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List of mathematical constants

From Wikipedia, the free encyclopedia

Amathematical constant is a keynumber whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., analphabet letter), or by mathematicians' names to facilitate using it across multiplemathematical problems.^[1] For example, the constantπ may be defined as the ratio of the length of a circle'scircumference to itsdiameter. The following list includes adecimal 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

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Name	Symbol	Decimal expansion	Formula	Year	Set
Name	Symbol	Decimal expansion	Formula	Year	$\mathbb {Q}$	$\mathbb {A}$	${\mathcal {P}}$
One	1	1	Multiplicative identity of $\mathbb {C}$ .	Prehistory	✓	✓	✓
Two	2	2		Prehistory	✓	✓	✓
One half	⁠1/2⁠	0.5	Multiplicative inverse of 2.	Prehistory	✓	✓	✓
Pi	$\pi$	3.14159 26535 89793 23846^{[Mw 1]}^{[OEIS 1]}	Ratio of a circle's circumference to its diameter.	1900 to 1600 BCE^[2]	✗	✗	✓
Tau	$\tau$	6.28318 53071 79586 47692^[3]^{[OEIS 2]}	Ratio of a circle's circumference to its radius. Equal to $2\pi$	1900 to 1600 BCE^[2]	✗	✗	✓
Square root of 2, Pythagoras constant^[4]	${\sqrt {2}}$	1.41421 35623 73095 04880^{[Mw 2]}^{[OEIS 3]}	Positive root of $x^{2}=2$	1800 to 1600 BCE^[5]	✗	✓	✓
Square root of 3, Theodorus' constant^[6]	${\sqrt {3}}$	1.73205 08075 68877 29352^{[Mw 3]}^{[OEIS 4]}	Positive root of $x^{2}=3$	465 to 398 BCE	✗	✓	✓
Square root of 5^[7]	${\sqrt {5}}$	2.23606 79774 99789 69640^{[OEIS 5]}	Positive root of $x^{2}=5$		✗	✓	✓
Phi,Golden ratio^[8]	$\varphi$ or $\phi$	1.61803 39887 49894 84820^{[Mw 4]}^{[OEIS 6]}	${\frac {1+{\sqrt {5}}}{2}}$	~300 BCE	✗	✓	✓
Silver ratio^[9]	$\sigma$	2.41421 35623 73095 04880^{[Mw 5]}^{[OEIS 7]}	${\sqrt {2}}+1$	~300 BCE	✗	✓	✓
Zero	0	0	Additive identity of $\mathbb {C}$ .	300 to 100 BCE^[10]	✓	✓	✓
Negative one	−1	−1	Additive inverse of 1.	300 to 200 BCE	✓	✓	✓
Cube root of 2, Delian constant	${\sqrt[{3}]{2}}$	1.25992 10498 94873 16476^{[Mw 6]}^{[OEIS 8]}	Real root of $x^{3}=2$	46 to 120 CE^[11]	✗	✓	✓
Cube root of 3	${\sqrt[{3}]{3}}$	1.44224 95703 07408 38232^{[OEIS 9]}	Real root of $x^{3}=3$		✗	✓	✓
Twelfth root of 2^[12]	${\sqrt[{12}]{2}}$	1.05946 30943 59295 26456^{[OEIS 10]}	Real positive root of $x^{12}=2$		✗	✓	✓
Supergolden ratio^[13]	$\psi$	1.46557 12318 76768 02665^{[OEIS 11]}	${\frac {1+{\sqrt[{3}]{\frac {29+3{\sqrt {93}}}{2}}}+{\sqrt[{3}]{\frac {29-3{\sqrt {93}}}{2}}}}{3}}$ Real root of $x^{3}=x^{2}+1$		✗	✓	✓
Imaginary unit^[14]	$i {\displaystyle i}$	0 + 1i	Principal root of $x^{2}=-1$ ^{[nb 1]}	1501 to 1576	✗	✓	✓
Connective constant for the hexagonal lattice^[15]^[16]	$\mu$	1.84775 90650 22573 51225^{[Mw 7]}^{[OEIS 12]}	${\sqrt {2+{\sqrt {2}}}}$ , as a root of the polynomial $x^{4}-4x^{2}+2=0$	1593^{[OEIS 12]}	✗	✓	✓
Kepler–Bouwkamp constant^[17]	$K^{'} {\displaystyle K'}$	0.11494 20448 53296 20070^{[Mw 8]}^{[OEIS 13]}	$\prod _{n=3}^{\infty }\cos \left({\frac {\pi }{n}}\right)=\cos \left({\frac {\pi }{3}}\right)\cos \left({\frac {\pi }{4}}\right)\cos \left({\frac {\pi }{5}}\right)...$	1596^{[OEIS 13]}	?	?	?
