infinity (redirected from Infinate)

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).

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infinity

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 x → ∞, which suggests that the variable x increases without bound. For example, the function f(x) = 1/x, or the reciprocal of x, tends toward 0 as 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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1.	(mathematics)	infinity - The size of something infinite. Using the word in the context of sets is sloppy, since different infinite sets aren't necessarily the same size cardinality as each other. See also aleph 0
2.	(programming)	infinity - The largest value that can be represented in a particular type of variable (register, memory location, data type, whatever). See also minus infinity.