infinity

infinity in mathematics, that which is not finite. A sequence of numbers, a₁ , a₂ , a₃ , … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e., are larger than some number, N, that may be chosen at will to be a million, a billion, or any other large number (see limit ). The term infinity is used in a somewhat different sense to refer to a collection of objects that does not contain a finite number of objects. For example, there are infinitely many points on a line, and Euclid demonstrated that there are infinitely many prime numbers. The German mathematician Georg Cantor showed that there are different orders of infinity, the infinity of points on a line being of a greater order than that of prime numbers (see transfinite number ). In geometry one may define a point at infinity, or ideal point, as the point of intersection of two parallel lines, and similarly the line at infinity is the locus of all such points; if homogeneous coordinates ( x₁ , x₂ , x₃ ) are used, the line at infinity is the locus of all points ( x₁ , x₂ , 0), where x₁ and x₂ are not both zero. (Homogeneous coordinates are related to Cartesian coordinates by x = x₁ / x₃ and y = x₂ / x₃ .)

Bibliography: See A. D. Aczel, The Mystery of the Aleph (2000); D. F. Wallace, Everything and More (2003).

infinity (symbol ∞) Abstract quantity that represents the magnitude of an object without limit or end. In geometry, the ‘point at infinity’ is where parallel lines can be considered as meeting. In algebra, 1/x approaches infinity as x approaches zero. In set theory, the set of all integers is an example of an infinite set.