Wallis's constant		2.09455 14815 42326 59148^{[Mw 9]}^{[OEIS 14]}	${\sqrt[{3}]{\frac {45-{\sqrt {1929}}}{18}}}+{\sqrt[{3}]{\frac {45+{\sqrt {1929}}}{18}}}$ Real root of $x^{3}-2x-5=0$	1616 to 1703	✗	✓	✓
Euler's number^[18]	$e {\displaystyle e}$	2.71828 18284 59045 23536^{[Mw 10]}^{[OEIS 15]}	$\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}=\sum _{n=0}^{\infty }{\frac {1}{n!}}=1+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}\cdots$	1618^[19]	✗	✗	?
Natural logarithm of 2^[20]	$\ln 2$	0.69314 71805 59945 30941^{[Mw 11]}^{[OEIS 16]}	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^[21] & 1668^[22]	✗	✗	✓
Lemniscate constant^[23]	$\varpi$	2.62205 75542 92119 81046^{[Mw 12]}^{[OEIS 17]}	$2\int _{0}^{1}{\frac {dt}{\sqrt {1-t^{4}}}}={\frac {1}{4}}{\sqrt {\frac {2}{\pi }}}\,\Gamma {\left({\frac {1}{4}}\right)^{2}}$ Ratio of the perimeter ofBernoulli's lemniscate to its diameter.	1718 to 1798	✗	✗	✓
Euler's constant	$\gamma$	0.57721 56649 01532 86060^{[Mw 13]}^{[OEIS 18]}	$\lim _{n\to \infty }\left(-\log n+\sum _{k=1}^{n}{\frac {1}{k}}\right)=\int _{1}^{\infty }\left(-{\frac {1}{x}}+{\frac {1}{\lfloor x\rfloor }}\right)\,dx$ Limiting difference between theharmonic series and thenatural logarithm.	1735	?	?	?
Erdős–Borwein constant^[24]	$E {\displaystyle E}$	1.60669 51524 15291 76378^{[Mw 14]}^{[OEIS 19]}	$\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}={\frac {1}{1}}\!+\!{\frac {1}{3}}\!+\!{\frac {1}{7}}\!+\!{\frac {1}{15}}\!+\!\cdots$	1749^[25]	✗	?	?
Omega constant	$\Omega$	0.56714 32904 09783 87299^{[Mw 15]}^{[OEIS 20]}	$W(1)={\frac {1}{\pi }}\int _{0}^{\pi }\log \left(1+{\frac {\sin t}{t}}e^{t\cot t}\right)dt$ where W is theLambert W function	1758 & 1783	✗	✗	?
Apéry's constant^[26]	$\zeta (3)$	1.20205 69031 59594 28539^{[Mw 16]}^{[OEIS 21]}	$\zeta (3)=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}={\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{3}}}+{\frac {1}{5^{3}}}+\cdots$ with theRiemann zeta function $\zeta (s)$ .	1780^{[OEIS 21]}	✗	?	✓
Laplace limit^[27]		0.66274 34193 49181 58097^{[Mw 17]}^{[OEIS 22]}	Real root of ${\frac {xe^{\sqrt {x^{2}+1}}}{{\sqrt {x^{2}+1}}+1}}=1$	~1782	✗	✗	?
Soldner constant^[28]^[29]	$\mu$	1.45136 92348 83381 05028^{[Mw 18]}^{[OEIS 23]}	$\mathrm {li} (x)=\int _{0}^{x}{\frac {dt}{\ln t}}=0$ ; root of thelogarithmic integral function.	1792^{[OEIS 23]}	?	?	?
