User:Atavoidirc/constants

List of mathematical constants

A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems.^[1] For example, the constant π may be defined as the ratio of the length of a circle's circumference to its diameter. The following list includes a decimal expansion and set containing each number, ordered by year of discovery.

The column headings may be clicked to sort the table alphabetically, by decimal value, or by set. Explanations of the symbols in the right hand column can be found by clicking on them.

List

Name	Symbol	Decimal expansion	Formula	Year	Set
Second Hermite constant^[2]	$\gamma _{2}$	1.15470 05383 ^{[Mw 1]}	${\frac {2}{\sqrt {3}}}$	1822 to 1901	$\mathbb {A}$
Liouville's constant^[3]	$L$	0.11000 10000 ^{[Mw 2]}^{[OEIS 1]}	$\sum _{n=1}^{\infty }{\frac {1}{10^{n!}}}={\frac {1}{10^{1!}}}+{\frac {1}{10^{2!}}}+{\frac {1}{10^{3!}}}+{\frac {1}{10^{4!}}}+\cdots$	Before 1844	$\mathbb {T}$
First continued fraction constant	$C_{1}$	0.69777 46579 ^{[Mw 3]}^{[OEIS 2]}	${\tfrac {1}{1+{\tfrac {1}{2+{\tfrac {1}{3+{\tfrac {1}{4+{\tfrac {1}{5+\cdots }}}}}}}}}}$ ${\frac {I_{1}(2)}{I_{0}(2)}}$ , where $I_{\alpha }(x)$ is the modified Bessel function	1855^[4]	$\mathbb {R} \setminus \mathbb {Q}$
Hermite–Ramanujan constant^[5]		262537412... ^{[Mw 4]}^{[OEIS 3]}	$e^{\pi {\sqrt {163}}}$	1859	$\mathbb {T}$
Weierstrass constant ^[6]		0.47494 93799 ^{[Mw 5]}^{[OEIS 4]}	${\frac {2^{5/4}{\sqrt {\pi }}\,e^{\pi /8}}{\Gamma ({\frac {1}{4}})^{2}}}$	1872 ?	$\mathbb {T}$
First Hafner–Sarnak–McCurley constant ^[7]	$D(1)$	0.60792 71018 ^{[Mw 6]}^{[OEIS 5]}	${\frac {6}{\pi ^{2}}}=\prod _{p{\text{ prime}}}{\left(1-{\frac {1}{p^{2}}}\right)}\!=\textstyle \left(1-{\frac {1}{2^{2}}}\right)\left(1-{\frac {1}{3^{2}}}\right)\cdots$	1883^{[Mw 6]}	$\mathbb {T}$
Second Favard constant^[8]	$K_{2}$	1.23370 05501 ^{[Mw 7]}^{[OEIS 6]}	${\frac {\pi ^{2}}{8}}=\sum _{n=0}^{\infty }{\frac {1}{(2n-1)^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\cdots$	1902 to 1965	$\mathbb {T}$
First Nielsen–Ramanujan constant^[9]	$a_{1}$	0.82246 70334 ^{[Mw 8]}^{[OEIS 7]}	${\frac {{\zeta }(2)}{2}}={\frac {\pi ^{2}}{12}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{2}}}={\frac {1}{1^{2}}}{-}{\frac {1}{2^{2}}}{+}{\frac {1}{3^{2}}}{-}{\frac {1}{4^{2}}}{+}\cdots$	1909	$\mathbb {T}$
