Hyperinteger

In non-standard analysis, a hyperinteger N is a hyperreal number equal to its own integer part.

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:

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

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

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

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, either 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 $^{*}\mathbb {N}$.

References

• H. Jerome Keisler: Elementary calculus: An Approach Using Infinitesimals. 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