In general, infinity is the quality or state of endlessness or having no limits in terms of time, space, or other quantity. In mathematics, infinity is the conceptual expression of such a "numberless" number. It is often symbolized by the lemniscate (also known as the *lemniscate of Bernoulli* ), which looks something like the numeral 8 written sideways ( ). This symbol for infinity was first used in the 1600s by the mathematician John Wallis.

Infinity can be defined as the limit of 1/ *x* as *x* approaches zero. Sometimes people say that 1/0 is equal to infinity, but technically, division by zero is not defined. Another notion is that infinity is a quantity *x* such that *x* + 1 = *x* . The idea is that the quantity is so large (either positive or negative) that increasing its value by 1 does not change it.

A set (see set theory ) can be defined as infinite if there exists a one-to-one correspondence between that set and a proper subset of itself. According to this definition, the set of integer s is infinite because its elements can be paired off one-to-one with all the even integers:

... | -3 | -2 | -1 | 1 | 2 | 3 | ... | |

... | -6 | -4 | -2 | 2 | 4 | 6 | ... |

The converse of the above statement is not always true. Some infinite sets have infinite proper subsets such that they cannot be paired off one-to-one. An example is the set of real number s and its proper subset, the set of integers.

In the 1800s, Georg Cantor defined infinity in terms of the cardinalities of infinite sets. The cardinality of a set is the number of elements in the set. In this sense, the cardinality of the set of integers is smaller than the cardinality of the set of real numbers, even though both sets are infinite. The set of integers is denumerable (its elements can all be accounted for by means of a listing scheme), while the set of real numbers is not denumerable.

In a more down-to-earth sense, the words "approaches infinity" are used in place of the words "increases without limit." Thus, it is said that the limit of 1/ *x* , as *x* approaches infinity, is equal to zero. In this context, infinity does not represent a defined quantity, but is merely a convenient expression.

Also see Mathematical Symbols .

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