Jump to content

Integral symbol: Difference between revisions

From Wikipedia, the free encyclopedia
Browse history interactively
← Previous editNext edit →
Content deleted Content added
VisualWikitext
0123456789
Tags: Reverted nowiki added Visual edit
→‎References: Performed cleanup.
(30 intermediate revisions by 18 users not shown)
Line 1: Line 1:
{{short description|Mathematical symbol used to denote integrals and antiderivatives}}
<nowiki>Insert non-formatted text here</nowiki>[[0]]
{{distinguish|Long s|Esh (letter)|Voiceless postalveolar fricative|text=}}
[[1]]
{{Infobox symbol
[[2]]
|sign=∫
[[3]]
|name=Integral symbol
[[4]]
|unicode={{unichar|222B|Integral|html=}}
[[5]]
|variant1=<math>\displaystyle \int</math>
[[6]]
|different from={{unichar|017F|Long s|nlink=Long s}}<br>{{unichar|0283|Esh|nlink=Esh (letter)}}
[[7]]
}}
[[8]]

[[9]]
The '''integral symbol''':
{{block indent | em = 1.5 | text = <span style=font-size:200%>∫</span> ([[Unicode]]), <math>\displaystyle \int</math> ([[LaTeX]])}}
is used to denote [[integral]]s and [[Antiderivative|antiderivatives]] in [[mathematics]], especially in [[calculus]].

==History==
{{main|Leibniz's notation}}

The notation was introduced by the [[Germany|German]] mathematician [[Gottfried Wilhelm Leibniz]] in 1675 in his private writings;<ref>Gottfried Wilhelm Leibniz, ''Sämtliche Schriften und Briefe, Reihe VII: Mathematische Schriften, vol. 5: Infinitesimalmathematik 1674–1676'', Berlin: Akademie Verlag, 2008, pp. [http://www.gwlb.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/VII5A.pdf 288–295] {{Webarchive|url=https://web.archive.org/web/20211009052830/https://www.gwlb.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/VII5A.pdf |date=2021-10-09 }} ("Analyseos tetragonisticae pars secunda", October 29, 1675) and [http://www.gwlb.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/VII5B.pdf 321–331] {{Webarchive|url=https://web.archive.org/web/20161003130340/http://www.gwlb.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/VII5B.pdf |date=2016-10-03 }} ("Methodi tangentium inversae exempla", November 11, 1675).</ref><ref>{{cite web |url=http://jeff560.tripod.com/calculus.html |author=Aldrich, John |title=Earliest Uses of Symbols of Calculus |access-date=20 April 2017}}</ref> it first appeared publicly in the article "{{lang|la|De Geometria Recondita et analysi indivisibilium atque infinitorum}}" (On a hidden geometry and analysis of indivisibles and infinites), published in ''[[Acta Eruditorum]]'' in June 1686.<ref>{{citation |first=Frank J. |last=Swetz |title=Mathematical Treasure: Leibniz's Papers on Calculus – Integral Calculus |publisher=[[Mathematical Association of America]] |url=http://www.maa.org/press/periodicals/convergence/mathematical-treasure-leibnizs-papers-on-calculus-integral-calculus |access-date=February 11, 2017 |series=Convergence}}</ref><ref>{{cite book |title=Mathematics and its History |url=https://archive.org/details/mathematicsitshi0000stil |url-access=registration |first=John |last=Stillwell |author-link=John Stillwell |publisher=Springer |year=1989 |page=[https://archive.org/details/mathematicsitshi0000stil/page/110 110] }}</ref> The symbol was based on the ſ ([[long s]]) character and was chosen because Leibniz thought of the integral as an infinite [[summation|sum]] of infinitesimal [[summand]]s.

== Typography in Unicode and LaTeX ==

===Fundamental symbol===
{{main|Integral calculus}}

The integral symbol is {{unichar|222B|INTEGRAL}} in [[Unicode]]<ref name="Unicode Mathematical Operators">{{cite web| title = Mathematical Operators – Unicode| url = https://www.unicode.org/charts/PDF/U2200.pdf| access-date = 2013-04-26}}</ref> and <code>\int</code> in [[LaTeX]]. In [[HTML]], it is written as <code>&amp;#x222b;</code> ([[hexadecimal]]), <code>&amp;#8747;</code> ([[decimal]]) and <code>&amp;int;</code> ([[List of XML and HTML character entity references#Character entity references in HTML|named entity]]).

