ดูต้นฉบับสำหรับ รายการค่าคงตัวทางคณิตศาสตร์

== ยุคโบราณ ==

{| class="wikitable sortable"
!ชื่อ
!สัญลักษณ์
!การขยายทศนิยม
!สูตร
!ปี
!เซต
|-
|[[1|หนึ่ง]]
|1
|1
|ไม่มี{{refn|group=nb|1 can be given as a primitive notion within [[Peano arithmetic]]. Alternatively, 0 can be a primitive notion in Peano arithmetic and 1 defined as the successor to 0. This article uses the former definition for pedagogical and chronological simplicity.}}
|ก่อนประวัติศาสตร์
|[[Natural number|<math>\mathbb{N}</math>]]
|-
|[[2|สอง]]
|2
|2
|<math>1+1</math>
|ก่อนประวัติศาสตร์
|[[Natural number|<math>\mathbb{N}</math>]]
|-
|[[One half]]
|1/2
|0.5
|<math>1/2</math>
|ก่อนประวัติศาสตร์
|[[Rational number|<math>\mathbb{Q}</math>]]
|-
|[[พาย (ค่าคงตัว)|พาย]]
|<math> \pi </math>
|3.14159 26535 89793 23846 <ref group="Mw">{{MathWorld|PiFormulas|Pi Formulas}}</ref><ref group="OEIS">{{OEIS2C|A000796}}</ref>
|อัตราส่วนของความยาวเส้นรอบวงของวงกลมต่อเส้นผ่านศูนย์กลาง
|1900 ถึง 1600 ปีก่อนคริสตกาล<ref name="Arndt_d">{{harvnb|Arndt|Haenel|2006|p=167}}</ref>
|'''''[[Transcendental number|<math>\mathbb{T}</math>]]'''''
|-
|[[รากที่สองของสอง]],
ค่าคงตัวของ[[พีทาโกรัส]]<ref>{{cite book|url=https://books.google.com/books?id=sPRbZACsXogC&pg=PA293&dq=pi+nested+radical#v=snippet|title=Mathematical sorcery: revealing the secrets of numbers|author=Calvin C Clawson|year=2001|isbn=978 0 7382 0496-3|page=IV}}{{ลิงก์เสีย|date=เมษายน 2023 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>
|<math>\sqrt{2}</math>
|1.41421 35623 73095 04880 <ref group="Mw">{{MathWorld|PythagorassConstant|Pythagoras's Constant}}</ref><ref group="OEIS">{{OEIS2C|A002193}}</ref>
|รากบวกของ <math>x^2=2 </math>
|<nowiki>1800 ถึง 1600 ปีก่อนคริสตกาล</nowiki><ref>Fowler and Robson, p. 368.

[http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html Photograph, illustration, and description of the ''root(2)'' tablet from the Yale Babylonian Collection] {{webarchive|url=https://web.archive.org/web/20120813054036/http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html|date=2012-08-13}}

