Infinity symbol

${\ displaystyle \ infty}$
Mathematical signs
arithmetic
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Analysis
Sum symbol Σ
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Difference sign , Nabla ,
Prime
Partial differential
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Concatenation characters
Infinity symbol
geometry
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Set theory
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Element character
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logic
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The infinity sign ( ) is a mathematical sign that is used to symbolize infinity . It resembles a lying digit eight . In the meaning of an infinitely large number it was introduced in 1655 by the English mathematician John Wallis . The reasons for this choice are not entirely clear; possibly it originated from a ligature ↀ of the Roman numeral CIƆ for the number 1000, or as a closed variant of the last Greek lowercase letter ω ( omega ). Depending on the font, the two loops are the same size or the left one is smaller. ${\ displaystyle \ infty}$

In modern mathematics , the infinity sign is mainly used to describe limit values for sequences and series . As a symbol, it is also used outside of mathematics with a transferred meaning.

history

The mathematician John Wallis is considered to be one of the pioneers of infinitesimal calculus . In his work he further developed the principle of the individual from Bonaventura Cavalieri . Right at the beginning of his work on conic sections De sectionibus conicis from 1655, written in Latin , he wrote:

“Suppono in limine (juxta Bonaventuræ Cavallerii Geometriam Indiviſibilium ) Planum quodlibet quaſi ex infinitis lineis parallelis conflari: Vel potius (quod ego mallem) ex infinitis parallelogrammis æque altis; quorum quidem ſingulorum altitudo fit totius altitudinis , ſive aliquota pars infinite para; (eſto enim nota numeri infiniti;) adeoque omnium ſimul altitudo æqualis altitunini figuræ. " ${\ displaystyle {\ tfrac {1} {\ infty}}}$${\ displaystyle \ infty}$

“At first I assume (according to Bonaventura Cavalieri's Geometry of Individuals ) that every flat figure is composed of an infinite number of parallel lines: Or rather (which I prefer) of an infinite number of parallelograms of the same height; Every single one of these heights makes up the total height, or an infinitely small portion (for this denote an infinitely large number;) therefore the height of all taken together is equal to the height of the figure. " ${\ displaystyle {\ tfrac {1} {\ infty}}}$${\ displaystyle \ infty}$

- John Wallis : De sectionibus conicis , 1655

At this point Wallis makes a significant modification of the cavalier principle. With him, a flat geometric figure does not consist of individual lines , but of parallelograms . He specifies its height as , i.e. as an infinitely small part of the total height of the figure. With the symbol he denotes an infinitely large number . ${\ displaystyle {\ tfrac {1} {\ infty}}}$${\ displaystyle \ infty}$

Roman numerals according to Liberius (1582)

It is not known exactly why Wallis chose this symbol. He probably knew it as a 7th century ligature ↀ of the Roman numeral CIƆ (also M) for the number 1000. The Dutch mathematician Bernard Nieuwentijt also used a lowercase m as a symbol for infinity in his work Analysis infinitorum in 1695 . According to other authors, the character originated from a closed variant of the last Greek lowercase letter ω ( omega ). Interpretations of the sign as a lemniscate , Möbius strip or number 8 lying on its side ( English ) are more modern in nature.

At the beginning of the 18th century, the infinity sign is mostly found in literature in connection with the concept of the infinitely small. For Gottfried Leibniz and Isaac Newton , its importance and admissibility were still considered a mathematical and philosophical problem. It was not until Leonhard Euler , who took a formal standpoint and, in contrast to Leibniz and Newton, rejected metaphysical legitimations of infinitely small sizes, that the infinity sign became an integral part of the mathematical symbolic language in the second half of the 18th century. In the course of the 19th century, the theory of infinitesimal quantities was replaced by the mathematically more stringent theory of differential and integral calculus . Since then, the infinity symbol has mainly been used to describe limit values for sequences and series .

use

In modern mathematics , the infinity sign is primarily used to represent potential infinity . If a sequence of numbers tends towards a limit value , this fact becomes through ${\ displaystyle a_ {1}, a_ {2}, \ ldots}$${\ displaystyle a}$

${\ displaystyle \ lim _ {n \ to \ infty} a_ {n} = a}$

written down. It symbolizes that the natural number should be arbitrarily large. However, the infinity sign itself is not a natural number. A series, i.e. an infinite sum of the terms of a sequence, is correspondingly through ${\ displaystyle n \ to \ infty}$ ${\ displaystyle n}$

${\ displaystyle \ sum _ {i = 1} ^ {\ infty} a_ {i} = \ lim _ {n \ to \ infty} \ sum _ {i = 1} ^ {n} a_ {i}}$

written down. For real number sequences , certain divergence is also defined and one then writes

${\ displaystyle \ lim _ {n \ to \ infty} a_ {n} = \ infty}$.

Correspondingly, an upwardly unbounded interval of real numbers is denoted by. In the integral calculus there are also improper integrals of the form ${\ displaystyle [a, \ infty)}$

${\ displaystyle \ int _ {a} ^ {\ infty} f (x) ~ dx = \ lim _ {b \ to \ infty} \ int _ {a} ^ {b} f (x) ~ dx}$

considered. Certain divergence can also take place after and therefore there are also downward or bilateral unlimited intervals and corresponding integrals. In the topology , an extension of the real numbers by the two elements and is considered, in which certain divergent sequences then also converge. With ${\ displaystyle - \ infty}$${\ displaystyle + \ infty}$${\ displaystyle - \ infty}$

${\ displaystyle \ | f \ | _ {\ infty} = \ lim _ {p \ to \ infty} \ | f \ | _ {p}}$

is also called the supremum norm of a (restricted) function, which arises as the limit value of the L p norms for . ${\ displaystyle p \ to \ infty}$

symbolism

The sign is also used outside of mathematics with different meanings, including as a symbol for ${\ displaystyle \ infty}$

It is used as a trademark for the loudspeaker brand Infinity , the car brand Infiniti and the Microsoft Visual Studio software. It can also be found in the label ♾ for acid-free and therefore long-lasting paper .

Lenses e.g. B. in photography must be focused using the distance adjustment . The axial setting relative to the film plane is not linear to the object distance. For large distances (depending on the focal length used) it is no longer necessary to set very precisely because the values ​​are very close together. From a certain distance - depending on the lens construction - all objects are perceived as sharp at the same time. This setting is usually marked with infinity ( ) on lenses . ${\ displaystyle \ infty}$

Coding

The infinity symbol in different fonts

The infinity sign is encoded in computer systems as follows:

Coding in Unicode, HTML and LaTeX
character Unicode designation HTML Latex
U+221E infinity infinity & # x221E; & # 8734; ? \infty

Variations of the infinity sign are the following signs:

Coding in Unicode, HTML and LaTeX
character Unicode designation HTML Latex
U+221D proportional to proportional to & # x221D; & # 8733; & prop; \propto
U+22DE infinity negated with vertical bar infinity negated with a vertical line & # x22DE; & # 8926;
U+267E permanent paper sign Acid-free paper sign & # x267E; & # 9854;
U+29DC incomplete infinity incomplete infinity & # x29DC; & # 10716;
U+29DD tie over infinity Arc over infinity & # x29DD; & # 10717;

literature

• Brian Clegg: A brief history of infinity . Constable & Robinson, 2013, ISBN 978-1-4721-0764-0 .
• Maria Reményi: History of the Symbol . In: Spectrum of Science Highlights 2/13 . Spektrum Verlag, 2013.
• Paolo Zellini: A Brief History of Infinity . CH Beck, 2010, ISBN 978-3-406-59092-4 .