The original [[IBM PC]] [[code page 437]] character set included a couple of characters ⌠ and ⌡ (codes 244 and 245 respectively) to build the integral symbol. These were deprecated in subsequent [[MS-DOS]] [[code page]]s, but they still remain in [[Unicode]] ([[Miscellaneous Technical#(2300–233F)|U+2320 and U+2321]] respectively) for compatibility.

The ∫ symbol is very similar to, but not to be confused with, the letter [[esh (letter)|ʃ]] ("[[esh (letter)|esh]]").

===Extensions of the symbol===
{{see also|Multiple integral}}

Related symbols include:<ref name="Unicode Mathematical Operators"/><ref name="Unicode Supplemental Mathematical Operators">{{cite web| title = Supplemental Mathematical Operators – Unicode| url = https://www.unicode.org/charts/PDF/U2A00.pdf| access-date = 2013-05-05}}</ref>


[[number]]
{| class="wikitable"
|+[[Number system|Number systems]]
|[[Complex number|Complex]]
|
{| class="wikitable"
|
{| class="wikitable"
|[[Real number|Real]]
|
{| class="wikitable"
|
{| class="wikitable"
|[[Rational number|Rational]]
|
{| class="wikitable"
|
{| class="wikitable"
|[[Integer]]
|
{| class="wikitable"
|
{| class="wikitable"
|[[Natural number|Natural]]
|
{| class="wikitable"
{| class="wikitable"
|[[Zero]]: 0
|-
|-
! Meaning
|[[One]]: 1
! scope="col" colspan="2"| [[Unicode]]
! scope="col" colspan="2"| [[LaTeX]]
|-
|-
! [[Double integral]]
|[[Prime number|Prime numbers]]
|style="font-size:20pt;"| ∬
|| U+222C
|| <math>\iint</math>
|| <code>\iint</code>
|-
|-
! [[Triple integral]]
|[[Composite number|Composite numbers]]
|style="font-size:20pt;"| ∭
|}
|| U+222D
|}
|| <math>\iiint</math>
|| <code>\iiint</code>
|-
|-
! [[Quadruple integral]]
|[[Negative integer|Negative integers]]
|style="font-size:20pt;"| ⨌
|}
|| U+2A0C
|}
|| <math>\iiiint</math>
|| <code>\iiiint</code>
|-
|-
! [[Contour integral]]
|
|style="font-size:20pt;"| ∮
{| class="wikitable"
|| U+222E
|[[Fraction]]
|| <math>\oint</math>
|
|| <code>\oint</code>
{| class="wikitable"
|[[Finite decimal]]
|-
|-
! [[Clockwise]] integral
|[[Dyadic rational|Dyadic (finite binary)]]
|style="font-size:20pt;"| ∱
|| U+2231
||
||
|-
|-
! [[Counterclockwise]] integral
|[[Repeating decimal]]
|style="font-size:20pt;"| ⨑
|
|| U+2A11
|}
|}
||
|}
||
|}
|-
|-
! Clockwise contour integral
|
|style="font-size:20pt;"| ∲
{| class="wikitable"
|| U+2232
|[[Irrational number|Irrational]]
|| {{Intorient|symbol=varoint}}
|
|| <code>\varointclockwise</code>
{| class="wikitable"
|[[Algebraic number|Algebraic]]
|-
|-
! Counterclockwise contour integral
|[[Transcendental number|Transcendental]]
|style="font-size:20pt;"| ∳
|}
|| U+2233
|}
|| {{Intorient|symbol=ointctr}}
|}
|| <code>\ointctrclockwise</code>
|}
|-
! [[closed manifold|Closed]] [[surface integral]]
|style="font-size:20pt;"| ∯
|| U+222F
|| {{oiint}}
|| <code>\oiint</code>
|-
|-
! Closed [[volume integral]]
|[[Imaginary number|Imaginary]]
|style="font-size:20pt;"| ∰
|}
|| U+2230
|| {{oiiint}}
|| <code>\oiiint</code>
|}
|}


