# Hyperinteger

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.