User:Atavoidirc/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	1	1		Prehistory	$\mathbb {N}$
Two	2	2		Prehistory	$\mathbb {N}$
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	$\mathbb {T}$
Golden angle	$g$	2.39996 32297 28653 32223	${\frac {2\pi }{\varphi ^{2}}}=\pi (3-{\sqrt {5}})$ or $180(3-{\sqrt {5}})=137.50776\ldots$ in degrees	1907	$\mathbb {T}$
Sierpiński's constant	$K$	2.58498 17595 79253 21706	${\begin{aligned}&\pi \left(2\gamma +\ln {\frac {4\pi ^{3}}{\Gamma ({\tfrac {1}{4}})^{4}}}\right)=\pi (2\gamma +4\ln \Gamma ({\tfrac {3}{4}})-\ln \pi )\\&=\pi \left(2\ln 2+3\ln \pi +2\gamma -4\ln \Gamma ({\tfrac {1}{4}})\right)\end{aligned}}$	1907
Landau–Ramanujan constant	$K$	0.76422 36535 89220 66299	${\frac {1}{\sqrt {2}}}\prod _{{p\equiv 3{\text{ mod }}4} \atop p\;{\rm {prime}}}{\left(1-{\frac {1}{p^{2}}}\right)^{-{\frac {1}{2}}}}\!\!={\frac {\pi }{4}}\prod _{{p\equiv 1{\text{ mod }}4} \atop p\;{\rm {prime}}}{\left(1-{\frac {1}{p^{2}}}\right)^{\frac {1}{2}}}$	1908
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	$\mathbb {T}$
Gieseking constant	$G$	1.01494 16064 09653 62502	${\frac {3{\sqrt {3}}}{4}}\left(1-\sum _{n=0}^{\infty }{\frac {1}{(3n+2)^{2}}}+\sum _{n=1}^{\infty }{\frac {1}{(3n+1)^{2}}}\right)=$ $\textstyle {\frac {3{\sqrt {3}}}{4}}\left(1-{\frac {1}{2^{2}}}+{\frac {1}{4^{2}}}-{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}-{\frac {1}{8^{2}}}+{\frac {1}{10^{2}}}\pm \cdots \right)$.	1912
Bernstein's constant	$\beta$	0.28016 94990 23869 13303	$\lim _{n\to \infty }2nE_{2n}(f)$, where En(f) is the error of the best uniform approximation to a real function f(x) on the interval [−1, 1] by real polynomials of no more than degree n, and f(x) = |x|	1913
Tribonacci constant		1.83928 67552 14161 13255	${\textstyle {\frac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}={\frac {1+4\cosh \left({\frac {1}{3}}\cosh ^{-1}\left(2+{\frac {3}{8}}\right)\right)}{3}}}$ Real root of $x^{3}-x^{2}-x-1=0$	1914 to 1963	$\mathbb {A}$
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	$\mathbb {A}$
Bloch's constant	$B$	$0.4332\leq B\leq 0.4719$	The best known bounds are ${\frac {\sqrt {3}}{4}}+2\times 10^{-4}\leq B\leq {\sqrt {\frac {{\sqrt {3}}-1}{2}}}\cdot {\frac {\Gamma ({\frac {1}{3}})\Gamma ({\frac {11}{12}})}{\Gamma ({\frac {1}{4}})}}$	1925
Z score for the 97.5 percentile point	$z_{.975}$	1.95996 39845 40054 23552	${\sqrt {2}}\operatorname {erf} ^{-1}(0.95)$ 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
Landau's constant	$L$	$0.5<L\leq 0.54326$	The best known bounds are $0.5<L\leq {\frac {\Gamma ({\frac {1}{3}})\Gamma ({\frac {5}{6}})}{\Gamma ({\frac {1}{6}})}}$	1929
Landau's third constant	$A$	$0.5<A\leq 0.7853$		1929
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	$\mathbb {T}$
Golomb–Dickman constant	$\lambda$	0.62432 99885 43550 87099	$\int _{0}^{1}e^{\mathrm {Li} (t)}dt=\int _{0}^{\infty }{\frac {\rho (t)}{t+2}}dt$ where Li(t) is the logarithmic integral, and ρ(t) is the Dickman function	1930 & 1964
Constant related to the asymptotic behavior of Lebesgue constants	$c$	0.98943 12738 31146 95174	$\lim _{n\to \infty }\!\!\left(\!{L_{n}{-}{\frac {4}{\pi ^{2}}}\ln(2n{+}1)}\!\!\right)\!{=}{\frac {4}{\pi ^{2}}}\!\left({\sum _{k=1}^{\infty }\!{\frac {2\ln k}{4k^{2}{-}1}}}{-}{\frac {\Gamma '({\tfrac {1}{2}})}{\Gamma ({\tfrac {1}{2}})}}\!\!\right)$	1930
