In
nonstandard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...
, a monad (also called halo)
is the set of points infinitesimally close to a given point.
Given a
hyperreal number
In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
''x'' in R
∗, the monad of ''x'' is the set
:
If ''x'' is finite (limited), the unique real number in the monad of ''x'' is called the
standard part
In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every suc ...
of ''x''.
See also
*
Infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
Notes
References
H. Jerome Keisler: Foundations of Infinitesimal Calculus, available for downloading
Nonstandard analysis
{{mathanalysis-stub