Skip to content

Infinity

Definition

Infinity refers to a concept that is unbounded, limitless, and endless. It is denoted by the symbol ∞,\infty\text{,} and it is not a (real) number, but rather an abstract idea used to describe unbounded growth or extent.

In contexts where the direction is implicit, such as limit of a sequence, the notation n→∞n\to\infty is commonly used. However, when a variable may increase or decrease without bound, the direction must be specified explicitly:

  • +∞+\infty denotes positive infinity, e.g., x→+∞;x\to+\infty\text{;}
  • −∞-\infty denotes negative infinity, e.g., f(x)→−∞.f(x)\to -\infty\text{.}

Interpretation

Infinity describes unbounded growth rather than a numerical value. Writing n→∞n\to\infty says that the index nn eventually passes every bound one names, however large; it does not say that nn reaches a number called ∞\infty.

No real number can play this role, since every real number xx is passed by x+1x+1; and a number with ∞+1=∞\infty+1=\infty would give 1=01=0 after subtracting it from both sides. This is why expressions such as ∞−∞\infty-\infty are given no value. When growth is possible in both directions, the sign must be stated: +∞+\infty for growth upwards, −∞-\infty for growth downwards.

For finite and infinite sets, infinity describes the absence of a finite number of elements. This is a different use from unbounded growth: an infinite set, such as {1/n ⁣:n≥1}\{1/n\colon n\ge1\}, can remain bounded, since all its elements lie between 00 and 11. Later, the extended real numbers add −∞-\infty and +∞+\infty to the number line as two endpoints.

Key Results

  • Infinity is not a real number, only a symbol: it cannot be used in real arithmetic (expressions such as ∞−∞\infty-\infty or ∞/∞\infty/\infty are undefined). Order comparisons such as 1<+∞1<+\infty make sense only in the extended real number system, never as statements about real numbers.

Examples

  • n→∞n\to\infty describes unbounded growth of a sequence.
  • lim⁡x→0+1x=+∞.\displaystyle \lim_{x\to 0^+}\frac{1}{x}=+\infty\text{.}

Prerequisites

Sources

Symbols: infinity

Last updated: 20 October 2026 03:02