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. 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
One half	1/2	0.5		Prehistory	$\mathbb {Q}$
Pi	$\pi$	3.14159 26535 89793 23846	Ratio of a circle's circumference to its diameter.	1900 to 1600 BCE	$\mathbb {T}$
Square root of 2, Pythagoras constant.	${\sqrt {2}}$	1.41421 35623 73095 04880	Positive root of $x^{2}=2$	1800 to 1600 BCE	$\mathbb {A}$
Square root of 3, Theodorus' constant	${\sqrt {3}}$	1.73205 08075 68877 29352	Positive root of $x^{2}=3$	465 to 398 BCE	$\mathbb {A}$
Square root of 5	${\sqrt {5}}$	2.23606 79774 99789 69640	Positive root of $x^{2}=5$		$\mathbb {A}$
Phi, Golden ratio	$\varphi$	1.61803 39887 49894 84820	${\frac {1+{\sqrt {5}}}{2}}$	~300 BCE	$\mathbb {A}$
Silver ratio	$\delta _{S}$	2.41421 35623 73095 04880	${\sqrt {2}}+1$	~300 BCE	$\mathbb {A}$
Zero	0	0		300 to 100 BCE	$\mathbb {Z}$
Negative one	−1	−1		300 to 200 BCE	$\mathbb {Z}$
Cube root of 2	${\sqrt[{3}]{2}}$	1.25992 10498 94873 16476	Real root of $x^{3}=2$	46 to 120 CE	$\mathbb {A}$
Cube root of 3	${\sqrt[{3}]{3}}$	1.44224 95703 07408 38232	Real root of $x^{3}=3$		$\mathbb {A}$
Twelfth root of 2	${\sqrt[{12}]{2}}$	1.05946 30943 59295 26456	Real root of $x^{12}=2$		$\mathbb {A}$
Supergolden ratio	$\psi$	1.46557 12318 76768 02665	${\frac {1+{\sqrt[{3}]{\frac {29+3{\sqrt {93}}}{2}}}+{\sqrt[{3}]{\frac {29-3{\sqrt {93}}}{2}}}}{3}}$Real root of $x^{3}=x^{2}+1$		$\mathbb {A}$
Imaginary unit	$i$	0 + 1i	Either of the two roots of $x^{2}=-1$	1501 to 1576	$\mathbb {C}$
Connective constant for the hexagonal lattice	$\mu$	1.84775 90650 22573 51225	${\sqrt {2+{\sqrt {2}}}}$, as a root of the polynomial $x^{4}-4x^{2}+2=0$	1593	$\mathbb {A}$
Kepler–Bouwkamp constant	$K'$	0.11494 20448 53296 20070	$\prod _{n=3}^{\infty }\cos \left({\frac {\pi }{n}}\right)=\cos \left({\frac {\pi }{3}}\right)\cos \left({\frac {\pi }{4}}\right)\cos \left({\frac {\pi }{5}}\right)...$	1596
Wallis's constant		2.09455 14815 42326 59148	${\sqrt[{3}]{\frac {45-{\sqrt {1929}}}{18}}}+{\sqrt[{3}]{\frac {45+{\sqrt {1929}}}{18}}}$Real root of $x^{3}-2x-5=0$	1616 to 1703	$\mathbb {A}$
Euler's number	$e$	2.71828 18284 59045 23536	$\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}=\sum _{n=0}^{\infty }{\frac {1}{n!}}=1+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}\cdots$	1618	$\mathbb {T}$
Natural logarithm of 2	$\ln 2$	0.69314 71805 59945 30941	Real root of $e^{x}=2$$\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+\cdots$	1619 & 1668	$\mathbb {T}$