Tribonacci constant^[10]		1.83928 67552 ^{[Mw 9]}^{[OEIS 8]}	${\textstyle {\frac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}={\frac {1+4\cosh \left({\frac {1}{3}}\cosh ^{-1}\left(2+{\frac {3}{8}}\right)\right)}{3}}}$ Real root of $x^{3}-x^{2}-x-1=0$	1914 to 1963	$\mathbb {A}$
Twin primes constant	$C_{2}$	0.66016 18158 ^{[Mw 10]}^{[OEIS 9]}	$\prod _{\textstyle {p\;{\rm {prime}} \atop p\geq 3}}\left(1-{\frac {1}{(p-1)^{2}}}\right)$	1922
Z score for the 97.5 percentile point^[11]^[12]^[13]^[14]	$z_{.975}$	1.95996 39845 ^{[Mw 11]}^{[OEIS 10]}	${\sqrt {2}}\operatorname {erf} ^{-1}(0.95)$ where $erf -1 (x)$ is the inverse error function Real number $z$ such that ${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{z}e^{-x^{2}/2}\,\mathrm {d} x=0.975$	1925
Champernowne constants^[15]	$C_{b}$	0.12345 67891 01112 13141 ^{[Mw 12]}^{[OEIS 11]}	Defined by concatenating representations of successive integers in base b	1933	$\mathbb {T}$
Mills' constant^[16]	$A$	1.30637 78838 63080 69046 ^{[Mw 13]}^{[OEIS 12]}	Smallest positive real number A such that $\lfloor A^{3^{n}}\rfloor$ is prime for all positive integers n	1947
Lieb's square ice constant^[17]		1.53960 07178 39002 03869 ^{[Mw 14]}^{[OEIS 13]}	$\left({\frac {4}{3}}\right)^{\frac {3}{2}}={\frac {8}{3{\sqrt {3}}}}$	1967	$\mathbb {A}$
Connective constant for the hexagonal lattice^[18]^[19]	$\mu$	1.84775 90650 22573 51225 ^{[Mw 15]}^{[OEIS 14]}	${\sqrt {2+{\sqrt {2}}}}$ , as a root of the polynomial $x^{4}-4x^{2}+2=0$	1982^[20]	$\mathbb {A}$
Brun's constant^[21]	$B_{2}$	1.90216 05831 04 ^{[Mw 16]}^{[OEIS 15]}	$\textstyle {\sum \limits _{p}({\frac {1}{p}}+{\frac {1}{p+2}})}=({\frac {1}{3}}\!+\!{\frac {1}{5}})+({\tfrac {1}{5}}\!+\!{\tfrac {1}{7}})+({\tfrac {1}{11}}\!+\!{\tfrac {1}{13}})+\cdots$ where the sum ranges over all primes p such that p + 2 is also a prime	1989^{[OEIS 15]}
Prime constant^[22]	$\rho$	0.41468 25098 51111 66024 ^{[OEIS 16]}	$\sum _{p{\text{ prime}}}{\frac {1}{2^{p}}}={\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{32}}+\cdots$	Before 2008	$\mathbb {R} \setminus \mathbb {Q}$
Copeland–Erdős constant^[23]	${\mathcal {C}}_{CE}$	0.23571 11317 19232 93137 ^{[Mw 17]}^{[OEIS 17]}	Defined by concatenating representations of successive prime numbers: 0.2 3 5 7 11 13 17 19 23 29 31 37 ...	Before 2012^[23]	$\mathbb {R} \setminus \mathbb {Q}$

