View source for Hyperinteger

{{Short description|A hyperreal number that is equal to its own integer part}}
In [[nonstandard analysis]], a '''hyperinteger''' ''n'' is a [[hyperreal number]] that is 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 [[infinite sequence|sequence]] {{nowrap|(1, 2, 3, ...)}} in the [[ultrapower]] construction of the hyperreals.

==Discussion==
The standard integer part [[Function (mathematics)|function]]: 
:<math>\lfloor x \rfloor</math>
is defined for all [[Real number|real]] ''x'' and equals the greatest integer not exceeding ''x''.  By the [[transfer principle]] of nonstandard analysis, there exists a natural extension: 
:<math>{}^*\! \lfloor \,\cdot\, \rfloor</math>
defined for all hyperreal ''x'', and we say that ''x'' is a hyperinteger if <math> x = {}^*\! \lfloor x \rfloor.</math> Thus the hyperintegers are the [[Image (mathematics)|image]] of the integer part function on the hyperreals.

==Internal sets==
The set <math>^*\mathbb{Z}</math> of all hyperintegers is an [[internal set|internal subset]] of the hyperreal line <math>^*\mathbb{R}</math>.  The set of all finite hyperintegers (i.e. <math>\mathbb{Z}</math> itself) is not an internal subset.  Elements of the complement <math>^*\mathbb{Z}\setminus\mathbb{Z}</math> are called, depending on the author, ''nonstandard'', ''unlimited'', or ''infinite'' hyperintegers. The reciprocal of an infinite hyperinteger is always an [[infinitesimal]].

Nonnegative hyperintegers are sometimes called ''hypernatural'' numbers.  Similar remarks apply to the sets <math>\mathbb{N}</math> and <math>^*\mathbb{N}</math>.  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

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[[Category:Nonstandard analysis]]
[[Category:Infinity]]
[[Category:Calculus]]