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.)

In mathematics, the useful concept of a process with no end. As represented by the symbol ∞, it is often mistakenly thought to be the largest number or a place on the real number line. Instead, it is the idea of a limit, as in the expression math.x → ∞, which suggests that the variable math.x increases without bound. For example, the function math.f(math.x) = 1/math.x, or the reciprocal of math.x, tends toward 0 as math.x approaches infinity as a limit. This process of approaching is crucial to the definition of the derivative and the integral in calculus, as well as to many other concepts of mathematical analysis.

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