Lemniscate constant	$\varpi$	2.62205 75542 92119 81046	$\pi \,{G}=4{\sqrt {\tfrac {2}{\pi }}}\,\Gamma {\left({\tfrac {5}{4}}\right)^{2}}={\tfrac {1}{4}}{\sqrt {\tfrac {2}{\pi }}}\,\Gamma {\left({\tfrac {1}{4}}\right)^{2}}$where $G$ is Gauss's constant	1718 to 1798	$\mathbb {T}$
Euler's constant	$\gamma$	0.57721 56649 01532 86060	$\lim _{n\to \infty }\left(-\log n+\sum _{k=1}^{n}{\frac {1}{k}}\right)=\int _{1}^{\infty }\left(-{\frac {1}{x}}+{\frac {1}{\lfloor x\rfloor }}\right)\,dx$	1735
Erdős–Borwein constant	$E$	1.60669 51524 15291 76378	$\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}={\frac {1}{1}}\!+\!{\frac {1}{3}}\!+\!{\frac {1}{7}}\!+\!{\frac {1}{15}}\!+\!\cdots$	1749	$\mathbb {R} \setminus \mathbb {Q}$
Omega constant	$\Omega$	0.56714 32904 09783 87299	$W(1)={\frac {1}{\pi }}\int _{0}^{\pi }\log \left(1+{\frac {\sin t}{t}}e^{t\cot t}\right)dt$where W is the Lambert W function	1758 & 1783	$\mathbb {T}$
Apéry's constant	$\zeta (3)$	1.20205 69031 59594 28539	$\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}={\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{3}}}+{\frac {1}{5^{3}}}+\cdots$	1780	$\mathbb {R} \setminus \mathbb {Q}$
Laplace limit		0.66274 34193 49181 58097	Real root of ${\frac {xe^{\sqrt {x^{2}+1}}}{{\sqrt {x^{2}+1}}+1}}=1$	~1782	$\mathbb {T}$
Ramanujan–Soldner constant	$\mu$	1.45136 92348 83381 05028	$\mathrm {li} (x)=\int _{0}^{x}{\frac {dt}{\ln t}}=0$; root of the logarithmic integral function.	1792
Gauss's constant	$G$	0.83462 68416 74073 18628	${\frac {1}{\mathrm {agm} (1,{\sqrt {2}})}}={\frac {\Gamma ({\frac {1}{4}})^{2}}{2{\sqrt {2\pi ^{3}}}}}={\frac {2}{\pi }}\int _{0}^{1}{\frac {dx}{\sqrt {1-x^{4}}}}$where agm is the arithmetic–geometric mean	1799	$\mathbb {T}$
Second Hermite constant	$\gamma _{2}$	1.15470 05383 79251 52901	${\frac {2}{\sqrt {3}}}$	1822 to 1901	$\mathbb {A}$
Liouville's constant	$L$	0.11000 10000 00000 00000 0001	$\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 64007 98201	${\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	$\mathbb {R} \setminus \mathbb {Q}$
Ramanujan's constant		262 53741 26407 68743 .99999 99999 99250 073	$e^{\pi {\sqrt {163}}}$	1859	$\mathbb {T}$
Glaisher–Kinkelin constant	$A$	1.28242 71291 00622 63687	$e^{{\frac {1}{12}}-\zeta ^{\prime }(-1)}=e^{{\frac {1}{8}}-{\frac {1}{2}}\sum \limits _{n=0}^{\infty }{\frac {1}{n+1}}\sum \limits _{k=0}^{n}\left(-1\right)^{k}{\binom {n}{k}}\left(k+1\right)^{2}\ln(k+1)}$	1860
