REAL NUMBER

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 involving the symbol i, or , 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. 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 = .16666 etc. or 2/7 = .285714285714 etc. The decimal formed as .42442444244442 etc. 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 . Some numbers, such as p and e, 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 of fractions determined in some regular way. The real numbers can be characterized by the important mathematical property of completeness, meaning that every 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 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, meaning that the rationals can be paired off with a subset of the irrationals, while the reverse pairing is not possible. The class of real numbers is generally extended to include the first transfinite number (q.v.), which is symbolized by the Hebrew letter aleph and that is identified with the number of integers.