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*hyperreal number*

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hyperreal number

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**hyperreal numbers**is a way of treating 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

1 + 1 + cdots + 1 (for any finite number of terms).

Such numbers are infinite, and their reciprocals are infinitesimals. The term "hyper-real" was introduced by Edwin Hewitt in 1948.Hewitt (1948), p. 74, as reported in Keisler (1994)The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic Law of Continuity. The transfer principle 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 sin({pi n})=0 for all integers

*n*, one also has sin({pi H})=0 for all hyperintegers

*H*. The transfer principle for ultrapowers is a consequence of ÅoÅ›' theorem of 1955.Concerns about the soundness of arguments involving infinitesimals date back to ancient Greek mathematics, with Archimedes replacing such proofs with ones using other techniques such as the method of exhaustion.Ball, p. 31 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 that Robinson delineated.The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called non-standard analysis. One immediate application is the definition of the basic concepts of analysis such as derivative and integral in a direct fashion, without passing via logical complications of multiple quantifiers. Thus, the derivative of

*f(x)*becomes f'(x) = {rm st}left( frac{f(x+Delta x)-f(x)}{Delta x} right) for an infinitesimal Delta x, where

*st(·)*denotes the standard part function, which "rounds off" each finite hyperreal to the nearest real. Similarly, the integral is defined as the standard part of a suitable infinite sum.

## The transfer principle

The idea of the hyperreal system is to extend the real numbers**R**to form a system *

**R**that includes infinitesimal and infinite numbers, but without changing any of the elementary axioms of algebra. Any statement of the form "for any number x..." that is true for the reals is also true for the hyperreals. For example, the axiom that states "for any number

*x*,

*x*+ 0 =

*x*" still applies. The same is true for quantification over several numbers, e.g., "for any numbers

*x*and

*y*,

*xy*=

*yx*." This ability to carry over statements from the reals to the hyperreals is called the transfer principle. However, statements of the form "for any

*set*of numbers S ..." may not carry over. The only properties that differ between the reals and the hyperreals are those that rely on quantification over sets, or other higher-level structures such as functions and relations, which are typically constructed out of sets. Each real set, function, and relation has its natural hyperreal extension, satisfying the same first-order properties. The kinds of logical sentences that obey this restriction on quantification are referred to as statements in first-order logic.The transfer principle, however, doesn't mean that

**R**and *

**R**have identical behavior. For instance, in *

**R**there exists an element

*ω*such that

1

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