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Definition: real number from Philip's Encyclopedia

Any number that is a rational number or an irrational number. Real numbers exclude imaginary numbers (the square roots of negative quantities). See also complex number

Summary Article: real number from The Penguin Dictionary of Mathematics

A number that can be written in the form ±n:a1a2a3…, where n is an integer and each ai is one of the digits 0 to 9, for example, 2, ⅓ = 0:333 …, -1:5, and π = 3:14159…. Real numbers are either rational or irrational. The set of all real numbers is denoted by R. The real numbers can be formally defined in terms of Dedekind cuts of the rational numbers or Cauchy sequences (see metric space) of rational numbers.

There is a one-to-one correspondence between the set of real numbers and the points of an infinite directed line containing a fixed origin. The positive real number +a corresponds to the point whose distance from the origin is a units measured in the positive direction, and the negative number -b corresponds to the point whose distance from the origin is b units measured in the negative direction. The number zero corresponds to the origin. In this context the line is called a number line or real line, and may be denoted by R1.

Compare complex number; see decimal; Euclidean space.

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