Feller–Tornier constant	${\mathcal {C}}_{\mathrm {FT} }$	0.66131 70494 69622 33528	${{\frac {1}{2}}\prod _{p{\text{ prime}}}\left(1-{\frac {2}{p^{2}}}\right)+{\frac {1}{2}}}={\frac {3}{\pi ^{2}}}\prod _{p{\text{ prime}}}\left(1-{\frac {1}{p^{2}-1}}\right)+{\frac {1}{2}}$	1932
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
Lévy's constant (1)	$\beta$	1.18656 91104 15625 45282	${\frac {\pi ^{2}}{12\,\ln 2}}$	1935
Lévy's constant (2)	$e^{\beta }$	3.27582 29187 21811 15978	$e^{\pi ^{2}/(12\ln 2)}$	1936
Copeland–Erdős constant	${\mathcal {C}}_{CE}$	0.23571 11317 19232 93137	Defined by concatenating representations of successive prime numbers: 0.2 3 5 7 11 13 17 19 23 29 31 37 ...	1946	$\mathbb {R} \setminus \mathbb {Q}$
Mills' constant	$A$	1.30637 78838 63080 69046	Smallest positive real number A such that $\lfloor A^{3^{n}}\rfloor$ is prime for all positive integers n	1947
Gompertz constant	$\delta$	0.59634 73623 23194 07434	$\int _{0}^{\infty }\!\!{\frac {e^{-x}}{1+x}}\,dn=\!\!\int _{0}^{1}\!\!{\frac {dx}{1-\ln x}}={\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {2}{1+{\tfrac {2}{1+{\tfrac {3}{1+3{/\cdots }}}}}}}}}}}}}$	Before 1948
de Bruijn–Newman constant	$\Lambda$	$0\leq \Lambda \leq 0.2$	The number Λ where for where $H(\lambda ,z)=\int _{0}^{\infty }e^{\lambda u^{2}}\Phi (u)\cos(zu)du$ has real zeros if and only if λ ≥ Λ. where $\Phi (u)=\sum _{n=1}^{\infty }(2\pi ^{2}n^{4}e^{9u}-3\pi n^{2}e^{5u})e^{-\pi n^{2}e^{4u}}$.	1950
Van der Pauw constant	${\frac {\pi }{\ln 2}}$	4.53236 01418 27193 80962		Before 1958	$\mathbb {R} \setminus \mathbb {Q}$
Magic angle	$\theta _{\mathrm {m} }$	0.95531 66181 245092 78163	$\arctan {\sqrt {2}}=\arccos {\tfrac {1}{\sqrt {3}}}\approx \textstyle {54.7356}^{\circ }$	Before 1959	$\mathbb {T}$
Artin's constant	$C_{\mathrm {Artin} }$	0.37395 58136 19202 28805	$\prod _{p{\text{ prime}}}\left(1-{\frac {1}{p(p-1)}}\right)$	Before 1961
Porter's constant	$C$	1.46707 80794 33975 47289	${\frac {6\ln 2}{\pi ^{2}}}\left(3\ln 2+4\,\gamma -{\frac {24}{\pi ^{2}}}\,\zeta '(2)-2\right)-{\frac {1}{2}}$ where γ is the Euler–Mascheroni constant and ζ '(2) is the derivative of the Riemann zeta function evaluated at s = 2	1961
Lochs constant	$L$	0.97027 01143 92033 92574	${\frac {6\ln 2\ln 10}{\pi ^{2}}}$	1964
DeVicci's tesseract constant		1.00743 47568 84279 37609	The largest cube that can pass through in an 4D hypercube. Positive root of $4x^{8}{-}28x^{6}{-}7x^{4}{+}16x^{2}{+}16=0$	1966	$\mathbb {A}$
Lieb's square ice constant		1.53960 07178 39002 03869	$\left({\frac {4}{3}}\right)^{\frac {3}{2}}={\frac {8}{3{\sqrt {3}}}}$	1967	$\mathbb {A}$
Niven's constant	$C$	1.70521 11401 05367 76428	$1+\sum _{n=2}^{\infty }\left(1-{\frac {1}{\zeta (n)}}\right)$	1969
Stephens' constant		0.57595 99688 92945 43964	$\prod _{p{\text{ prime}}}\left(1-{\frac {p}{p^{3}-1}}\right)$	1969
Regular paperfolding sequence	$P$	0.85073 61882 01867 26036	$\sum _{n=0}^{\infty }{\frac {8^{2^{n}}}{2^{2^{n+2}}-1}}=\sum _{n=0}^{\infty }{\cfrac {\tfrac {1}{2^{2^{n}}}}{1-{\tfrac {1}{2^{2^{n+2}}}}}}$	1970	$\mathbb {T}$
