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infinity, in mathematics, that which is not finite; it is often indicated by the symbol *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).

*The Columbia Electronic Encyclopedia,* 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.