Hyperinteger

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In non-standard analysis, a hyperinteger N is a hyperreal number equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An example of an infinite hyperinteger is given by the class of the sequence (1,2,3,...) in the ultrapower construction of the hyperreals.

Discussion

The standard integer part function:

$[x]$

is defined for all real x and equals the greatest integer not exceeding x. By the transfer principle of non-standard analysis, there exists a natural extension:

$^{*}[\,\cdot \,]$

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

$x={}^{*}\![x]$ .

Thus the hyperintegers are the image of the integer part function on the hyperreals.

Internal sets

The set $^{*}{\mathbb {Z}}$ of all hyperintegers is an internal subset of the hyperreal line $^{*}{\mathbb {R}}$ . The set of all finite hyperintegers (i.e. ${\mathbb {Z}}$ itself) is not an internal subset. Elements of the complement

$^{*}{\mathbb {Z}}\setminus {\mathbb {Z}}$

are called, depending on the author, non-standard, unlimited, or infinite hyperintegers. The reciprocal of an infinite hyperinteger is an infinitesimal.

Positive hyperintegers are sometimes called hypernatural numbers. Similar remarks apply to the sets ${\mathbb {N}}$ and $^{*}{\mathbb {N}}$ . Note that the latter gives a non-standard model of arithmetic in the sense of Skolem.

References

Howard Jerome Keisler: 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

v t e Number systems

Countable sets	Natural numbers ( $\scriptstyle {\mathbb {N}}$ ) Integers ( $\scriptstyle {\mathbb {Z}}$ ) Rational numbers ( $\scriptstyle {\mathbb {Q}}$ ) Algebraic numbers ( $\scriptstyle \overline {{\mathbb {Q}}}$ ) Periods Computable numbers Arithmetical numbers

Real numbers and their extensions	Real numbers ( $\scriptstyle {\mathbb {R}}$ ) Complex numbers ( $\scriptstyle {\mathbb {C}}$ ) Quaternions ( $\scriptstyle {\mathbb {H}}$ ) Octonions ( $\scriptstyle {\mathbb {O}}$ ) Sedenions ( $\scriptstyle {\mathbb {S}}$ ) Cayley–Dickson construction Dual numbers Split-complex numbers Hypercomplex numbers Superreal numbers Irrational numbers Transcendental numbers Hyperreal numbers Surreal numbers

Other systems	Cardinal numbers Ordinal numbers p-adic numbers Supernatural numbers

v t e Infinitesimals

History	Adequality Infinitesimal calculus Leibniz's notation Integral symbol Criticism of non-standard analysis The Analyst The Method of Mechanical Theorems Cavalieri's principle

Related branches	Non-standard analysis Non-standard calculus Internal set theory Synthetic differential geometry Constructive non-standard analysis

Formalizations	Differentials Hyperreal numbers Dual numbers Surreal numbers

Individual concepts	Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of Continuity Overspill Microcontinuity Transcendental law of homogeneity

Scientists	Gottfried Wilhelm Leibniz Abraham Robinson Pierre de Fermat Augustin-Louis Cauchy Leonhard Euler

Physics Engineering	Infinitesimal strain theory Infinitesimal oscillations of a pendulum

Textbooks	Analyse des Infiniment Petits Elementary Calculus Cours d'Analyse

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