Gauss's constant^[30]	$G {\displaystyle G}$	0.83462 68416 74073 18628^{[Mw 19]}^{[OEIS 24]}	${\frac {1}{\mathrm {agm} (1,{\sqrt {2}})}}={\frac {1}{4\pi }}{\sqrt {\frac {2}{\pi }}}\Gamma \left({\frac {1}{4}}\right)^{2}={\frac {\varpi }{\pi }}$ whereagm is thearithmetic–geometric mean and $\varpi$ is thelemniscate constant.	1799^[31]	✗	✗	?
SecondHermite constant^[32]	$\gamma _{2}$	1.15470 05383 79251 52901^{[Mw 20]}^{[OEIS 25]}	${\frac {2}{\sqrt {3}}}$	1822 to 1901	✗	✓	✓
Liouville's constant^[33]	$L {\displaystyle L}$	0.11000 10000 00000 00000 0001^{[Mw 21]}^{[OEIS 26]}	$\sum _{n=1}^{\infty }{\frac {1}{10^{n!}}}={\frac {1}{10^{1!}}}+{\frac {1}{10^{2!}}}+{\frac {1}{10^{3!}}}+{\frac {1}{10^{4!}}}+\cdots$	Before 1844	✗	✗	?
Firstcontinued fraction constant	$C_{1}$	0.69777 46579 64007 98201^{[Mw 22]}^{[OEIS 27]}	$C_{1}=[0;1,2,3,4,5,...]={\frac {I_{1}(2)}{I_{0}(2)}}$ , (seeBessel functions). $C_{1}\notin \mathbb {A} .$ ^[34]	1855^[35]	✗	✗	?
Ramanujan's constant^[36]		262 53741 26407 68743 .99999 99999 99250 073^{[Mw 23]}^{[OEIS 28]}	$e^{\pi {\sqrt {163}}}$	1859	✗	✗	?
Glaisher–Kinkelin constant	$A {\displaystyle A}$	1.28242 71291 00622 63687^{[Mw 24]}^{[OEIS 29]}	$e^{{\frac {1}{12}}-\zeta ^{\prime }(-1)}=e^{{\frac {1}{8}}-{\frac {1}{2}}\sum \limits _{n=0}^{\infty }{\frac {1}{n+1}}\sum \limits _{k=0}^{n}\left(-1\right)^{k}{\binom {n}{k}}\left(k+1\right)^{2}\ln(k+1)}$	1860^{[OEIS 29]}	?	?	?
Catalan's constant^[37]^[38]^[39]	$G {\displaystyle G}$	0.91596 55941 77219 01505^{[Mw 25]}^{[OEIS 30]}	$\beta (2)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2}}}={\frac {1}{1^{2}}}-{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}-{\frac {1}{7^{2}}}+{\frac {1}{9^{2}}}+\cdots$ with theDirichlet beta function $\beta (s)$ .	1864	?	?	✓
Dottie number^[40]	$D {\displaystyle D}$	0.73908 51332 15160 64165^{[Mw 26]}^{[OEIS 31]}	Real root of $\cos x=x$	1865^{[Mw 26]}	✗	✗	?
Meissel–Mertens constant^[41]	$M {\displaystyle M}$	0.26149 72128 47642 78375^{[Mw 27]}^{[OEIS 32]}	$\lim _{n\to \infty }\left(\sum _{p\leq n}{\frac {1}{p}}-\ln \ln n\right)=\gamma +\sum _{p}\left(\ln \left(1-{\frac {1}{p}}\right)+{\frac {1}{p}}\right)$ whereγ is theEuler–Mascheroni constant andp is prime	1866 & 1873	?	?	?
Universal parabolic constant^[42]	$P {\displaystyle P}$	2.29558 71493 92638 07403^{[Mw 28]}^{[OEIS 33]}	$\ln(1+{\sqrt {2}})+{\sqrt {2}}\;=\;\operatorname {arsinh} (1)+{\sqrt {2}}$	Before 1891^[43]	✗	✗	✓
Cahen's constant^[44]	$C {\displaystyle C}$	0.64341 05462 88338 02618^{[Mw 29]}^{[OEIS 34]}	$\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{s_{k}-1}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{6}}-{\frac {1}{42}}+{\frac {1}{1806}}{\,\pm \cdots }$ wheres_k is thekth term ofSylvester's sequence 2, 3, 7, 43, 1807, ...	1891	✗	✗	?
