Infinity symbol

John Wallis introduced the infinity symbol to mathematical literature.

The shape of a sideways figure eight has a long pedigree; for instance, it appears in the cross of Saint Boniface, wrapped around the bars of a Latin cross.[1] However, John Wallis is credited with introducing the infinity symbol with its mathematical meaning in 1655, in his De sectionibus conicis.[1][2][3][4][5] Wallis did not explain his choice of this symbol, but it has been conjectured to be a variant form of a Roman numeral for 1,000 (originally CIƆ, also CƆ,[6] which was sometimes used to mean "many"), or a variant of the Greek letter ω (omega)—the last letter in the Greek alphabet.[7]

Symbol used by Euler to denote infinity

Leonhard Euler used an open variant of the symbol[8] in order to denote "absolutus infinitus". Euler freely performed various operations on infinity, such as taking its logarithm. This symbol is not used anymore, and is not encoded as a separate character in Unicode.

In mathematics, the infinity symbol is used more often to represent a potential infinity,[1] rather than an actually infinite quantity as included in the extended real numbers, the ordinal numbers and the cardinal numbers (which use other notations).[9] For instance, in mathematical expressions with summations and limits such as the one below:

${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{2^{n}}}=\lim _{x\to \infty }{\frac {2^{x}-1}{2^{x-1}}}=2,}$

the infinity sign is conventionally interpreted as meaning that the variable grows arbitrarily large towards infinity—rather than actually taking an infinite value.

The infinity symbol may also be used to represent a point at infinity, especially when there is only one such point under consideration. This usage includes, in particular, the infinite point of a projective line,[10][11] and the point added to a topological space ${\displaystyle T}$ to form its one-point compactification ${\displaystyle T_{\infty }}$.[12]

In areas other than mathematics, the infinity symbol may take on other related meanings. For instance, it has been used in bookbinding to indicate that a book is printed on acid-free paper and will therefore be long-lasting.[13]

The infinity symbol appears on several cards of the Rider–Waite tarot deck

In modern mysticism, the infinity symbol has become identified with a variation of the ouroboros, an ancient image of a snake eating its own tail that has also come to symbolize the infinite, and the ouroboros is sometimes drawn in figure-eight form to reflect this identification—rather than in its more traditional circular form.[14]

In the works of Vladimir Nabokov, including The Gift and Pale Fire, the figure-eight shape is used symbolically to refer to the Möbius strip and the infinite, as is the case in these books' descriptions of the shapes of bicycle tire tracks and of the outlines of half-remembered people. The poem after which Pale Fire is entitled explicitly refers to "the miracle of the lemniscate".[15]

The well-known shape and meaning of the infinity symbol have made it a common typographic element of graphic design. For instance, the Métis flag, used by the Canadian Métis people since the early 19th century, is based around this symbol.[16] A rainbow-coloured infinity symbol is also used by the neurodiversity movement.[17] In modern commerce, corporate logos featuring this symbol have been used by, among others, the 2022 FIFA World Cup.[18]

The symbol is encoded in Unicode at U+221E ∞ INFINITY[19] and in LaTeX as \infty: ${\displaystyle \infty }$.[20] An encircled version is encoded for use either as a symbol for acid-free paper or as an emoji.[21]

The Unicode set of symbols also includes several variant forms of the infinity symbol that are less frequently available in fonts in the block Miscellaneous Mathematical Symbols-B.[34]

Wikimedia Commons has media related to Infinity symbols.

• History of mathematical notation