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# real number

/ree"euhl, reel/, Math.
a rational number or the limit of a sequence of rational numbers, as opposed to a complex number. Also called real.
[1905-10]

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In mathematics, a quantity that can be expressed as a finite or infinite decimal expansion.

The counting numbers, integers, rational numbers, and irrational numbers are all real numbers. Real numbers are used in measuring continuously varying quantities (e.g., size, time), in contrast to measurements that result from counting. The word real distinguishes them from the imaginary numbers.

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in mathematics, a quantity that can be expressed as an infinite decimal expansion. Real numbers are used in measurements of continuously varying quantities such as size and time, in contrast to the natural numbers 1, 2, 3, …, arising from counting. The word real distinguishes them from the complex numbers (complex number) involving the symbol i, or √(−1) , used to simplify the mathematical interpretation of effects such as those occurring in electrical phenomena. The real numbers include the positive and negative integers and fractions (or rational numbers) and also the irrational numbers (irrational number). The irrational numbers have decimal expansions that do not repeat themselves, in contrast to the rational numbers, the expansions of which always contain a digit or group of digits that repeats itself, as 1/6 = 0.16666… or 2/7 = 0.285714285714…. The decimal formed as 0.42442444244442… has no regularly repeating group and is thus irrational.

The most familiar irrational numbers are algebraic numbers, which are the roots of algebraic equations with integer coefficients. For example, the solution to the equation x2 − 2 = 0 is an algebraic irrational number, indicated by √2. Some numbers, such as π (pi) and e (exponential function), are not the solutions of any such algebraic equation and are thus called transcendental irrational numbers. These numbers can often be represented as an infinite sum (infinite series) of fractions determined in some regular way, indeed the decimal expansion is one such sum.

The real numbers can be characterized by the important mathematical property of completeness, meaning that every nonempty set that has an upper bound has a smallest such bound, a property not possessed by the rational numbers. For example, the set of all rational numbers the squares of which are less than 2 has no smallest upper bound, because √2 is not a rational number. The irrational and rational numbers are both infinitely numerous, but the infinity of irrationals is “greater” than the infinity of rationals, in the sense that the rationals can be paired off with a subset of the irrationals, while the reverse pairing is not possible.

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Universalium. 2010.

### Look at other dictionaries:

• real number — n. Math. any rational or irrational number not having an imaginary part …   English World dictionary

• Real number — For the real numbers used in descriptive set theory, see Baire space (set theory). For the computing datatype, see Floating point number. A symbol of the set of real numbers …   Wikipedia

• real number — noun any rational or irrational number • Syn: ↑real • Hypernyms: ↑complex number, ↑complex quantity, ↑imaginary number, ↑imaginary • Hyponyms: ↑ …   Useful english dictionary

• real number — noun a) An element of the set of real numbers.; the set of real numbers include the rational numbers and the irrational numbers, but not all complex numbers. Every integer is a real number, but not vice versa. b) A floating point number. Even if… …   Wiktionary

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• real number — noun Date: circa 1909 a number that has no imaginary part < the set of all real numbers comprises the rationals and the irrationals > …   New Collegiate Dictionary

• real number — /ril ˈnʌmbə/ (say reel numbuh) noun 1. a number which can be the result of a measurement (opposed to complex number). 2. → decimal number …   Australian English dictionary

• real number — See number …   Philosophy dictionary

• real number — re′al num′ber [[t]ˈri əl, ril[/t]] n. math. a rational number or the limit of a sequence of rational numbers • Etymology: 1905–10 …   From formal English to slang

• real number line — noun A geometric representation of the real number system. Syn: real line …   Wiktionary