[http://www.math.ubc.ca/%7Ecass/Euclid/ybc/ybc.html High resolution photographs, descriptions, and analysis of the ''root(2)'' tablet (YBC 7289) from the Yale Babylonian Collection]</ref>
|'''''[[Algebraic number|<math>\mathbb{A}</math>]]'''''
|-
|[[รากที่สองของสาม]],
ค่าคงตัวของธีโอโดรัส<ref>{{cite book|url=https://books.google.com/books?id=xAwukpHCqH0C&pg=PA15&dq=1.732050807#v=onepage|title=Figuring Out Mathematics|author=Vijaya AV|publisher=Dorling Kindcrsley (India) Pvt. Lid.|year=2007|isbn=978-81-317-0359-5|page=15}}</ref>
|<math>\sqrt{3} </math>
|1.73205 08075 68877 29352 <ref group="Mw">{{MathWorld|TheodorussConstant|Theodorus's Constant}}</ref><ref group="OEIS">{{OEIS2C|A002194}}</ref>
|รากบวกของ <math> x^2=3 </math>
| 465 ถึง 398 ปีก่อนคริสตกาล
|'''''[[Algebraic number|<math>\mathbb{A}</math>]]'''''
|-
|[[รากที่สองของห้า]]<ref>{{cite book|url=https://books.google.com/books?id=KjMVx6ljh6YC&pg=PA24&dq=2.236067977#v=onepage|title=Essential Mathematics 9|author=P A J Lewis|publisher=Ratna Sagar|year=2008|isbn=9788183323673|page=24|quote=}}</ref>
|<math> \sqrt{5} </math>
|2.23606 79774 99789 69640<ref group="OEIS">{{OEIS2C|A002163}}</ref>
|รากบวกของ <math> x^2=5</math>
|
|'''''[[Algebraic number|<math>\mathbb{A}</math>]]'''''
|-
|ฟี, [[อัตราส่วนทอง]]<ref name=":0" /><ref>{{cite book|url=https://books.google.com/books?id=ZOfUsvemJDMC&lpg=PA316&dq=1.618033988749894848&pg=PA316#v=onepage|title=The Princeton Companion to Mathematics|author1=Timothy Gowers|author2=June Barrow-Green|author3=Imre Leade|publisher=Princeton University Press|year=2007|isbn=978-0-691-11880-2|page=316}}</ref>
|<math>{\varphi}</math>
|1.61803 39887 49894 84820 <ref group="Mw">{{MathWorld|GoldenRatio|Golden Ratio}}</ref><ref group="OEIS">{{OEIS2C|A001622}}</ref>
|รากบวกของ <math>x^2-x-1=0 </math>
| ~300 ปีก่อนคริสตกาล
|'''''[[Algebraic number|<math>\mathbb{A}</math>]]'''''
|-
|[[0|ศูนย์]]
|0
|0
|The additive identity: <math>x+0=x</math>
|300-100 ศตวรรษ ก่อนคริสตกาล<ref>Kim Plofker (2009), Mathematics in India, Princeton University Press, {{ISBN|978-0-691-12067-6}}, pp. 54–56. Quote – "In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, [ ...] Pingala's use of a zero symbol [śūnya] as a marker seems to be the first known explicit reference to zero." Kim Plofker (2009), Mathematics in India, Princeton University Press, {{ISBN|978-0-691-12067-6}}, 55–56. "In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, there are five questions concerning the possible meters for any value "n". [ ...] The answer is (2)<sup>7</sup> = 128, as expected, but instead of seven doublings, the process (explained by the sutra) required only three doublings and two squarings – a handy time saver where "n" is large. Pingala's use of a zero symbol as a marker seems to be the first known explicit reference to zero.</ref>
|[[Integer|<math>\mathbb{Z}</math>]]
|-
|[[−1|ลบหนึ่ง]]
|  -1
|  -1
|<math>1-2</math>
|300-200 ปีก่อนคริสตกาล
|[[Integer|<math>\mathbb{Z}</math>]]
|-
|[[รากที่สาม]]ของสอง ([[Delian constant|Delian Constant]])
|<math>\sqrt[3]{2}</math>
|1.25992 10498 94873 16476 <ref group="Mw">{{MathWorld|DelianConstant|Delian Constant}}</ref><ref group="OEIS">{{OEIS2C|A002580}}</ref>
|รากจริงของ <math>x^3=2</math>
|ค.ศ. 46-120

<ref>{{cite book|author=Plutarch|title=Quaestiones convivales VIII.ii|url=http://ebooks.adelaide.edu.au/p/plutarch/symposiacs/chapter8.html#section80|section=718ef|quote=And therefore Plato himself dislikes Eudoxus, Archytas, and Menaechmus for endeavoring to bring down the [[doubling the cube]] to mechanical operations|access-date=2020-10-10|archive-date=2019-07-28|archive-url=https://web.archive.org/web/20190728214640/http://ebooks.adelaide.edu.au/p/plutarch/symposiacs/chapter8.html#section80|url-status=dead}}</ref>
|'''''[[Algebraic number|<math>\mathbb{A}</math>]]'''''
|-
|[[รากที่สาม]]ของสาม
|<math>\sqrt[3]3</math>
|1.44224 95703 07408 38232<ref group="OEIS">{{OEIS2C|A002581}}</ref>
|รากจริงของ <math>x^3=3</math>
|
|<math>\mathbb{A}</math>
|}