Gelfond's constant^[45]	$e^{\pi }$	23.14069 26327 79269 0057^{[Mw 30]}^{[OEIS 35]}	$(-1)^{-i}=i^{-2i}=\sum _{n=0}^{\infty }{\frac {\pi ^{n}}{n!}}=1+{\frac {\pi ^{1}}{1}}+{\frac {\pi ^{2}}{2}}+{\frac {\pi ^{3}}{6}}+\cdots$	1900^[46]	✗	✗	?
Gelfond–Schneider constant^[47]	$2^{\sqrt {2}}$	2.66514 41426 90225 18865^{[Mw 31]}^{[OEIS 36]}	$2^{\sqrt {2}}$	Before 1902^{[OEIS 36]}	✗	✗	?
SecondFavard constant^[48]	$K_{2}$	1.23370 05501 36169 82735^{[Mw 32]}^{[OEIS 37]}	${\frac {\pi ^{2}}{8}}=\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\cdots$	1902 to 1965	✗	✗	✓
Golden angle^[49]	$g {\displaystyle g}$	2.39996 32297 28653 32223^{[Mw 33]}^{[OEIS 38]}	${\frac {2\pi }{\varphi ^{2}}}=\pi (3-{\sqrt {5}})$ or $180(3-{\sqrt {5}})=137.50776\ldots$ in degrees	1907	✗	✗	✓
Sierpiński's constant^[50]	$K {\displaystyle K}$	2.58498 17595 79253 21706^{[Mw 34]}^{[OEIS 39]}	${\begin{aligned}&\pi \left(2\gamma +\ln {\frac {4\pi ^{3}}{\Gamma ({\tfrac {1}{4}})^{4}}}\right)=\pi (2\gamma +4\ln \Gamma ({\tfrac {3}{4}})-\ln \pi )\\&=\pi \left(2\ln 2+3\ln \pi +2\gamma -4\ln \Gamma ({\tfrac {1}{4}})\right)\end{aligned}}$	1907	?	?	?
Landau–Ramanujan constant^[51]	$K {\displaystyle K}$	0.76422 36535 89220 66299^{[Mw 35]}^{[OEIS 40]}	${\frac {1}{\sqrt {2}}}\prod _{{p\equiv 3{\text{ mod }}4} \atop p\;{\rm {prime}}}{\left(1-{\frac {1}{p^{2}}}\right)^{-{\frac {1}{2}}}}\!\!={\frac {\pi }{4}}\prod _{{p\equiv 1{\text{ mod }}4} \atop p\;{\rm {prime}}}{\left(1-{\frac {1}{p^{2}}}\right)^{\frac {1}{2}}}$	1908^{[OEIS 40]}	?	?	?
FirstNielsen–Ramanujan constant^[52]	$a_{1}$	0.82246 70334 24113 21823^{[Mw 36]}^{[OEIS 41]}	${\frac {{\zeta }(2)}{2}}={\frac {\pi ^{2}}{12}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{2}}}={\frac {1}{1^{2}}}{-}{\frac {1}{2^{2}}}{+}{\frac {1}{3^{2}}}{-}{\frac {1}{4^{2}}}{+}\cdots$	1909	✗	✗	✓
Gieseking constant^[53]	$V {\displaystyle V}$	1.01494 16064 09653 62502^{[Mw 37]}^{[OEIS 42]}	${\frac {3{\sqrt {3}}}{4}}\left(1-\sum _{n=0}^{\infty }{\frac {1}{(3n+2)^{2}}}+\sum _{n=1}^{\infty }{\frac {1}{(3n+1)^{2}}}\right)$ $={\frac {\sqrt {3}}{3}}\left({\frac {\psi _{1}(1/3)}{2}}-{\frac {\pi ^{2}}{3}}\right)$ with thetrigamma function $\psi _{1}$ .	1912	?	?	✓
Bernstein's constant^[54]	$\beta$	0.28016 94990 23869 13303^{[Mw 38]}^{[OEIS 43]}	$\lim _{n\to \infty }2nE_{2n}(f)$ , whereE_n(f) is the error of the bestuniform approximation to areal functionf(x) on the interval [−1, 1] by real polynomials of no more than degreen, andf(x) = \|x\|	1913	?	?	?