Catalan's constant	$G$	0.91596 55941 77219 01505	$\int _{0}^{1}\!\!\int _{0}^{1}\!\!{\frac {dx\,dy}{1{+}x^{2}y^{2}}}=\!\sum _{n=0}^{\infty }\!{\frac {(-1)^{n}}{(2n{+}1)^{2}}}\!=\!{\frac {1}{1^{2}}}{-}{\frac {1}{3^{2}}}{+}{\cdots }$	1864
Dottie number		0.73908 51332 15160 64165	Real root of $\cos x=x$	1865	$\mathbb {T}$
Meissel–Mertens constant	$M$	0.26149 72128 47642 78375	$\lim _{n\to \infty }\left(\sum _{p\leq n}{\frac {1}{p}}\ln \ln n\right)=\gamma +\sum _{p}\left(\ln \left(1-{\frac {1}{p}}\right)+{\frac {1}{p}}\right)$where γ is the Euler–Mascheroni constant and p is prime	1866 & 1873
Universal parabolic constant	$P$	2.29558 71493 92638 07403	$\ln(1+{\sqrt {2}})+{\sqrt {2}}\;=\;\operatorname {arsinh} (1)+{\sqrt {2}}$	Before 1891	$\mathbb {T}$
Cahen's constant	$C$	0.64341 05462 88338 02618	$\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{s_{k}-1}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{6}}-{\frac {1}{42}}+{\frac {1}{1806}}{\,\pm \cdots }$where sk is the kth term of Sylvester's sequence 2, 3, 7, 43, 1807, ...	1891	$\mathbb {T}$
Gelfond's constant	$e^{\pi }$	23.14069 26327 79269 0057	$(-1)^{-i}=i^{-2i}=\sum _{n=0}^{\infty }{\frac {\pi ^{n}}{n!}}=1+{\frac {\pi ^{1}}{1}}+{\frac {\pi ^{2}}{2}}+{\frac {\pi ^{3}}{6}}+\cdots$	1900	$\mathbb {T}$
Gelfond–Schneider constant	$2^{\sqrt {2}}$	2.66514 41426 90225 18865		Before 1902	$\mathbb {T}$
Second Favard constant	$K_{2}$	1.23370 05501 36169 82735	${\frac {\pi ^{2}}{8}}=\sum _{n=0}^{\infty }{\frac {1}{(2n-1)^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\cdots$	1902 to 1965	$\mathbb {T}$
First Nielsen–Ramanujan constant	$a_{1}$	0.82246 70334 24113 21823	${\frac {{\zeta }(2)}{2}}={\frac {\pi ^{2}}{12}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{2}}}={\frac {1}{1^{2}}}{-}{\frac {1}{2^{2}}}{+}{\frac {1}{3^{2}}}{-}{\frac {1}{4^{2}}}{+}\cdots$	1909	$\mathbb {T}$
Brun's constant	$B_{2}$	1.90216 05831 04	$\textstyle {\sum \limits _{p}({\frac {1}{p}}+{\frac {1}{p+2}})}=({\frac {1}{3}}\!+\!{\frac {1}{5}})+({\tfrac {1}{5}}\!+\!{\tfrac {1}{7}})+({\tfrac {1}{11}}\!+\!{\tfrac {1}{13}})+\cdots$where the sum ranges over all primes p such that p + 2 is also a prime	1919
Twin primes constant	$C_{2}$	0.66016 18158 46869 57392	$\prod _{\textstyle {p\;{\rm {prime}} \atop p\geq 3}}\left(1-{\frac {1}{(p-1)^{2}}}\right)$	1922
Plastic number	$\rho$	1.32471 79572 44746 02596	${\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\cdots }}}}}}=\textstyle {\sqrt[{3}]{{\frac {1}{2}}+{\frac {\sqrt {69}}{18}}}}+{\sqrt[{3}]{{\frac {1}{2}}-{\frac {\sqrt {69}}{18}}}}$Real root of $x^{3}=x+1$	1924	$\mathbb {A}$
