irrational number

Math.

a number that cannot be exactly expressed as a ratio of two integers.

[1545-55]

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Among the real numbers, any of those that cannot be represented as quotients of integers.

In decimal form, irrational numbers are represented by nonterminating, nonrepeating decimals. Examples include square roots of prime numbers and such transcendental numbers as π and e.

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▪ mathematics

      any real number that cannot be expressed as the quotient of two integers. For example, there is no number among integers and fractions that equals the square root of 2. A counterpart problem in measurement would be to find the length of the diagonal of a square whose side is one unit long; there is no subdivision of the unit length that will divide evenly into the length of the diagonal. (See Sidebar: Incommensurables.) It thus became necessary, early in the history of mathematics, to extend the concept of number to include irrational numbers. Each irrational number can be expressed as an infinite decimal expansion with no regularly repeating digit or group of digits. Together with the rational numbers, they form the real numbers.

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