Reciprocal Fibonacci constant	$\psi$	3.35988 56662 43177 55317	$\sum _{n=1}^{\infty }{\frac {1}{F_{n}}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{8}}+{\frac {1}{13}}+\cdots$ where Fn is the nth Fibonacci number	1974	$\mathbb {R} \setminus \mathbb {Q}$
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
Feigenbaum constant δ	$\delta$	4.66920 16091 02990 67185	$\lim _{n\to \infty }{\frac {x_{n+1}-x_{n}}{x_{n+2}-x_{n+1}}}$ where the sequence xn is given by $x_{n+1}=ax_{n}(1-x_{n})$	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
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
Conway's constant	$\lambda$	1.30357 72690 34296 39125	Real root of the polynomial: ${\begin{smallmatrix}x^{71}-x^{69}-2x^{68}-x^{67}+2x^{66}+2x^{65}+x^{64}-x^{63}-x^{62}-x^{61}-x^{60}\\-x^{59}+2x^{58}+5x^{57}+3x^{56}-2x^{55}-10x^{54}-3x^{53}-2x^{52}+6x^{51}+6x^{50}\\+x^{49}+9x^{48}-3x^{47}-7x^{46}-8x^{45}-8x^{44}+10x^{43}+6x^{42}+8x^{41}-5x^{40}\\-12x^{39}+7x^{38}-7x^{37}+7x^{36}+x^{35}-3x^{34}+10x^{33}+x^{32}-6x^{31}-2x^{30}\\-10x^{29}-3x^{28}+2x^{27}+9x^{26}-3x^{25}+14x^{24}-8x^{23}-7x^{21}+9x^{20}\\+3x^{19}\!-4x^{18}\!-10x^{17}\!-7x^{16}\!+12x^{15}\!+7x^{14}\!+2x^{13}\!-12x^{12}\!-4x^{11}\!-2x^{10}\\+5x^{9}+x^{7}-7x^{6}+7x^{5}-4x^{4}+12x^{3}-6x^{2}+3x-6\ =\ 0\quad \quad \quad \end{smallmatrix}}$	1987	$\mathbb {A}$
Hafner–Sarnak–McCurley constant	$\sigma$	0.35323 63718 54995 98454	$\prod _{p{\text{ prime}}}{\left(1-\left(1-\prod _{n\geq 1}\left(1-{\frac {1}{p^{n}}}\right)\right)^{2}\right)}\!$	1991
Backhouse's constant	$B$	1.45607 49485 82689 67139	$\lim _{k\to \infty }\left|{\frac {q_{k+1}}{q_{k}}}\right\vert \quad \scriptstyle {\text{where:}}\displaystyle \;\;Q(x)={\frac {1}{P(x)}}=\!\sum _{k=1}^{\infty }q_{k}x^{k}$ $P(x)=1+\sum _{k=1}^{\infty }{p_{k}x^{k}}=1+2x+3x^{2}+5x^{3}+\cdots$where pk is the kth prime number	1995
Viswanath constant		1.13198 82487 943	$\lim _{n\to \infty }|f_{n}|^{\frac {1}{n}}$      where fn = fn−1 ± fn−2, where the signs + or − are chosen at random with equal probability 1/2	1997
Komornik–Loreti constant	$q$	1.78723 16501 82965 93301	Real number $q$ such that $1=\sum _{n=1}^{\infty }{\frac {t_{k}}{q^{k}}}$, or $\prod _{n=0}^{\infty }\left(1-{\frac {1}{q^{2^{n}}}}\right)+{\frac {q-2}{q-1}}=0$ where tk is the kth term of the Thue–Morse sequence	1998	$\mathbb {T}$
Embree–Trefethen constant	$\beta ^{\star }$	0.70258		1999
Heath-Brown–Moroz constant	$C$	0.00131 76411 54853 17810	$\prod _{p{\text{ prime}}}\left(1-{\frac {1}{p}}\right)^{7}\left(1+{\frac {7p+1}{p^{2}}}\right)$	1999
MRB constant	$S$	0.18785 96424 62067 12024	$\sum _{n=1}^{\infty }(-1)^{n}(n^{1/n}-1)=-{\sqrt[{1}]{1}}+{\sqrt[{2}]{2}}-{\sqrt[{3}]{3}}+\cdots$	1999
Prime constant	$\rho$	0.41468 25098 51111 66024	$\sum _{p{\text{ prime}}}{\frac {1}{2^{p}}}={\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{32}}+\cdots$	1999	$\mathbb {R} \setminus \mathbb {Q}$
Somos' quadratic recurrence constant	$\sigma$	1.66168 79496 33594 12129	$\prod _{n=1}^{\infty }n^{{1/2}^{n}}={\sqrt {1{\sqrt {2{\sqrt {3\cdots }}}}}}=1^{1/2}\;2^{1/4}\;3^{1/8}\cdots$	1999