Tribonacci constant^[55]		1.83928 67552 14161 13255^{[Mw 39]}^{[OEIS 44]}	${\textstyle {\frac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}={\frac {1+4\cosh \left({\frac {1}{3}}\cosh ^{-1}\left(2+{\frac {3}{8}}\right)\right)}{3}}}$ Real root of $x^{3}-x^{2}-x-1=0$	1914 to 1963	✗	✓	✓
Brun's constant^[56]	$B_{2}$	1.90216 05831 04^{[Mw 40]}^{[OEIS 45]}	$\textstyle {\sum \limits _{p}({\frac {1}{p}}+{\frac {1}{p+2}})}=({\frac {1}{3}}\!+\!{\frac {1}{5}})+({\tfrac {1}{5}}\!+\!{\tfrac {1}{7}})+({\tfrac {1}{11}}\!+\!{\tfrac {1}{13}})+\cdots$ where the sum ranges over all primesp such thatp + 2 is also a prime	1919^{[OEIS 45]}	?	?	?
Twin primes constant	$C_{2}$	0.66016 18158 46869 57392^{[Mw 41]}^{[OEIS 46]}	$\prod _{\textstyle {p\;{\rm {prime}} \atop p\geq 3}}\left(1-{\frac {1}{(p-1)^{2}}}\right)$	1922	?	?	?
Plastic ratio^[57]	$\rho$	1.32471 79572 44746 02596^{[Mw 42]}^{[OEIS 47]}	${\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\cdots }}}}}}=\textstyle {\sqrt[{3}]{{\frac {1}{2}}+{\frac {\sqrt {69}}{18}}}}+{\sqrt[{3}]{{\frac {1}{2}}-{\frac {\sqrt {69}}{18}}}}$ Real root of $x^{3}=x+1$	1924^{[OEIS 47]}	✗	✓	✓
Bloch's constant^[58]	$B {\displaystyle B}$	$0.4332\leq B\leq 0.4719$ ^{[Mw 43]}^{[OEIS 48]}	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^{[OEIS 48]}	?	?	?
Z score for the 97.5 percentile point^[59]^[60]^[61]^[62]	$z_{.975}$	1.95996 39845 40054 23552^{[Mw 44]}^{[OEIS 49]}	${\sqrt {2}}\operatorname {erf} ^{-1}(0.95)$ whereerf⁻¹(x) is theinverse error function Real number $z {\displaystyle z}$ such that ${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{z}e^{-x^{2}/2}\,\mathrm {d} x=0.975$	1925	?	?	?
Landau's constant^[58]	$L {\displaystyle L}$	$0.5<L\leq 0.54326$ ^{[Mw 45]}^{[OEIS 50]}	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^[58]	$A {\displaystyle A}$	$0.5<A\leq 0.7853$		1929	?	?	?
Prouhet–Thue–Morse constant^[63]	$\tau$	0.41245 40336 40107 59778^{[Mw 46]}^{[OEIS 51]}	$\sum _{n=0}^{\infty }{\frac {t_{n}}{2^{n+1}}}={\frac {1}{4}}\left[2-\prod _{n=0}^{\infty }\left(1-{\frac {1}{2^{2^{n}}}}\right)\right]$ where ${t_{n}}$ is then^th term of theThue–Morse sequence	1929^{[OEIS 51]}	✗	✗	?
Golomb–Dickman constant^[64]	$\lambda$	0.62432 99885 43550 87099^{[Mw 47]}^{[OEIS 52]}	$\int _{0}^{1}e^{\mathrm {Li} (t)}dt=\int _{0}^{\infty }{\frac {\rho (t)}{t+2}}dt$ where Li(t) is the logarithmic integral, andρ(t) is theDickman function	1930 & 1964	?	?	?
Constant related to the asymptotic behavior ofLebesgue constants^[65]	$c {\displaystyle c}$	0.98943 12738 31146 95174^{[Mw 48]}^{[OEIS 53]}	$\lim _{n\to \infty }\!\!\left(\!{L_{n}{-}{\frac {4}{\pi ^{2}}}\ln(2n{+}1)}\!\!\right)\!{=}{\frac {4}{\pi ^{2}}}\!\left({-}{\frac {\Gamma '({\tfrac {1}{2}})}{\Gamma ({\tfrac {1}{2}})}}{+}{\sum _{k=1}^{\infty }\!{\frac {2\ln k}{4k^{2}{-}1}}}\right)$	1930^{[Mw 48]}	?	?	?