Notes

References

^ Weisstein, Eric W. "Constant". mathworld.wolfram.com. Retrieved 2020-08-08.
^ Steven Finch (2014). Errata and Addenda to Mathematical Constants (PDF). Harvard.edu. Archived from the original (PDF) on 2016-03-16. Retrieved 2013-12-17.
^ Calvin C. Clawson (2003). Mathematical Traveler: Exploring the Grand History of Numbers. Perseus. p. 187. ISBN 978-0-7382-0835-0.
^ Amoretti, F. (1855). "Sur la fraction continue [0,1,2,3,4,...]". Nouvelles annales de mathématiques. 1 (14): 40–44.
^ L. J. Lloyd James Peter Kilford (2008). Modular Forms: A Classical and Computational Introduction. Imperial College Press. p. 107. ISBN 978-1-84816-213-6.
^ Eric W. Weisstein (2003). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. p. 151. ISBN 978-1-58488-347-0.
^ Holger Hermanns; Roberto Segala (2000). Process Algebra and Probabilistic Methods. Springer-Verlag. p. 270. ISBN 978-3-540-67695-9.
^ Helmut Brass; Knut Petras (2010). Quadrature Theory: The Theory of Numerical Integration on a Compact Interval. AMS. p. 274. ISBN 978-0-8218-5361-0.
^ Mauro Fiorentini. Nielsen – Ramanujan (costanti di).
^ Agronomof, M. (1914). "Sur une suite récurrente". Mathesis. 4: 125–126.
^ Rees, DG (1987), Foundations of Statistics, CRC Press, p. 246, ISBN 0-412-28560-6, Why 95% confidence? Why not some other confidence level? The use of 95% is partly convention, but levels such as 90%, 98% and sometimes 99.9% are also used.
^ "Engineering Statistics Handbook: Confidence Limits for the Mean". National Institute of Standards and Technology. Archived from the original on 5 February 2008. Retrieved 4 February 2008. Although the choice of confidence coefficient is somewhat arbitrary, in practice 90%, 95%, and 99% intervals are often used, with 95% being the most commonly used.
^ Olson, Eric T; Olson, Tammy Perry (2000), Real-Life Math: Statistics, Walch Publishing, p. 66, ISBN 0-8251-3863-9, While other stricter, or looser, limits may be chosen, the 95 percent interval is very often preferred by statisticians.
^ Swift, MB. "Comparison of Confidence Intervals for a Poisson Mean - Further Considerations". Communications in Statistics - Theory and Methods. Vol. 38, no. 5. pp. 748–759. doi:10.1080/03610920802255856. In modern applied practice, almost all confidence intervals are stated at the 95% level.
^ Michael J. Dinneen; Bakhadyr Khoussainov; Prof. Andre Nies (2012). Computation, Physics and Beyond. Springer. p. 110. ISBN 978-3-642-27653-8.
^ Laith Saadi (2004). Stealth Ciphers. Trafford Publishing. p. 160. ISBN 978-1-4120-2409-9.
^ Robin Whitty. Lieb's Square Ice Theorem (PDF).
^ Mireille Bousquet-Mélou. Two-dimensional self-avoiding walks (PDF). CNRS, LaBRI, Bordeaux, France.
^ Hugo Duminil-Copin & Stanislav Smirnov (2011). The connective constant of the honeycomb lattice √ (2 + √ 2) (PDF). Université de Geneve.
^ B. Nienhuis (1982). "Exact critical point and critical exponents of O(n) models in two dimensions". Phys. Rev. Lett. 49 (15): 1062–1065. Bibcode:1982PhRvL..49.1062N. doi:10.1103/PhysRevLett.49.1062.
^ Thomas Koshy (2007). Elementary Number Theory with Applications. Elsevier. p. 119. ISBN 978-0-12-372-487-8.
^ Hardy, G. H. (2008). An introduction to the theory of numbers. E. M. Wright, D. R. Heath-Brown, Joseph H. Silverman (6th ed.). Oxford: Oxford University Press. ISBN 978-0-19-921985-8. OCLC 214305907.
^ ^a ^b Yann Bugeaud (2012). Distribution Modulo One and Diophantine Approximation. Cambridge University Press. p. 87. ISBN 978-0-521-11169-0.

Site MathWorld Wolfram.com

^ Weisstein, Eric W. "Hermite Constants". MathWorld.
^ Weisstein, Eric W. "Liouville's Constant". MathWorld.
^ Weisstein, Eric W. "Continued Fraction Constants". MathWorld.
^ Weisstein, Eric W. "Ramanujan Constant". MathWorld.
^ Weisstein, Eric W. "Weierstrass Constant". MathWorld.
^ ^a ^b Weisstein, Eric W. "Relatively Prime". MathWorld.
^ Weisstein, Eric W. "Favard Constants". MathWorld.
^ Weisstein, Eric W. "Nielsen-Ramanujan Constants". MathWorld.
^ Weisstein, Eric W. "Tribonacci Constant". MathWorld.
^ Weisstein, Eric W. "Twin Primes Constant". MathWorld.
^ Weisstein, Eric W. "Confidence Interval". MathWorld.
^ Weisstein, Eric W. "Champernowne Constant". MathWorld.
^ Weisstein, Eric W. "Mills Constant". MathWorld.
^ Weisstein, Eric W. "Liebs Square Ice Constant". MathWorld.
^ Weisstein, Eric W. "Self-Avoiding Walk Connective Constant". MathWorld.
^ Weisstein, Eric W. "Brun's Constant". MathWorld.
^ Weisstein, Eric W. "Copeland-Erdos Constant". MathWorld.