== Typography in other languages ==
{| class="wikitable"
[[File:Integral Uprightness.svg|150px|thumb|Regional variations ([[English language|English]], [[German language|German]], and [[Russian language|Russian]] from left to right) of the integral symbol]]
!
In other languages, the shape of the integral symbol differs slightly from the shape commonly seen in English-language textbooks. While the [[English language|English]] integral symbol leans to the right, the [[German language|German]] symbol (used throughout [[Central Europe]]) is upright, and the [[Russian language|Russian]] variant leans slightly to the left to occupy less horizontal space.<ref>{{cite web |title=Russian Typographical Traditions in Mathematical Literature |url=https://giftbot.toolforge.org/deref.fcgi?url=https%3A%2F%2F1.800.gay%3A443%2Fhttp%2Fwww.uni-giessen.de%2Fpartosch%2Feurotex99%2Fzaitsev.pdf |publisher=giftbot.toolforge.org |access-date=11 October 2021 |archive-url=https://web.archive.org/web/20120928083018/http://www.uni-giessen.de/partosch/eurotex99/zaitsev.pdf |archive-date=28 September 2012}}</ref>
![[Natural number|Natural]]

|0, 1, 2, 3, 4, 5, ... or 1, 2, 3, 4, 5, ...
Another difference is in the placement of [[limits of integration|limits]] for [[definite integral]]s. Generally, in English-language books, limits go to the right of the integral symbol:


<math display="block">
\int_0^5 f(t) \,\mathrm{d}t, \quad
\int_{g(t) = a}^{g(t) = b} f(t) \,\mathrm{d}t.
</math>


By contrast, in German and Russian texts, the limits are placed above and below the integral symbol, and, as a result, the notation requires larger line spacing but is more compact horizontally, especially when using longer expressions in the limits:
or  are sometimes used.

|-
<math display="block">
!
\int\limits_0^T f(t) \,\mathrm{d}t, \quad
![[Integer]]
\int\limits_{\!\!\!\!\!g(t) = a\!\!\!\!\!}^{\!\!\!\!\!g(t) = b\!\!\!\!\!} f(t) \,\mathrm{d}t.
|..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ...
</math><!-- the limits should use \mathclap, but as of 2019, en.wiki does not support it -->
|-

!
== See also ==
![[Rational number|Rational]]

|''a''/''b'' where ''a'' and ''b'' are integers and ''b'' is not 0
* [[Capital sigma notation]]
|-
* [[Capital pi notation]]
!

![[Real number|Real]]
== Notes ==
|The limit of a convergent sequence of rational numbers
{{Reflist}}
|-

!
== References==
![[Complex number|Complex]]
* {{Cite book |last=Stewart |first=James |year=2003 |chapter=Integrals |chapter-url-access=registration |chapter-url=https://archive.org/details/singlevariableca00stew/page/381 |title=Single Variable Calculus: Early Transcendentals |edition=5th |location=Belmont, CA |publisher=Brooks/Cole |isbn=0-534-39330-6 |page=[https://archive.org/details/singlevariableca00stew/page/381 381] }}
|''a'' + ''bi'' where ''a'' and ''b'' are real numbers and ''i'' is a formal square root of −1
* {{Cite conference |last1=Zaitcev |first1=V. |last2=Janishewsky |first2=A. |last3=Berdnikov |first3=A. |date=20 September 1999 |title=Russian Typographical Traditions in Mathematical Literature |url=https://www.staff.uni-giessen.de/~g029/eurotex99/zaitsev.pdf |archive-url=https://web.archive.org/web/20170224213730/https://www.staff.uni-giessen.de/~g029/eurotex99/zaitsev.pdf |archive-date=24 February 2017 |url-status=dead |location=Heidelberg |conference=EuroTeX'99 |work=EuroTeX'99 Proceedings}}
|}

== External links ==
* [https://www.fileformat.info/info/unicode/char/222b/index.htm Fileformat.info]

{{Infinitesimals}}

[[Category:History of calculus]]
[[Category:Mathematical symbols]]
[[Category:Gottfried Wilhelm Leibniz]]

Revision as of 16:08, 12 January 2024

Not to be confused with Long s, Esh (letter), or Voiceless postalveolar fricative.
∫
Integral symbol
In UnicodeU+222B ∫ INTEGRAL (&int;, &Integral;)
Graphical variants
Different from
Different fromU+017F ſ LATIN SMALL LETTER LONG S
U+0283 ʃ LATIN SMALL LETTER ESH

The integral symbol:

∫ (Unicode), (LaTeX)

is used to denote integrals and antiderivatives in mathematics, especially in calculus.