Foias constant	$\alpha$	1.18745 23511 26501 05459	$x_{n+1}=\left(1+{\frac {1}{x_{n}}}\right)^{n}{\text{ for }}n=1,2,3,\ldots$ Foias constant is the unique real number such that if x1 = α then the sequence diverges to infinity	2000
Logarithmic capacity of the unit disk		0.59017 02995 08048 11302	${\frac {\Gamma ({\tfrac {1}{4}})^{2}}{4\pi ^{3/2}}}$	Before 2003	$\mathbb {T}$
Taniguchi constant		0.67823 44919 17391 97803	$\prod _{p{\text{ prime}}}\left(1-{\frac {3}{p^{3}}}+{\frac {2}{p^{4}}}+{\frac {1}{p^{5}}}-{\frac {1}{p^{6}}}\right)$	Before 2005

Mathematical constants sorted by their representations as continued fractions

The following list includes the continued fractions of some constants and is sorted by their representations. Continued fractions with more than 20 known terms have been truncated, with an ellipsis to show that they continue. Rational numbers have two continued fractions; the version in this list is the shorter one. Decimal representations are rounded or padded to 10 places if the values are known.

Name	Symbol	Set	Decimal expansion	Continued fraction	Notes
Zero	0	$\mathbb {Z}$	0.00000 00000	[0; ]
Golomb–Dickman constant	$\lambda$		0.62432 99885	[0; 1, 1, 1, 1, 1, 22, 1, 2, 3, 1, 1, 11, 1, 1, 2, 22, 2, 6, 1, 1, …]	E. Weisstein noted that the continued fraction has an unusually large number of 1s.
Cahen's constant	$C_{2}$	$\mathbb {T}$	0.64341 05463	[0; 1, 1, 1, 22, 32, 132, 1292, 252982, 4209841472, 2694251407415154862, …]	All terms are squares and truncated at 10 terms due to large size. Davison and Shallit used the continued fraction expansion to prove that the constant is transcendental.
Euler–Mascheroni constant	$\gamma$		0.57721 56649	[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, 1, …]	Using the continued fraction expansion, it was shown that if γ is rational, its denominator must exceed 10244663.
First continued fraction constant	$C_{1}$	$\mathbb {R} \setminus \mathbb {Q}$	0.69777 46579	[0; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, …]	Equal to the ratio $I_{1}(2)/I_{0}(2)$ of modified Bessel functions of the first kind evaluated at 2.
Catalan's constant	$G$		0.91596 55942	[0; 1, 10, 1, 8, 1, 88, 4, 1, 1, 7, 22, 1, 2, 3, 26, 1, 11, 1, 10, 1, …]	Computed up to 4851389025 terms by E. Weisstein.
One half	1/2	$\mathbb {Q}$	0.50000 00000	[0; 2]
Prouhet–Thue–Morse constant	$\tau$	$\mathbb {T}$	0.41245 40336	[0; 2, 2, 2, 1, 4, 3, 5, 2, 1, 4, 2, 1, 5, 44, 1, 4, 1, 2, 4, 1, …]	Infinitely many partial quotients are 4 or 5, and infinitely many partial quotients are greater than or equal to 50.
Copeland–Erdős constant	${\mathcal {C}}_{CE}$	$\mathbb {R} \setminus \mathbb {Q}$	0.23571 11317	[0; 4, 4, 8, 16, 18, 5, 1, 1, 1, 1, 7, 1, 1, 6, 2, 9, 58, 1, 3, 4, …]	Computed up to 1011597392 terms by E. Weisstein. He also noted that while the Champernowne constant continued fraction contains sporadic large terms, the continued fraction of the Copeland-Erdős Constant do not exhibit this property.
Base 10 Champernowne constant	$C_{10}$	$\mathbb {T}$	0.12345 67891	[0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, 4.57540×10165, 6, 1, …]	Champernowne constants in any base exhibit sporadic large numbers; the 40th term in $C_{10}$ has 2504 digits.