== ยุคกลางและต้นสมัยใหม่ ==
{| class="wikitable sortable"
!ชื่อ
!สัญลักษณ์
!การขยายทศนิยม
!สูตร
!ปี
!เซต
|-
|[[หน่วยจินตภาพ]] <ref name=":0" /><ref>{{cite book|url=https://books.google.com/books?id=IKmMKOtSI50C&pg=PA66&dq=%22This+leads+to+some+amazing+results.+For+example,+Euler+discovered%22#v=onepage|title=Mathematics: The New Golden Age|author=Keith J. Devlin|publisher=Columbia University Press|year=1999|isbn=978-0-231-11638-1|page=66}}</ref>
|<math>{i}</math>
|{{mvar|0 + 1i}}
|รากทั้งสองของ <math>x^2=-1</math>{{refn|group=nb|Both {{math|i}} and {{math|-i}} are roots of this equation, though neither root is truly "positive" nor more fundamental than the other as they are algebraically equivalent. The distinction between signs of {{math|i}} and {{math|-i}} is in some ways arbitrary, but a useful notational device. See [[imaginary unit]] for more information.}}
|ค.ศ. 1501 ถึง 1576
|[[Complex number|<math>\mathbb{C}</math>]]
|-
|[[John Wallis|Wallis]] Constant
|<math> W </math>
|2.09455 14815 42326 59148 <ref group="Mw">{{MathWorld|WallissConstant|Wallis's Constant}}</ref><ref group="OEIS">{{OEIS2C|A007493}}</ref>
|<math> \sqrt[3]{\frac{45-\sqrt{1929}}{18}}+\sqrt[3]{\frac{45+\sqrt{1929}}{18}}</math>
|ค.ศ. 1616 <br> ถึง <br> 1703
|'''''[[Algebraic number|<math>\mathbb{A}</math>]]'''''
|-
|[[e (mathematical constant)|Euler's number]]<ref name=":0" /><ref>{{cite book|url=https://books.google.com/books?id=zdBHMHV3m5YC&pg=PA76&dq=2.7182818284590452353602874#v=onepage|title=Mathematics and the Imagination|author=E.Kasner y J.Newman.|publisher=Conaculta|year=2007|isbn=978-968-5374-20-0|page=77}}</ref>
|<math>{e}</math>
|2.71828 18284 59045 23536 <ref group="Mw">{{MathWorld|e|e}}</ref><ref group="OEIS">{{OEIS2C|A001113}}</ref>
|<math> \lim_{n \to \infty} \! \left( \! 1 \! + \! \frac {1}{n}\right)^n </math>{{refn|group=nb|Can also be defined by the infinite series  <math> \! \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1} + \frac{1}{2!} + \frac{1}{3!} + \textstyle \cdots </math>}}
|ค.ศ. 1618<ref name="OConnor2">{{cite web|url=<!-- http://www.gap-system.org/~history/PrintHT/e.html -->http://www-history.mcs.st-and.ac.uk/HistTopics/e.html|title=The number ''e''|last1=O'Connor|first1=J J|last2=Robertson|first2=E F|publisher=MacTutor History of Mathematics}}</ref>
|'''''[[Transcendental number|<math>\mathbb{T}</math>]]'''''
|-
|[[Natural logarithm of 2]] <ref>{{cite book|url=https://books.google.com/books?id=DQtpJaEs4NIC&lpg=PA182&dq=0.6931471805599&pg=PA182#v=onepage|title=Handbook of Continued Fractions for Special Functions|author1=Annie Cuyt|author2=Vigdis Brevik Petersen|author3=Brigitte Verdonk|author4=Haakon Waadeland|author5=William B. Jones|publisher=Springer|year=2008|isbn=978-1-4020-6948-2|page=182}}</ref>
|<math>\ln 2</math>
|0.69314 71805 59945 30941 <ref group="Mw">{{MathWorld|NaturalLogarithmof2|Natural Logarithm of 2}}</ref><ref group="OEIS">{{OEIS2C|A002162}}</ref>
|<math> \sum_{n=1}^\infty \frac{1}{n 2^n} = 