Feller–Tornier constant^[66]	${\mathcal {C}}_{\mathrm {FT} }$	0.66131 70494 69622 33528^{[Mw 49]}^{[OEIS 54]}	${{\frac {1}{2}}\prod _{p{\text{ prime}}}\left(1-{\frac {2}{p^{2}}}\right)+{\frac {1}{2}}}={\frac {3}{\pi ^{2}}}\prod _{p{\text{ prime}}}\left(1-{\frac {1}{p^{2}-1}}\right)+{\frac {1}{2}}$	1932	?	?	?
Base 10Champernowne constant^[67]	$C_{10}$	0.12345 67891 01112 13141^{[Mw 50]}^{[OEIS 55]}	Defined by concatenating representations of successive integers: 0.1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...	1933	✗	✗	?
Salem constant^[68]	$\sigma _{10}$	1.17628 08182 59917 50654^{[Mw 51]}^{[OEIS 56]}	Largest real root of $x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1=0$	1933^{[OEIS 56]}	✗	✓	✓
Khinchin's constant^[69]	$K_{0}$	2.68545 20010 65306 44530 ^{[Mw 52]}^{[OEIS 57]}	$\prod _{n=1}^{\infty }\left[{1+{1 \over n(n+2)}}\right]^{\log _{2}(n)}$	1934	?	?	?
Lévy's constant (1)^[70]	$\beta$	1.18656 91104 15625 45282^{[Mw 53]}^{[OEIS 58]}	${\frac {\pi ^{2}}{12\,\ln 2}}$	1935	?	?	?
Lévy's constant (2)^[71]	$e^{\beta }$	3.27582 29187 21811 15978^{[Mw 54]}^{[OEIS 59]}	$e^{\pi ^{2}/(12\ln 2)}$	1936	?	?	?
Copeland–Erdős constant^[72]	${\mathcal {C}}_{CE}$	0.23571 11317 19232 93137^{[Mw 55]}^{[OEIS 60]}	Defined by concatenating representations of successive prime numbers: 0.2 3 5 7 11 13 17 19 23 29 31 37 ...	1946^{[OEIS 60]}	✗	?	?
Mills' constant^[73]	$A {\displaystyle A}$	1.30637 78838 63080 69046^{[Mw 56]}^{[OEIS 61]}	Smallest positive real numberA such that $\lfloor A^{3^{n}}\rfloor$ is prime for all positive integersn	1947	?	?	?
Gompertz constant^[74]	$\delta$	0.59634 73623 23194 07434^{[Mw 57]}^{[OEIS 62]}	$\int _{0}^{\infty }\!\!{\frac {e^{-x}}{1+x}}\,dx=\!\!\int _{0}^{1}\!\!{\frac {dx}{1-\ln x}}={\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {2}{1+{\tfrac {2}{1+{\tfrac {3}{1+3{/\cdots }}}}}}}}}}}}}$	Before 1948^{[OEIS 62]}	?	?	?
de Bruijn–Newman constant	$\Lambda$	$0\leq \Lambda \leq 0.2$	The number Λ such that $H(\lambda ,z)=\int _{0}^{\infty }e^{\lambda u^{2}}\Phi (u)\cos(zu)du$ 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^{[OEIS 63]}	${\frac {\pi }{\ln 2}}$	Before 1958^{[OEIS 64]}	✗	?	?
Magic angle^[75]	$\theta _{\mathrm {m} }$	0.95531 66181 245092 78163^{[OEIS 65]}	$\arctan {\sqrt {2}}=\arccos {\tfrac {1}{\sqrt {3}}}\approx \textstyle {54.7356}^{\circ }$	Before 1959^[76]^[75]	✗	✗	✓
Artin's constant^[77]	$C_{\mathrm {Artin} }$	0.37395 58136 19202 28805^{[Mw 58]}^{[OEIS 66]}	$\prod _{p{\text{ prime}}}\left(1-{\frac {1}{p(p-1)}}\right)$	Before 1961^{[OEIS 66]}	?	?	?