History

Main article: Leibniz's notation

The notation was introduced by the German mathematician Gottfried Wilhelm Leibniz in 1675 in his private writings;[1][2] it first appeared publicly in the article "De Geometria Recondita et analysi indivisibilium atque infinitorum" (On a hidden geometry and analysis of indivisibles and infinites), published in Acta Eruditorum in June 1686.[3][4] The symbol was based on the ſ (long s) character and was chosen because Leibniz thought of the integral as an infinite sum of infinitesimal summands.

Typography in Unicode and LaTeX

Fundamental symbol

Main article: Integral calculus

The integral symbol is U+222B ∫ INTEGRAL in Unicode[5] and \int in LaTeX. In HTML, it is written as &#x222b; (hexadecimal), &#8747; (decimal) and &int; (named entity).

The original IBM PC code page 437 character set included a couple of characters ⌠ and ⌡ (codes 244 and 245 respectively) to build the integral symbol. These were deprecated in subsequent MS-DOS code pages, but they still remain in Unicode (U+2320 and U+2321 respectively) for compatibility.

The ∫ symbol is very similar to, but not to be confused with, the letter ʃ ("esh").

Extensions of the symbol

See also: Multiple integral

Related symbols include:[5][6]

Meaning Unicode LaTeX
Double integral ∬ U+222C \iint
Triple integral ∭ U+222D \iiint
Quadruple integral ⨌ U+2A0C \iiiint
Contour integral ∮ U+222E \oint
Clockwise integral ∱ U+2231
Counterclockwise integral ⨑ U+2A11
Clockwise contour integral ∲ U+2232 \varointclockwise \varointclockwise
Counterclockwise contour integral ∳ U+2233 \ointctrclockwise \ointctrclockwise
Closed surface integral ∯ U+222F \oiint \oiint
Closed volume integral ∰ U+2230 \oiiint \oiiint

Typography in other languages

Regional variations (English, German, and Russian from left to right) of the integral symbol

In other languages, the shape of the integral symbol differs slightly from the shape commonly seen in English-language textbooks. While the English integral symbol leans to the right, the German symbol (used throughout Central Europe) is upright, and the Russian variant leans slightly to the left to occupy less horizontal space.[7]

Another difference is in the placement of limits for definite integrals. Generally, in English-language books, limits go to the right of the integral symbol:

By contrast, in German and Russian texts, the limits are placed above and below the integral symbol, and, as a result, the notation requires larger line spacing but is more compact horizontally, especially when using longer expressions in the limits:

See also

Notes

  1. ^ Gottfried Wilhelm Leibniz, Sämtliche Schriften und Briefe, Reihe VII: Mathematische Schriften, vol. 5: Infinitesimalmathematik 1674–1676, Berlin: Akademie Verlag, 2008, pp. 288–295 Archived 2021-10-09 at the Wayback Machine ("Analyseos tetragonisticae pars secunda", October 29, 1675) and 321–331 Archived 2016-10-03 at the Wayback Machine ("Methodi tangentium inversae exempla", November 11, 1675).
  2. ^ Aldrich, John. "Earliest Uses of Symbols of Calculus". Retrieved 20 April 2017.
  3. ^ Swetz, Frank J., Mathematical Treasure: Leibniz's Papers on Calculus – Integral Calculus, Convergence, Mathematical Association of America, retrieved February 11, 2017
  4. ^ Stillwell, John (1989). Mathematics and its History. Springer. p. 110.
  5. ^ a b "Mathematical Operators – Unicode" (PDF). Retrieved 2013-04-26.
  6. ^ "Supplemental Mathematical Operators – Unicode" (PDF). Retrieved 2013-05-05.
  7. ^ "Russian Typographical Traditions in Mathematical Literature" (PDF). giftbot.toolforge.org. Archived from the original (PDF) on 28 September 2012. Retrieved 11 October 2021.

References

Retrieved from "https://en.wikipedia.org/w/index.php?title=Integral_symbol&oldid=1195161520"