mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperrealnumber x {\displaystyle
the standard real number system, but they do exist in other number systems, such as the surreal number system and the hyperrealnumber system, which can
numbers correspond to the same cardinal number. Hyperreal numbers are used in non-standard analysis. The hyperreals, or nonstandard reals (usually denoted
sentences in first-order logic as the real numbers themselves. The set of hyperreal numbers satisfies the same first order sentences as R {\displaystyle \mathbb