Prouhet–Thue–Morse constant	$\tau$	0.41245 40336 40107 59778	$\sum _{n=0}^{\infty }{\frac {t_{n}}{2^{n+1}}}={\frac {1}{4}}\left[2-\prod _{n=0}^{\infty }\left(1-{\frac {1}{2^{2^{n}}}}\right)\right]$where ${t_{n}}$ is the nth term of the Thue–Morse sequence	1929	$\mathbb {T}$
Base 10 Champernowne constant	$C_{10}$	0.12345 67891 01112 13141	Defined by concatenating representations of successive integers: 0.1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...	1933	$\mathbb {T}$
Salem constant	$\sigma _{10}$	1.17628 08182 59917 50654	Largest real root of $x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1=0$	1933	$\mathbb {A}$
Khinchin's constant	$K_{0}$	2.68545 20010 65306 44530	$\prod _{n=1}^{\infty }\left[{1+{1 \over n(n+2)}}\right]^{\log _{2}(n)}$	1934
Chvátal–Sankoff constant for the binary alphabet	$\gamma _{2}$	$0.788071\leq \gamma _{2}\leq 0.826280$	$\lim _{n\to \infty }{\frac {\operatorname {E} [\lambda _{n,2}]}{n}}$where E[λn,2] is the expected longest common subsequence of two random length-n binary strings	1975
Chaitin's constants	$\Omega$	In general they are uncomputable numbers. But one such number is 0.00787 49969 97812 3844.	$\sum _{p\in P}2^{-|p|}$ p: Halted program |p|: Size in bits of program p P: Domain of all programs that stop. See also: Halting problem	1975	$\mathbb {T}$
Robbins constant	$\Delta (3)$	0.66170 71822 67176 23515	${\frac {4\!+\!17{\sqrt {2}}\!-6{\sqrt {3}}\!-7\pi }{105}}\!+\!{\frac {\ln(1\!+\!{\sqrt {2}})}{5}}\!+\!{\frac {2\ln(2\!+\!{\sqrt {3}})}{5}}$	1977	$\mathbb {T}$
Weierstrass constant		0.47494 93799 87920 65033	${\frac {2^{5/4}{\sqrt {\pi }}\,e^{\pi /8}}{\Gamma ({\frac {1}{4}})^{2}}}$	Before 1978	$\mathbb {T}$
Fransén–Robinson constant	$F$	2.80777 02420 28519 36522	$\int _{0}^{\infty }{\frac {dx}{\Gamma (x)}}=e+\int _{0}^{\infty }{\frac {e^{-x}}{\pi ^{2}+\ln ^{2}x}}\,dx$	1978
Fractal dimension of the Cantor set	$\log _{3}2$	0.63092 97535 71457 43709	${\frac {\log 2}{\log 3}}$	Before 1979	$\mathbb {T}$
Feigenbaum constant α	$\alpha$	2.50290 78750 95892 82228	Ratio between the width of a tine and the width of one of its two subtines in a bifurcation diagram	1979
Second du Bois-Reymond constant	$C_{2}$	0.19452 80494 65325 11361	${\frac {e^{2}-7}{2}}=\int _{0}^{\infty }\left|{{\frac {d}{dt}}\left({\frac {\sin t}{t}}\right)^{n}}\right|\,dt-1$	1983	$\mathbb {T}$
Erdős–Tenenbaum–Ford constant	$\delta$	0.86071 33205 59342 06887	$1-{\frac {1+\log \log 2}{\log 2}}$	1984