Site OEIS.com

Site OEIS Wiki

Bibliography

Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 2013-06-05. English translation by Catriona and David Lischka.
Jensen, Johan Ludwig William Valdemar (1895), "Note numéro 245. Deuxième réponse. Remarques relatives aux réponses du MM. Franel et Kluyver", L'Intermédiaire des Mathématiciens, II: 346–347
Cuyt, Annie A.M.; Petersen, Vigdis; Verdonk, Brigitte; Waadeland, Haakon; Jones, William B. (2008). "Mathematical constants". Handbook of Continued Fractions for Special Functions. Dordrecht, Netherlands: Springer Science + Business Media. ISBN 9781402069499.
Borwein, Jonathan; van der Poorten, Alf; Shallit, Jeffrey; Zudilin, Wadim (2014). Neverending Fractions: An Introduction to Continued Fractions. Australian Mathematical Society Lecture Series. Vol. 23. Cambridge, United Kingdom: Cambridge University Press. ISBN 9780521186490. ISSN 0950-2815.

External links

[[Category:Mathematical constants|*] [[Category:Mathematics-related lists|mathematical constants] [[Category:Mathematical tables|Constants] [[Category:Articles containing video clips] [[Category:Number-related lists|constants] [[Category:Continued fractions]

[1] Weisstein, Eric W. "Constant". mathworld.wolfram.com. Retrieved 2020-08-08.

[2] Steven Finch (2014). Errata and Addenda to Mathematical Constants (PDF). Harvard.edu. Archived from the original (PDF) on 2016-03-16. Retrieved 2013-12-17.

[4] Calvin C. Clawson (2003). Mathematical Traveler: Exploring the Grand History of Numbers. Perseus. p. 187. ISBN 978-0-7382-0835-0.

[9] Amoretti, F. (1855). "Sur la fraction continue [0,1,2,3,4,...]". Nouvelles annales de mathématiques. 1 (14): 40–44.

[10] L. J. Lloyd James Peter Kilford (2008). Modular Forms: A Classical and Computational Introduction. Imperial College Press. p. 107. ISBN 978-1-84816-213-6.

[13] Eric W. Weisstein (2003). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. p. 151. ISBN 978-1-58488-347-0.

[16] Holger Hermanns; Roberto Segala (2000). Process Algebra and Probabilistic Methods. Springer-Verlag. p. 270. ISBN 978-3-540-67695-9.

[19] Helmut Brass; Knut Petras (2010). Quadrature Theory: The Theory of Numerical Integration on a Compact Interval. AMS. p. 274. ISBN 978-0-8218-5361-0.

[22] Mauro Fiorentini. Nielsen – Ramanujan (costanti di).

[agr-25] Agronomof, M. (1914). "Sur une suite récurrente". Mathesis. 4: 125–126.

[30] Rees, DG (1987), Foundations of Statistics, CRC Press, p. 246, ISBN 0-412-28560-6, Why 95% confidence? Why not some other confidence level? The use of 95% is partly convention, but levels such as 90%, 98% and sometimes 99.9% are also used.

[31] "Engineering Statistics Handbook: Confidence Limits for the Mean". National Institute of Standards and Technology. Archived from the original on 5 February 2008. Retrieved 4 February 2008. Although the choice of confidence coefficient is somewhat arbitrary, in practice 90%, 95%, and 99% intervals are often used, with 95% being the most commonly used.

[32] Olson, Eric T; Olson, Tammy Perry (2000), Real-Life Math: Statistics, Walch Publishing, p. 66, ISBN 0-8251-3863-9, While other stricter, or looser, limits may be chosen, the 95 percent interval is very often preferred by statisticians.

[33] Swift, MB. "Comparison of Confidence Intervals for a Poisson Mean - Further Considerations". Communications in Statistics - Theory and Methods. Vol. 38, no. 5. pp. 748–759. doi:10.1080/03610920802255856. In modern applied practice, almost all confidence intervals are stated at the 95% level.

[36] Michael J. Dinneen; Bakhadyr Khoussainov; Prof. Andre Nies (2012). Computation, Physics and Beyond. Springer. p. 110. ISBN 978-3-642-27653-8.

[39] Laith Saadi (2004). Stealth Ciphers. Trafford Publishing. p. 160. ISBN 978-1-4120-2409-9.