\sum_{n=1}^\infty \frac{({-}1)^{n+1}}{n} 
= \frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+{\cdots} </math>
|ค.ศ. 1619,<ref>{{cite book|url=https://books.google.com/books?id=mGJRjIC9fZgC|title=A History of Mathematics|author-last=Cajori|author-first=Florian|date=1991|publisher=AMS Bookstore|isbn=0-8218-2102-4|edition=5th|page=152|author-link=Florian Cajori}}</ref>ค.ศ.  1668<ref>{{cite web|url=http://www-history.mcs.st-and.ac.uk/HistTopics/e.html|title=The number e|author-last1=O'Connor|author-first1=J. J.|date=September 2001|publisher=The MacTutor History of Mathematics archive|access-date=2009-02-02|author-first2=E. F.|author-last2=Robertson}}</ref>
|'''''[[Transcendental number|<math>\mathbb{T}</math>]]'''''
|-
|[[Sophomore's dream]]<sub>1</sub> <br> J.[[Johann Bernoulli|Bernoulli]] <ref>{{cite book|url=https://books.google.com/books?id=QnXSqvTiEjYC&pg=PA51&lpg=PA51&dq=0.7834305107#v=onepage|title=The Calculus Gallery: Masterpieces from Newton to Lebesgue|author=William Dunham|publisher=Princeton University Press|year=2005|isbn=978-0-691-09565-3|page=51}}</ref>
|<math>{I}_{1}</math>
|0.78343 05107 12134 40705 <ref group="OEIS">{{OEIS2C|A083648}}</ref>
|<math>\int_0^1 \! x^{x}\,dx = \sum_{n = 1}^\infty \frac{(-1)^{n+1}}{n^n} = \frac{1}{1^1} - \frac{1}{2^2} + \frac{1}{3^3} - {\cdots} </math>
|ค.ศ. 1697
|
|-
|[[Sophomore's dream]]<sub>2</sub> <br> J.[[Johann Bernoulli|Bernoulli]] <ref>{{cite book|url=http://math.eretrandre.org/tetrationforum/attachment.php?aid=788|title=SOPHOMORE'S DREAM FUNCTION|author=Jean Jacquelin|year=2010|isbn=}}</ref>
|<math>{I}_{2}</math>
|1.29128 59970 62663 54040 <ref group="Mw">{{MathWorld|SophomoresDream|Sophomore's Dream}}</ref><ref group="OEIS">{{OEIS2C|A073009}}</ref>
|<math> \int_0^1 \! \frac{1}{x^x}\, dx 
= \sum_{n = 1}^\infty \frac{1}{n^n} =  \frac{1}{1^1} + \frac{1}{2^2} + \frac{1}{3^3}  + \frac{1}{4^4}+ \cdots</math>
|ค.ศ. 1697
|
|-
|[[Lemniscate constant]]<ref>{{cite book|url=https://books.google.com/books?id=aKQhpm1h770C&pg=PA333&dq=2.6220575#v=onepage|title=L-Functions and Arithmetic|author1=J. Coates|author2=Martin J. Taylor|publisher=Cambridge University Press|year=1991|isbn=978-0-521-38619-7|page=333}}</ref>
|<math>{\varpi} </math>
|2.62205 75542 92119 81046 <ref group="Mw">{{MathWorld|LemniscateConstant|Lemniscate Constant}}</ref><ref group="OEIS">{{OEIS2C|A062539}}</ref>
|<math> \pi \, {G} = 4 \sqrt{\tfrac2\pi}\,\Gamma{\left(\tfrac54 \right)^2} = \tfrac14 \sqrt{\tfrac{2}{\pi}}\,\Gamma {\left(\tfrac14 \right)^2} = 4 \sqrt{\tfrac2\pi}\left(\tfrac14 !\right)^2</math>
|ค.ศ. 1718 ถึง 1798
|'''''[[Transcendental number|<math>\mathbb{T}</math>]]'''''
|-
|[[Euler–Mascheroni constant]]<ref>{{Cite web|date=2020-03-20|title=Greek/Hebrew/Latin-based Symbols in Mathematics|url=https://mathvault.ca/hub/higher-math/math-symbols/greek-hebrew-latin-symbols/|access-date=2020-08-08|website=Math Vault|language=en-US}}</ref>
|<math>{\gamma}</math>
