Standard part function

The standard part function maps each finite hyperreal number to the unique real number infinitesimally close to it. It is undefined for infinite hyperreal numbers.

Connect

Hyperreal number: An element of an ordered field extending the real numbers and containing nonzero infinitesimals and infinite numbers. The function takes finite hyperreal numbers as inputs.
Ordered field: A field equipped with an order compatible with addition and multiplication. Order lets hyperreal numbers be classified as finite, infinitesimal, or infinite.
Nonstandard calculus: A formulation of calculus using hyperreal numbers, infinitesimals, and infinite numbers. Taking standard parts extracts real derivatives and integrals from hyperreal calculations.
Nonstandard analysis: A rigorous approach to analysis that extends the real numbers with infinitesimal and infinite elements. The standard-part function is a central bridge from its extended numbers to the reals.
Finite hyperreal number: A hyperreal number whose absolute value is less than some positive real number. Exactly these inputs have standard parts.
Completeness of the real numbers: The property that every nonempty bounded-above set of real numbers has a least upper bound. Completeness ensures each finite hyperreal lies infinitesimally close to a real number.
Nonstandard definition of limit: A characterization of limits using hyperreal inputs infinitesimally close to a point. Standard parts convert infinitesimal input and output relations into ordinary limit statements.
Real analysis: The study of real-valued functions, sequences, limits, and related structures. Standard parts translate certain hyperreal arguments into real-analysis results.
Infinitesimal: A nonzero hyperreal number whose absolute value is smaller than every positive real number. The output differs from its input by an infinitesimal.
Equivalence relation: A relation that is reflexive, symmetric, and transitive, partitioning objects into equivalence classes. Infinitesimal closeness groups finite hyperreals around the same real value.