List

Name	Symbol	Formula	Year	Set
Metallic mean		${\frac {n+{\sqrt {n^{2}+4}}}{2}}$		$\mathbb {A}$
Hermite constant	$\gamma _{n}$	For a lattice L in Euclidean space Rn with unit covolume, i.e. vol(Rn/L) = 1, let λ1(L) denote the least length of a nonzero element of L. Then √γnn is the maximum of λ1(L) over all such lattices L.	1822 to 1901	$\mathbb {A}$
Hafner–Sarnak–McCurley constant	$D(n)$	$D(n)=\prod _{k=1}^{\infty }\left\{1-\left[1-\prod _{j=1}^{n}(1-p_{k}^{-j})\right]^{2}\right\}$	1883
Favard constants	$K_{r}$	${\frac {4}{\pi }}\sum _{n=0}^{\infty }\left({\frac {(-1)^{n}}{2n+1}}\right)^{\!r+1}={\frac {4}{\pi }}\left({\frac {(-1)^{0(r+1)}}{1^{r}}}+{\frac {(-1)^{1(r+1)}}{3^{r}}}+{\frac {(-1)^{2(r+1)}}{5^{r}}}+{\frac {(-1)^{3(r+1)}}{7^{r}}}+\cdots \right)$	1902 to 1965	$\mathbb {T}$
Nielsen–Ramanujan constant	$a_{1}$	$a_{k}=\int _{1}^{2}{\frac {(\ln x)^{k}}{x-1}}dx$	1909
Brun's constant	$B_{n}$	${\sum \limits _{p}({\frac {1}{p}}+{\frac {1}{p+n}})}$where the sum ranges over all primes p such that p + n is also a prime	1919
Champernowne constants	$C_{b}$	Defined by concatenating representations of successive integers in base b. $C_{b}=\sum _{n=1}^{\infty }{\frac {n}{b^{\left(\sum _{k=1}^{n}\lceil \log _{b}(k+1)\rceil \right)}}}$	1933	$\mathbb {T}$
Beraha constants	$B(n)$	$2+2\cos \left({\frac {2\pi }{n}}\right)$		$\mathbb {T}$
Chvátal–Sankoff constants	$\gamma _{k}$	$\lim _{n\to \infty }{\frac {E[\lambda _{n,k}]}{n}}$
Feller's coin-tossing constants	$\alpha _{k},\beta _{k}$	$\alpha _{k}$ is the smallest positive real root of $x^{k+1}=2^{k+1}(x-1),\beta _{k}={\frac {2-\alpha _{k}}{k+1-k\alpha _{k}}}$
Bernoulli number	$B_{n}^{\pm }$	${\frac {t}{2}}\left(\operatorname {coth} {\frac {t}{2}}\pm 1\right)=\sum _{m=0}^{\infty }{\frac {B_{m}^{\pm {}}t^{m}}{m!}}$		$\mathbb {Q}$
Gregory coefficients	$G_{n}$	${\frac {1}{n!}}\int _{0}^{1}x(x-1)(x-2)\cdots (x-n+1)\,dx=\int _{0}^{1}{\binom {x}{n}}\,dx$
Harmonic number	$H_{n}$	$\sum _{k=1}^{n}{\frac {1}{k}}$		$\mathbb {Q}$
Hyperharmonic number	$H_{n}^{(r))}$	$\sum _{k=1}^{n}H_{k}^{(r-1)}$ and $H_{n}^{(0)}={\frac {1}{n}}$		$\mathbb {Q}$
Ramanujan's sum	$c_{q}$	$\sum _{1\leq a\leq q \atop gcd(a,q)=1}e^{2\pi i{\tfrac {a}{q}}n}$
Stoneham number	$\alpha _{b,c}$	$\sum _{n=c^{k}>1}{\frac {1}{b^{n}n}}=\sum _{k=1}^{\infty }{\frac {1}{b^{c^{k}}c^{k}}}$ if b,c are coprime integers greater than one	1973
Stieltjes constants	$\gamma _{n}$	${\frac {(-1)^{n}n!}{2\pi }}\int _{0}^{2\pi }e^{-nix}\zeta \left(e^{ix}+1\right)dx.$
Markov constant	$M(\alpha )$	$\limsup _{k\to \infty }{\frac {1}{k^{2}\left\vert \alpha -{\frac {\lfloor \alpha k\rfloor }{k}}\right\vert }}$
Lagrange number	$L(n)$	${\sqrt {9-{\frac {4}{{m_{n}}^{2}}}}}$ where $m_{n}$ is the nth smallest number such that $m^{2}+x^{2}+y^{2}=3mxy\,$ has positive (x,y)
Gregory number	$G_{x}$	$\sum _{n=0}^{\infty }(-1)^{n}{\frac {1}{(2n+1)x^{2n+1}}}$ for rational x greater than one

See also

• Invariant (mathematics)
• List of mathematical symbols
• List of mathematical symbols by subject
• List of numbers
• List of physical constants
• Particular values of the Riemann zeta function
• Physical constant

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.

Further reading

• Wolfram, Stephen. "4: Systems Based on Numbers". Section 5: Mathematical Constants — Continued fractions. {{cite book}}: |work= ignored (help)

External links

• Inverse Symbolic Calculator, Plouffe's Inverter
• Constants - from Wolfram MathWorld
• On-Line Encyclopedia of Integer Sequences (OEIS)
• Steven Finch's page of mathematical constants
• Xavier Gourdon and Pascal Sebah's page of numbers, mathematical constants and algorithms

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