infinity
infinity, in mathematics, that which is not finite; it is often indicated by the symbol ∞art/infinity.gif. A sequence of numbers, a _{1}, a 2, a 3, … , 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 ( x 1, x 2, x 3) are used, the line at infinity is the locus of all points ( x 1, x 2, 0), where x 1 and x 2 are not both zero. (Homogeneous coordinates are related to Cartesian coordinates by x = x 1/ x 3 and y = x 2/ x 3.) See A. D. Aczel, The Mystery of the Aleph (2000); D. F. Wallace, Everything and More (2003).
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