infty

Infty is a common informal shorthand for the mathematical concept of infinity, a notion used to describe quantities without bound or end. It is often denoted by the symbol ∞ and appears in many areas of mathematics to indicate unbounded growth, limits that grow without bound, or sets that are not finite in size.

Origin and symbol: The infinity symbol ∞ was introduced by John Wallis in 1655, derived from a sideways figure eight called a lemniscate. The word infinity itself comes from Latin infinitus, formed from in- “not” and finis “end.” Over time, the concept of infinity has become a central and formal part of analysis, set theory, and related fields.

Mathematical use: Infinity appears in limits, where a function or sequence grows without bound as its variable increases or decreases without bound. It also arises in improper integrals and infinite series, where summations or area calculations extend indefinitely. In the extended real number line, mathematicians sometimes include +∞ and −∞ as convenient boundary points to express certain limits, with specific conventions for arithmetic involving infinity and rules that avoid undefined expressions like ∞ − ∞.

Philosophical and set-theoretic perspectives: A key distinction is between potential infinity (an ongoing process with no final result) and actual infinity (a completed infinite object, such as the set of all natural numbers). In set theory, infinite sets are categorized by size or cardinality. Cantor showed that some infinite sets are larger than others (for example, the set of real numbers is uncountable, while the set of natural numbers is countable). This led to concepts such as aleph-null and the continuum, and to questions like the continuum hypothesis.

Computing and conventions: In computing and some mathematical software, infinity can be represented explicitly (for example, as +∞ or −∞) to model unbounded values. However, operations involving infinity follow specific rules and can be undefined in certain contexts.