Innonstandard analysis, ahyperintegern is ahyperreal number that is equal to its owninteger part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinaryinteger. An example of an infinite hyperinteger is given by the class of thesequence(1, 2, 3, ...) in theultrapower construction of the hyperreals.

The standard integer partfunction:

${\displaystyle \lfloor x\rfloor }$

is defined for allrealx and equals the greatest integer not exceedingx. By thetransfer principle of nonstandard analysis, there exists a natural extension:

${\displaystyle {}^{\ast }\lfloor \cdot \rfloor }$

defined for all hyperrealx, and we say thatx is a hyperinteger if${\displaystyle x={}^{\ast }\lfloor x\rfloor .}$ Thus, the hyperintegers are theimage of the integer part function on the hyperreals.

The set${\displaystyle {}^{\ast }\mathbb{Z}}$ of all hyperintegers is aninternal subset of the hyperreal line${\displaystyle {}^{\ast }\mathbb{R}}$. The set of all finite hyperintegers (i.e.${\displaystyle \mathbb{Z}}$ itself) is not an internal subset. Elements of the complement${\displaystyle {}^{\ast }\mathbb{Z}\setminus \mathbb{Z}}$ are called, depending on the author,nonstandard,unlimited, orinfinite hyperintegers. The reciprocal of an infinite hyperinteger is always aninfinitesimal.

Nonnegative hyperintegers are sometimes calledhypernatural numbers. Similar remarks apply to the sets${\displaystyle \mathbb{N}}$ and${\displaystyle {}^{\ast }\mathbb{N}}$. Note that the latter gives anon-standard model of arithmetic in the sense ofSkolem.

• 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 athttp://www.math.wisc.edu/~keisler/calc.html

• Nonstandard analysis
• Infinity
• Numbers

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