A real number is a member of the field containing all rational and irrational numbers. The set of real numbers is also called the continuum, denoted . The set of reals is called Reals in the Wolfram Language, and a number can be tested to see if it is a member of the reals using the command Element[x, Reals], and expressions that are real numbers have the Head of Real.

The real numbers can be extended with the addition of the imaginary number i, equal to . Numbers of the form , where and are both real, are called complex numbers, which also form a field. Another extension which includes both the real numbers and the infinite ordinal numbers of Georg Cantor is the surreal numbers.

The historical database "Plouffe's Inverter" cataloged real numbers related to mathematical constants and functions.

Almost all real numbers are lexicons, meaning that they do not obey probability laws such as the law of large numbers (Gruber 1991; Calude and Zamfirescu 1998; Trott 2004, p. 69).

See also

Affinely Extended Real Numbers, Complex Number, Constant, Continuum, i, Imaginary Number, Integer Relation, Projectively Extended Real Numbers, Rational Number, Real Number Picking, Real Part, Surreal Number Explore this topic in the MathWorld classroom

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References

Borwein, J. and Borwein, P. A Dictionary of Real Numbers. Pacific Grove, CA: Brooks/Cole, 1990.Calude, C. S. and Zamfirescu, T. "The Typical Number Is a Lexicon." New Zealand J. Math. 27, 7-13, 1998.Gruber, P. M. "An Overview of the Geometry of Numbers Including Aspects of Construction and Computation." Rendiconti Sem. Mat. Messina Ser. II, 1, 21-28, 1991.Jeffreys, H. and Jeffreys, B. S. "Real Numbers." §1.03 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 5-6, 1988.Plouffe, S. "Plouffe's Inverter." https://web.archive.org/web/20050812010306/http://pi.lacim.uqam.ca/eng/.Shamos, M. I. A Catalog of Real Numbers. In preparation? https://web.archive.org/web/20011005032001/http://www.ecom.cmu.edu/people/faculty/mshamos/resshort.htm.Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004. https://www.mathematicaguidebooks.org/.

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Real Number

Cite this as:

Weisstein, Eric W. "Real Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RealNumber.html

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