# infinity

**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 infinityif 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.

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