nonstandard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta p ...

, a hyperinteger ''n'' is a

hyperreal number In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains number ...

that is equal to its own

integer part In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least inte ...

. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...

. An example of an infinite hyperinteger is given by the class of the

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...

in the

ultrapower The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factors ...

construction of the hyperreals.

Discussion

The standard integer part function: :

\lfloor x \rfloor

is defined for all

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

''x'' and equals the greatest integer not exceeding ''x''. By the

transfer principle In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the first- ...

of nonstandard analysis, there exists a natural extension: :

^*\! \lfloor \,\cdot\, \rfloor

defined for all hyperreal ''x'', and we say that ''x'' is a hyperinteger if

x = ^*\! \lfloor x \rfloor.

Thus the hyperintegers are the

image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...

of the integer part function on the hyperreals.

Internal sets

The set

^*\mathbb

of all hyperintegers is an internal subset of the hyperreal line

^*\mathbb

. The set of all finite hyperintegers (i.e.

\mathbb

itself) is not an internal subset. Elements of the complement

^*\mathbb\setminus\mathbb

are called, depending on the author, ''nonstandard'', ''unlimited'', or ''infinite'' hyperintegers. The reciprocal of an infinite hyperinteger is always an

infinitesimal In mathematics, an 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 ref ...

. Nonnegative hyperintegers are sometimes called ''hypernatural'' numbers. Similar remarks apply to the sets

\mathbb

and

^*\mathbb

. Note that the latter gives a

non-standard model of arithmetic Standardization or standardisation is the process of implementing and developing technical standards based on the consensus of different parties that include firms, users, interest groups, standards organizations and governments. Standardization ...

in the sense of

Skolem Thoralf Albert Skolem (; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory. Life Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skol ...

References

Howard Jerome Keisler Howard Jerome Keisler (born 3 December 1936) is an American mathematician, currently professor emeritus at University of Wisconsin–Madison. His research has included model theory and non-standard analysis. His Ph.D. advisor was Alfred Tarski a ...

: '' Elementary Calculus: An Infinitesimal Approach''. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html {{Infinitesimal navbox Nonstandard analysis Infinity Calculus