Hyperreal number

A hyperreal number belongs to an ordered field extending the real numbers and containing nonzero infinitesimals and infinitely large elements. Hyperreals provide a rigorous setting for infinitesimal reasoning in calculus.

Real number: A real number is an element of the complete ordered field representing points on the number line. The hyperreal field extends the real numbers while preserving their ordinary arithmetic and order.
Ultrapower: An ultrapower is a structure formed from sequences by identifying those that agree on a chosen ultrafilter. An ultrapower of the reals supplies a standard construction of the hyperreal field.
Nonstandard analysis: Nonstandard analysis is a mathematical framework that extends classical mathematics with infinitesimals and infinite quantities. It is the framework in which hyperreal numbers make infinitesimal arguments rigorous.
Real number system: The real number system is the complete ordered field of finite numbers used in classical analysis. Unlike the reals, the hyperreals contain nonzero infinitesimals and infinite elements.
Ordered field: An ordered field is a field equipped with an order compatible with addition and multiplication. Hyperreal numbers retain field operations and a compatible order.
Ultrafilter: An ultrafilter is a maximal proper filter on a set, selecting one side of every subset and its complement. A nonprincipal ultrafilter determines which sequences count as equivalent in the ultrapower construction.
Derivative: A derivative measures a function's instantaneous rate of change at a point. A derivative can be expressed as the standard part of a hyperreal difference quotient.
Surreal number: A surreal number belongs to a vast ordered field constructed recursively from simpler numbers. Surreals also include infinitesimals, but their construction and scope differ from the hyperreals.
Infinitesimal: An infinitesimal is a nonzero quantity whose absolute value is smaller than every positive real number. Nonzero infinitesimals distinguish the hyperreal system from the real numbers.
Hyperreal field: A hyperreal field is an ordered field extending the reals with infinitesimal and infinite elements. This is the algebraic structure whose elements are hyperreal numbers.

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