|0.57721 56649 01532 86060 <ref group="Mw">{{MathWorld|Euler-MascheroniConstant|Euler–Mascheroni Constant}}</ref><ref group="OEIS">{{OEIS2C|A001620}}</ref>
|<math> \sum_{n=1}^\infty \sum_{k=0}^\infty \frac{(-1)^k}{2^n+k} 
 = \sum_{n=1}^\infty \left(\frac{1}{n} -\ln \left(1+\frac{1}{n}\right)\right) </math> <br> 
<math>= \int_{0}^{1} -\ln \left(\ln \frac{1}{x}\right)\, dx = -\Gamma'(1) = -\Psi(1)</math>
|ค.ศ. 1735
|'''''[[Irrational number|<math>\mathbb{R} \setminus \mathbb{Q}</math>]]?'''''
|-
|[[Erdős–Borwein constant]]<ref>{{cite arXiv|eprint=0806.4410|class=math.CA|author=Robert Baillie|title=Summing The Curious Series of Kempner and Irwin|year=2013}}</ref>
|<math>{E}_{\,B}</math>
|1.60669 51524 15291 76378 <ref group="Mw">{{MathWorld|Erdos-BorweinConstant|Erdos-Borwein Constant}}</ref><ref group="OEIS">{{OEIS2C|A065442}}</ref>
|<math>\sum_{m=1}^{\infty} \sum_{n=1}^{\infty}\frac{1}{2^{mn}} =\sum_{n=1}^{\infty}\frac{1}{2^n-1} = \frac{1}{1} \! + \! \frac{1}{3} \! + \! \frac{1}{7} \! + \! \frac{1}{15} \! + \! ...</math>
|ค.ศ. 1749<ref>{{cite book|url=http://www.math.dartmouth.edu/~euler/pages/E190.html|title=Consideratio quarumdam serierum, quae singularibus proprietatibus sunt praeditae|author=Leonhard Euler|year=1749|page=108}}</ref>
|'''''[[Irrational number|<math>\mathbb{R} \setminus \mathbb{Q}</math>]]'''''
|-
|[[Laplace limit]] <ref>{{cite book|title=Orbital Mechanics for Engineering Students|title-link=Orbital Mechanics for Engineering Students|author=Howard Curtis|publisher=Elsevier|year=2014|isbn=978-0-08-097747-8|page=159}}</ref>
|<math>{\lambda}</math>
|0.66274 34193 49181 58097 <ref group="Mw">{{MathWorld|LaplaceLimit|Laplace Limit}}</ref><ref group="OEIS">{{OEIS2C|A033259}}</ref>
|<math> \frac{ x \; e^\sqrt{x^2+1}}{\sqrt{x^2+1}+1} = 1</math>
|~ค.ศ. 1782
|'''''[[Transcendental number|<math>\mathbb{T}</math>]]'''''?
|-
|[[Gauss's constant]] <ref>{{cite book|url=https://books.google.com/books?id=UrSnNeJW10YC&pg=PA647&dq=%22Gauss%27s+constant%22#v=onepage|title=An Atlas of Functions: With Equator, the Atlas Function Calculator|author1=Keith B. Oldham|author2=Jan C. Myland|author3=Jerome Spanier|publisher=Springer|year=2009|isbn=978-0-387-48806-6|page=15}}</ref>
|<math>{G}</math>
|0.83462 68416 74073 18628 <ref name=":4" group="Mw">{{MathWorld|GausssConstant|Gauss's Constant}}</ref><ref group="OEIS">{{OEIS2C|A014549}}</ref>
|<math>  \frac{1}{\mathrm{agm}(1, \sqrt{2})} = \frac{4 \sqrt{2} \,(\tfrac14 !)^2}{\pi ^{3/2}}= \frac{2}{\pi}\int_0^1\frac{dx}{\sqrt{1 - x^4}}</math>

เมื่อ ''agm'' = [[Arithmetic–geometric mean]]
|ค.ศ. 1799<ref>{{Cite book|title=Undergraduate convexity : problems and solutions|last=Nielsen, Mikkel Slot.|date = July 2016|publisher=|isbn=9789813146211|location=|pages=162|oclc=951172848}}</ref>
|[[Transcendental number|<math>\mathbb{T}</math>]]?
|}

== อ้างอิง ==
{{รายการอ้างอิง}}
=== Site MathWorld Wolfram.com ===
{{reflist|group=Mw|2}}

=== Site OEIS.com ===
{{reflist|group=OEIS|2}}

=== Site OEIS Wiki ===
{{reflist|group=Ow|2}}

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