Hyperinteger

In non-standard analysis, a hyperinteger n is a hyperreal number that is 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:

\lfloor x\rfloor

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:

{}^{*}\!\lfloor \,\cdot \,\rfloor

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

x={}^{*}\!\lfloor x\rfloor .

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.

References

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 at http://www.math.wisc.edu/~keisler/calc.html

Number systems
Countable sets	Natural numbers (ℕ) Integers (ℤ) Rational numbers (ℚ) Constructible numbers Algebraic numbers (𝔸) Periods Computable numbers Definable real numbers Arithmetical numbers Gaussian integers
Division algebras	Real numbers (ℝ) Complex numbers (ℂ) Quaternions (ℍ) Octonions (𝕆)
Split Composition algebras	over ℝ: Split-complex numbers Split-quaternions Split-octonions over ℂ: Bicomplex numbers Biquaternions Bioctonions
Other hypercomplex	Dual numbers Dual quaternions Hyperbolic quaternions Sedenions (𝕊) Split-biquaternions Multicomplex numbers
Other types	Cardinal numbers Irrational numbers Fuzzy numbers Hyperreal numbers Levi-Civita field Surreal numbers Transcendental numbers Ordinal numbers p-adic numbers Supernatural numbers Superreal numbers
Classification List

Infinitesimals
History	Adequality Leibniz's notation Integral symbol Criticism of non-standard analysis The Analyst The Method of Mechanical Theorems Cavalieri's principle Method of indivisibles
Related branches	Non-standard analysis Non-standard calculus Internal set theory Synthetic differential geometry Constructive non-standard analysis Infinitesimal strain theory (physics)
Formalizations	Differentials Hyperreal numbers Dual numbers Surreal numbers
Individual concepts	Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of Continuity Overspill Microcontinuity Transcendental law of homogeneity
Scientists	Gottfried Wilhelm Leibniz Abraham Robinson Pierre de Fermat Augustin-Louis Cauchy Leonhard Euler
Textbooks	Analyse des Infiniment Petits Elementary Calculus Cours d'Analyse

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