One	1	$\mathbb {N}$	1.00000 00000	[1; ]
Phi, Golden ratio	$\varphi$	$\mathbb {A}$	1.61803 39887	[1; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …]	The convergents are ratios of successive Fibonacci numbers.
Brun's constant	$B_{2}$		1.90216 05831	[1; 1, 9, 4, 1, 1, 8, 3, 4, 7, 1, 3, 3, 1, 2, 1, 1, 12, 4, 2, 1, …]	The nth roots of the denominators of the nth convergents are close to Khinchin's constant, suggesting that $B_{2}$ is irrational. If true, this will prove the twin prime conjecture.
Square root of 2	${\sqrt {2}}$	$\mathbb {A}$	1.41421 35624	[1; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, …]	The convergents are ratios of successive Pell numbers.
Two	2	$\mathbb {N}$	2.00000 00000	[2; ]
Euler's number	$e$	$\mathbb {T}$	2.71828 18285	[2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, …]	The continued fraction expansion has the pattern [2; 1, 2, 1, 1, 4, 1, ..., 1, 2n, 1, ...].
Khinchin's constant	$K_{0}$		2.68545 20011	[2; 1, 2, 5, 1, 1, 2, 1, 1, 3, 10, 2, 1, 3, 2, 24, 1, 3, 2, 3, 1, …]	For almost all real numbers x, the coefficients of the continued fraction of x have a finite geometric mean known as Khinchin's constant.
Three	3	$\mathbb {N}$	3.00000 00000	[3; ]
Pi	$\pi$	$\mathbb {T}$	3.14159 26536	[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, …]	The first few convergents (3, 22/7, 333/106, 355/113, ...) are among the best-known and most widely used historical approximations of π.

List of series of Mathematical Constants

Name	Symbol	Formula	Year	Set
Harmonic number	$H_{n}$	$\sum _{k=1}^{n}{\frac {1}{k}}$	Antiquity	$\mathbb {Q}$
Gregory coefficients	$G_{n}$	${\frac {1}{n!}}\int _{0}^{1}x(x-1)(x-2)\cdots (x-n+1)\,dx=\int _{0}^{1}{\binom {x}{n}}\,dx$	1670	$\mathbb {Q}$
Bernoulli number	$B_{n}^{\pm }$	${\frac {t}{2}}\left(\operatorname {coth} {\frac {t}{2}}\pm 1\right)=\sum _{m=0}^{\infty }{\frac {B_{m}^{\pm {}}t^{m}}{m!}}$	1689	$\mathbb {Q}$
Hermite constants	$\gamma _{n}$	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	$\mathbb {R}$
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	$\mathbb {R}$
Stieltjes constants	$\gamma _{n}$	${\frac {(-1)^{n}n!}{2\pi }}\int _{0}^{2\pi }e^{-nix}\zeta \left(e^{ix}+1\right)dx.$	before 1894	$\mathbb {R}$
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	$\mathbb {R}$
Generalized 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	$\mathbb {R}$
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	$\mathbb {T}$
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).	before 1957	$\mathbb {A}$
Feller's coin-tossing constants	$\alpha _{k},\beta _{k}$	$\alpha _{k}$ is the smallest positive real root of $x^{k+1}=2^{k+1}(x-1),\beta _{k}={\frac {2-\alpha _{k}}{k+1-k\alpha _{k}}}$	1968	$\mathbb {R}$
Stoneham number	$\alpha _{b,c}$	$\sum _{n=c^{k}>1}{\frac {1}{b^{n}n}}=\sum _{k=1}^{\infty }{\frac {1}{b^{c^{k}}c^{k}}}$ where b,c are coprime integers.	1973	$\mathbb {R}$
Beraha constants	$B(n)$	$2+2\cos \left({\frac {2\pi }{n}}\right)$	1974	$\mathbb {R}$
Chvátal–Sankoff constants	$\gamma _{k}$	$\lim _{n\to \infty }{\frac {E[\lambda _{n,k}]}{n}}$	1975	$\mathbb {R}$
Hyperharmonic number	$H_{n}^{(r))}$	$\sum _{k=1}^{n}H_{k}^{(r-1)}$ and $H_{n}^{(0)}={\frac {1}{n}}$	1995	$\mathbb {Q}$
Gregory number	$G_{x}$	$\sum _{n=0}^{\infty }(-1)^{n}{\frac {1}{(2n+1)x^{2n+1}}}$ for rational x greater than one.	before 1996	$\mathbb {R}$
Metallic mean		${\frac {n+{\sqrt {n^{2}+4}}}{2}}$	before 1998	$\mathbb {A}$

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

References

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