Hyperreal Number

The hyperreal numbers are the elements of a field extension of the real numbers that includes certain classes of infinite and infinitesimal numbers; a hyperreal number is called finite when its absolute value is less than some integer, and infinitesimal when its absolute value is less than one over every positive integer. The term hyper-real was introduced by Edwin Hewitt in 1948, and the hyperreals satisfy the transfer principle, a rigorous version of Leibniz's heuristic law of continuity, under which any true first-order statement about the real numbers, such as the commutative law of addition, remains true of the hyperreals as well. In the 1960s Abraham Robinson proved that the hyperreal numbers are logically consistent exactly when the real numbers are, putting to rest longstanding concerns, dating back to ancient Greek mathematics, about whether reasoning with infinitesimals could be trusted; applying the hyperreals and the transfer principle to problems of analysis is called nonstandard analysis, and it lets basic concepts such as the derivative and the integral be defined directly, without the multiple layers of quantifiers that ordinary limit-based definitions require. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/