4401102215908890
 "Nonstandard Methods and Applications in Mathematics" (2007) Lecture Notes in Logic 25, Association for Symbolic Logic. http://www.aslonline.org/bookslnl_25.html
 "The Strength of Nonstandard Analysis" (2007) Springer.http://www.springer.com/west/home/springerwiennewyork/mathematics?SGWID=440638221737057220
See also
infinitesimally in Arabic: عدد لامتناهي
infinitesimally in Czech: Infinitezimální
hodnota
infinitesimally in German:
Infinitesimalzahl
infinitesimally in Spanish: Infinitesimal
infinitesimally in French: Infiniment
petit
infinitesimally in Galician: Infinitesimal
infinitesimally in Korean: 무한소
infinitesimally in Italian: Infinitesimo
infinitesimally in Hebrew: אינפיניטסימל
infinitesimally in Dutch: Infinitesimaal
infinitesimally in Japanese: 無限小
infinitesimally in Polish: Nieskończenie
małe
infinitesimally in Portuguese:
Infinitesimal
infinitesimally in Romanian: Infinitezimal
infinitesimally in Russian: Бесконечно малая
величина
infinitesimally in Slovenian:
Infinitezimala
infinitesimally in Serbian:
Инфинитезималан
infinitesimally in Finnish: Mielivaltaisen pieni
positiivinen luku
infinitesimally in Swedish: Infinitesimal
infinitesimally in Ukrainian:
Інфінітозимальний