## hyperreal numberAny of a colossal set of numbers, also known as nonstandard reals,
that includes not only all the real numbers
but also certain classes of infinitely large (see infinity)
and infinitesimal numbers as well.
Hyperreals emerged in the 1960s from the work of Abraham Robinson
who showed how infinitely large and infinitesimal numbers can be rigorously
defined and developed in what is called nonstandard
analysis. Because hyperreals represent an extension of the real numbers,
R, they are usually denoted by *R. Hyperreals include all the reals (in the technical sense that they form an ordered field containing the reals as a subfield) and also contain infinitely many other numbers that are either infinitely large (numbers whose absolute value is greater than any positive real number) or infinitely small (numbers whose absolute value is less than any positive real number). No infinitely large number exists in the real number system and the only real infinitesimal is zero. But in the hyperreal system, it turns out that that each real number is surrounded by a cloud of hyperreals that are infinitely close to it; the cloud around zero consists of the infinitesimals themselves. Conversely, every (finite) hyperreal number x is infinitely close to exactly
one real number, which is called its standard part, st(x).
In other words, there exists one and only one real number st(x) such
that x – st(x) is infinitesimal. ## Related category• TYPES OF NUMBERS | |||||

Home • About • Copyright © The Worlds of David Darling • Encyclopedia of Alternative Energy • Contact |