Hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal
numbers. A hyperreal number 
is finite iff 
for some ordinary positive
integer 
. It is infinitesimal iff 
for every ordinary positive
integer 
(Keisler 2007, p. 2). This convention includes 0 among the infinitesimals.
Hyperreal Number
See also
Ax-Kochen Isomorphism Theorem, Nonstandard AnalysisExplore with Wolfram|Alpha
References
Apps, P. "The Hyperreal Line." http://members.tripod.com/PhilipApps/line.html.Goldblatt, R. Lectures on the Hyperreals: An Introduction to Nonstandard Analysis. New York: Springer-Verlag, 1998. https://doi.org/10.1007/978-1-4612-0615-6.Keisler, H. J. Foundations of Infinitesimal Calculus. Online ed., 2007. Revised Jun. 19, 2022. https://people.math.wisc.edu/~hkeisler/foundations.pdf.Keisler, H. J. "The Hyperreal Line." In Real Numbers, Generalizations of the Reals, and Theories of Continua (Ed. P. Ehrlich). Norwell, MA: Kluwer, 1994.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002.Referenced on Wolfram|Alpha
Hyperreal NumberCite this as:
Weisstein, Eric W. "Hyperreal Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HyperrealNumber.html