Infinity symbol

"∞" redirects here. For other uses, seeinfinity (disambiguation)andinfinity sign (disambiguation).

$\infty$
Infinity symbol
InUnicode	U+221E∞INFINITY(∞)
Different from
Different from	U+267E♾PERMANENT PAPER SIGN U+26AD⚭MARRIAGE SYMBOL U+221D∝PROPORTIONAL TO

Theinfinity symbol(∞) is amathematical symbolrepresenting the concept ofinfinity.This symbol is also called a lemniscate,after thelemniscatecurves of a similar shape studied inalgebraic geometry,or "lazy eight", in the terminology oflivestock branding.

This symbol was first used mathematically byJohn Wallisin the 17th century, although it has a longer history of other uses. In mathematics, it often refers to infinite processes (potential infinity) rather than infinite values (actual infinity). It has other related technical meanings, such as the use of long-lasting paper inbookbinding,and has been used for its symbolic value of the infinite in modern mysticism and literature. It is a common element ofgraphic design,for instance in corporate logos as well as in older designs such as theMétis flag.

Both the infinity symbol itself and several variations of the symbol are available in variouscharacter encodings.

History

First known usage of the infinity symbol, byJohn Wallisin 1655

John Wallis in 1701

The lemniscate has been a common decorative motif since ancient times; for instance, it is commonly seen onViking Agecombs.

The English mathematicianJohn Wallisis credited with introducing the infinity symbol with its mathematical meaning in 1655, in hisDe sectionibus conicis.Wallis did not explain his choice of this symbol. It has been conjectured to be a variant form of aRoman numeral,but which Roman numeral is unclear. One theory proposes that the infinity symbol was based on the numeral for 100 million, which resembled the same symbol enclosed within a rectangular frame.Another proposes instead that it was based on the notation CIↃ used to represent 1,000.Instead of a Roman numeral, it may alternatively be derived from a variantofω,the lower-case form ofOmega,the last letter in theGreek Alpha bet.

Perhaps in some cases because of typographic limitations, other symbols resembling the infinity sign have been used for the same meaning.One paper byLeonhard Eulerwas typeset with an open letterform more closely resembling a reflected and sideways S than a lemniscate (somethinglikeS),and even"O–O"has been used as a stand-in for the infinity symbol itself.

Usage

Mathematics

In mathematics, the infinity symbol is typically used to represent apotential infinity.For instance, in mathematical expressions withsummationsandlimitssuch as

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

the infinity sign is conventionally interpreted as meaning that the variable growsarbitrarily largetowards infinity, rather than actually taking an infinite value, although other interpretations are possible.

When quantifyingactual infinity,infinite entities taken as objects per se, other notations are typically used. For example,$\aleph _{0}$(aleph-nought) denotes thecardinal numberrepresenting the size of the set ofnatural numbers,and$\ Omega$(Omega) denotes the smallestordinal numberwhich is larger than all natural numbers.

The infinity symbol may also be used to represent apoint at infinity,especially when there is only one such point under consideration. This usage includes, in particular, the infinite point of aprojective line,and the point added to atopological spaceto form itsone-point compactification.

Other technical uses

Side view of a camera lens, showing infinity symbol on the focal length indicator

In areas other than mathematics, the infinity symbol may take on other related meanings. For instance, it has been used inbookbindingto indicate that a book is printed onacid-free paperand will therefore be long-lasting.Oncamerasand theirlenses,the infinity symbol indicates that the lens'sfocal lengthisset to an infinite distance,and is "probably one of the oldest symbols to be used on cameras".

Symbolism and literary uses

The infinity symbol appears on several cards of theRider–Waite tarot deck.

In modern mysticism, the infinity symbol has become identified with a variation of theouroboros,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.

In the works ofVladimir Nabokov,includingThe GiftandPale Fire,the figure-eight shape is used symbolically to refer to theMöbius stripand 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. Nabokov's poem after which he entitledPale Fireexplicitly refers to "the miracle of the lemniscate".Other authors whose works use this shape with its symbolic meaning of the infinite includeJames Joyce,inUlysses,andDavid Foster Wallace,inInfinite Jest.

Graphic design

The well-known shape and meaning of the infinity symbol have made it a commontypographicelement ofgraphic design.For instance, theMétis flag,used by the CanadianMétispeople since the early 19th century, is based around this symbol.Different theories have been put forward for the meaning of the symbol on this flag, including the hope for an infinite future for Métis culture and its mix of European andFirst Nationstraditions,but also evoking the geometric shapes of Métic dances,,Celtic knots,orPlains First Nations Sign Language.

Arainbow-coloured infinity symbol is also used by theautism rights movement,as a way to symbolize the infinite variation of the people in the movement and of human cognition.TheBakelitecompany took up this symbol in its corporate logo to refer to the wide range of varied applications of the synthetic material they produced.Versions of this symbol have been used in other trademarks, corporate logos, and emblems including those ofFujitsu,Cell Press,and the2022 FIFA World Cup.

Encoding

The$\infty$symbol in severaltypefaces

The symbol is encoded inUnicodeatU+221E∞INFINITYand inLaTeXas\infty:$\infty$.An encircled version is encoded for use as a symbol foracid-free paper.

Character information

Preview	∞		♾
Unicode name	INFINITY		PERMANENT PAPER SIGN
Encodings	decimal	hex	dec	hex
Unicode	8734	U+221E	9854	U+267E
UTF-8	226 136 158	E2 88 9E	226 153 190	E2 99 BE
GB 18030	161 222	A1 DE	129 55 174 56	81 37 AE 38
Numeric character reference	∞	∞	♾	♾
Named character reference	∞
OEM-437 (Alt Code)	236	EC
Mac OS Roman	176	B0
Symbol Font encoding	165	A5
Shift JIS	129 135	81 87
EUC-JP	161 231	A1 E7
EUC-KR/UHC	161 196	A1 C4
EUC-KPS-9566	162 172	A2 AC
Big5	161 219	A1 DB
LaTeX	\infty		\acidfree
CLDRtext-to-speech name	infinity sign		infinity

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

Character information

Preview	⧜		⧝		⧞
Unicode name	INCOMPLETE INFINITY		TIE OVER INFINITY		INFINITY NEGATED WITH VERTICAL BAR
Encodings	decimal	hex	dec	hex	dec	hex
Unicode	10716	U+29DC	10717	U+29DD	10718	U+29DE
UTF-8	226 167 156	E2 A7 9C	226 167 157	E2 A7 9D	226 167 158	E2 A7 9E
Numeric character reference	⧜	⧜	⧝	⧝	⧞	⧞
Named character reference	⧜		⧝		⧞
LaTeX	\iinfin		\tieinfty		\nvinfty

See also

• Aleph number
• History of mathematical notation
• Lazy Eight (disambiguation)