The system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. Such quantities had been in widespread use in various forms, for several centuries prior to the introduction of hyperreal numbers. 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. 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. 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

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