The system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form
Such a number is infinite, and its inverse is infinitesimal. According to Keisler (1994), the term "hyperreal" was introduced by Edwin Hewitt (1948, p. 74), who spelled it with a dash: "hyper-real".
The hyperreal numbers satisfy the transfer principle, which states that true first order statements about R are also valid in *R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals; since R is a real closed field, so is *R. Since sinπn = 0 for all integer n, one also has sinπH = 0 for all hyperinteger H. The transfer principle for ultrapowers is a consequence of Łoś' theorem of 1955.
Concerns about the logical soundness of arguments involving infinitesimals date back to ancient Greek mathematics, with Euclid replacing such proofs with ones using other techniques such as the method of exhaustion. In the 1960s Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were. This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules which Robinson delineated.
The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called non-standard analysis; some find it more intuitive than standard real analysis.
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