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Hyperinteger
 
 
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In non-standard analysis
Non-standard analysis

Non-standard analysis is a branch of mathematics that formulates mathematical analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson....
, a hyperinteger N is a hyperreal number
Hyperreal number

The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Isaac Newton and Gottfried Leibniz....
 equal to its own integer part.

standard integer part function: is defined for all real x and equals the greatest integer not exceeding x. By the extension principle of non-standard analysis, there exists a natural extension: defined for all hyperreal x, and we say that x is a hyperinteger if: .

set of all hyperintegers is an internal subset
Internal set

In mathematical logic, in particular in model theory and non-standard analysis, an internal set is a set that is a member of a model.Internal set is the key tool in formulating the transfer principle, which concerns the logical relation between the properties of the real numbers R, and the properties of a larger field denoted *R called the...
 of the hyperreal line .






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In non-standard analysis
Non-standard analysis

Non-standard analysis is a branch of mathematics that formulates mathematical analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson....
, a hyperinteger N is a hyperreal number
Hyperreal number

The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Isaac Newton and Gottfried Leibniz....
 equal to its own integer part.

Discussion

The standard integer part function: is defined for all real x and equals the greatest integer not exceeding x. By the extension principle of non-standard analysis, there exists a natural extension: defined for all hyperreal x, and we say that x is a hyperinteger if: .

Internal sets

The set of all hyperintegers is an internal subset
Internal set

In mathematical logic, in particular in model theory and non-standard analysis, an internal set is a set that is a member of a model.Internal set is the key tool in formulating the transfer principle, which concerns the logical relation between the properties of the real numbers R, and the properties of a larger field denoted *R called the...
 of the hyperreal line . The set of all finite hyperintegers (i.e. itself) is not an internal subset. Elements of the complement

are called, depending on the author, either unbounded or infinite hyperintegers.