[42] Robin Whitty. Lieb's Square Ice Theorem (PDF).

[45] Mireille Bousquet-Mélou. Two-dimensional self-avoiding walks (PDF). CNRS, LaBRI, Bordeaux, France.

[:13-46] Hugo Duminil-Copin & Stanislav Smirnov (2011). The connective constant of the honeycomb lattice √ (2 + √ 2) (PDF). Université de Geneve.

[Nienhuis1982-49] B. Nienhuis (1982). "Exact critical point and critical exponents of O(n) models in two dimensions". Phys. Rev. Lett. 49 (15): 1062–1065. Bibcode:1982PhRvL..49.1062N. doi:10.1103/PhysRevLett.49.1062.

[:53-50] Thomas Koshy (2007). Elementary Number Theory with Applications. Elsevier. p. 119. ISBN 978-0-12-372-487-8.

[53] Hardy, G. H. (2008). An introduction to the theory of numbers. E. M. Wright, D. R. Heath-Brown, Joseph H. Silverman (6th ed.). Oxford: Oxford University Press. ISBN 978-0-19-921985-8. OCLC 214305907.

[:40-55] Yann Bugeaud (2012). Distribution Modulo One and Diophantine Approximation. Cambridge University Press. p. 87. ISBN 978-0-521-11169-0.

[3] Weisstein, Eric W. "Hermite Constants". MathWorld.

[5] Weisstein, Eric W. "Liouville's Constant". MathWorld.

[7] Weisstein, Eric W. "Continued Fraction Constants". MathWorld.

[11] Weisstein, Eric W. "Ramanujan Constant". MathWorld.

[14] Weisstein, Eric W. "Weierstrass Constant". MathWorld.

[:3-17] Weisstein, Eric W. "Relatively Prime". MathWorld.

[20] Weisstein, Eric W. "Favard Constants". MathWorld.

[23] Weisstein, Eric W. "Nielsen-Ramanujan Constants". MathWorld.

[26] Weisstein, Eric W. "Tribonacci Constant". MathWorld.

[28] Weisstein, Eric W. "Twin Primes Constant". MathWorld.

[34] Weisstein, Eric W. "Confidence Interval". MathWorld.

[37] Weisstein, Eric W. "Champernowne Constant". MathWorld.

[40] Weisstein, Eric W. "Mills Constant". MathWorld.

[43] Weisstein, Eric W. "Liebs Square Ice Constant". MathWorld.

[47] Weisstein, Eric W. "Self-Avoiding Walk Connective Constant". MathWorld.

[Brun's_Constant-51] Weisstein, Eric W. "Brun's Constant". MathWorld.

[56] Weisstein, Eric W. "Copeland-Erdos Constant". MathWorld.

[6] OEIS: A012245

[8] OEIS: A052119

[12] OEIS: A060295

[15] OEIS: A094692

[18] OEIS: A059956

[21] OEIS: A111003

[24] OEIS: A072691

[27] OEIS: A058265

[29] OEIS: A005597

[35] OEIS: A220510

[38] OEIS: A033307

[41] OEIS: A051021

[44] OEIS: A118273

[48] OEIS: A179260

[:0-52] OEIS: A065421

[54] OEIS: A051006

[57] OEIS: A033308

[1]

[2]

[Mw 1]

[3]

[Mw 2]

[OEIS 1]

[Mw 3]

[OEIS 2]

[4]

[5]

[Mw 4]

[OEIS 3]

[6]

[Mw 5]

[OEIS 4]

[7]

[Mw 6]

[OEIS 5]

[8]

[Mw 7]

[OEIS 6]

[9]

[Mw 8]

[OEIS 7]

[10]

[Mw 9]

[OEIS 8]

[Mw 10]

[OEIS 9]

[11]

[12]

[13]

[14]

[Mw 11]

[OEIS 10]

[15]

[Mw 12]

[OEIS 11]

[16]

[Mw 13]

[OEIS 12]

[17]

[Mw 14]

[OEIS 13]

[18]

[19]

[Mw 15]

[OEIS 14]

[20]

[21]

[Mw 16]

[OEIS 15]

[22]

[OEIS 16]

[23]

[Mw 17]

[OEIS 17]