Definitions for hyperreal number
hy·per·re·al num·ber

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1. hyperreal numbernoun

   Any of a set of numbers which extends the set of real numbers by including infinitesimals and infinities, and is used in non-standard analysis.

WikipediaRate this definition:0.0 / 0 votes

1. Hyperreal number

   In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) 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 + ⋯ + 1 {\displaystyle 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.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 ⁡ ( π n ) = 0 {\displaystyle \sin({\pi n})=0} for all integers n, one also has sin ⁡ ( π H ) = 0 {\displaystyle \sin({\pi H})=0} for all hyperintegers H {\displaystyle H} . The transfer principle for ultrapowers is a consequence of Łoś' 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. 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 nonstandard analysis. One immediate application is the definition of the basic concepts of analysis such as the derivative and integral in a direct fashion, without passing via logical complications of multiple quantifiers. Thus, the derivative of f(x) becomes f ′ ( x ) = st ⁡ ( f ( x + Δ x ) − f ( x ) Δ x ) {\displaystyle f'(x)=\operatorname {st} \left({\frac {f(x+\Delta x)-f(x)}{\Delta x}}\right)} for an infinitesimal Δ x {\displaystyle \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.

WikidataRate this definition:0.0 / 0 votes

1. Hyperreal number

   The system of hyperreal numbers is a way 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 reciprocal is infinitesimal. The term "hyper-real" was introduced by Edwin Hewitt in 1948. 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 for all integers n, one also has for all hyperintegers H. The transfer principle for ultrapowers is a consequence of Łoś' 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. 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.

How to pronounce hyperreal number?

1. Alex

   US English

   David

   US English

   Mark

   US English

   Daniel

   British

   Libby

   British

   Mia

   British

   Karen

   Australian

   Hayley

   Australian

   Natasha

   Australian

   Veena

   Indian

   Priya

   Indian

   Neerja

   Indian

   Zira

   US English

   Oliver

   British

   Wendy

   British

   Fred

   US English

   Tessa

   South African

How to say hyperreal number in sign language?

Numerology

1. Chaldean Numerology

   The numerical value of hyperreal number in Chaldean Numerology is: 2

2. Pythagorean Numerology

   The numerical value of hyperreal number in Pythagorean Numerology is: 1

References

1. ^ Wiktionary
   https://en.wiktionary.org/wiki/Hyperreal_Number
2. ^ Wikipedia
   https://en.wikipedia.org/wiki/Hyperreal_Number
3. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=hyperreal number