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 Skip Navigation LinksMath Help > Sets, Set theory, Number systems > Number systems > Hyperreal

Hyperreal numbers, used in Nonstandard Analysis, extend the real numbers using infinitesimal and unbounded quantities, corresponding to sequences of reals.

From Internet Encyclopedia of Science,

Any 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 they 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.

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Internet references

Dr. Math: Nonstandard Analysis and the Hyperreals 

Wikipedia: Hyperreal number

Mathworld: Hyperreal Number and Nonstandard Analysis

Fact archive: Hyperreal number 

Internet Encyclopedia of Science: hyperreal number

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