Porter's constant^[78]	$C {\displaystyle C}$	1.46707 80794 33975 47289^{[Mw 59]}^{[OEIS 67]}	${\frac {6\ln 2}{\pi ^{2}}}\left(3\ln 2+4\,\gamma -{\frac {24}{\pi ^{2}}}\,\zeta '(2)-2\right)-{\frac {1}{2}}$ whereγ is theEuler–Mascheroni constant andζ '(2) is the derivative of theRiemann zeta function evaluated ats = 2	1961^{[OEIS 67]}	?	?	?
Lochs constant^[79]	$L {\displaystyle L}$	0.97027 01143 92033 92574^{[Mw 60]}^{[OEIS 68]}	${\frac {6\ln 2\ln 10}{\pi ^{2}}}$	1964	?	?	?
DeVicci's tesseract constant		1.00743 47568 84279 37609^{[OEIS 69]}	The largest cube that can pass through a 4D hypercube. Positive root of $4x^{8}{-}28x^{6}{-}7x^{4}{+}16x^{2}{+}16=0$	1966^{[OEIS 69]}	✗	✓	✓
Lieb's square ice constant^[80]		1.53960 07178 39002 03869^{[Mw 61]}^{[OEIS 70]}	$\left({\frac {4}{3}}\right)^{\frac {3}{2}}={\frac {8}{3{\sqrt {3}}}}$	1967	✗	✓	✓
Niven's constant^[81]	$C {\displaystyle C}$	1.70521 11401 05367 76428^{[Mw 62]}^{[OEIS 71]}	$1+\sum _{n=2}^{\infty }\left(1-{\frac {1}{\zeta (n)}}\right)$	1969	?	?	?
Stephens' constant^[82]	$C_{S}$	0.57595 99688 92945 43964^{[Mw 63]}^{[OEIS 72]}	$\prod _{p{\text{ prime}}}\left(1-{\frac {p}{p^{3}-1}}\right)$	1969^{[OEIS 72]}	?	?	?
Regular paperfolding sequence^[83]^[84]	$P {\displaystyle P}$	0.85073 61882 01867 26036^{[Mw 64]}^{[OEIS 73]}	$\sum _{n=0}^{\infty }{\frac {8^{2^{n}}}{2^{2^{n+2}}-1}}=\sum _{n=0}^{\infty }{\cfrac {\tfrac {1}{2^{2^{n}}}}{1-{\tfrac {1}{2^{2^{n+2}}}}}}$	1970^{[OEIS 73]}	✗	✗	?
Reciprocal Fibonacci constant^[85]	$\psi$	3.35988 56662 43177 55317^{[Mw 65]}^{[OEIS 74]}	$\sum _{n=1}^{\infty }{\frac {1}{F_{n}}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{8}}+{\frac {1}{13}}+\cdots$ whereF_n is then^thFibonacci number	1974^{[OEIS 74]}	✗	?	?
Chvátal–Sankoff constant for the binary alphabet	$\gamma _{2}$	$0.788071\leq \gamma _{2}\leq 0.826280$	$\lim _{n\to \infty }{\frac {\operatorname {E} [\lambda _{n,2}]}{n}}$ whereE[λ_n,2] is theexpected longest common subsequence of two random length-n binary strings	1975	?	?	?
Feigenbaum constant δ^[86]	$\delta$	4.66920 16091 02990 67185^{[Mw 66]}^{[OEIS 75]}	$\lim _{n\to \infty }{\frac {a_{n+1}-a_{n}}{a_{n+2}-a_{n+1}}}$ where the sequencea_n is given by n-th period-doubling bifurcation oflogistic map $x_{k+1}=ax_{k}(1-x_{k})$ or any other one-dimensional map with a single quadratic maximum	1975	?	?	?
Chaitin's constants^[87]	$\Omega$	In general they areuncomputable numbers. But one such number is 0.00787 49969 97812 3844. ^{[Mw 67]}^{[OEIS 76]}	$\sum _{p\in P}2^{-\|p\|}$ p: Halted program \|p\|: Size in bits of programp P: Domain of all programs that stop. See also:Halting problem	1975	✗	✗	✗
Robbins constant^[88]	$\Delta (3)$	0.66170 71822 67176 23515^{[Mw 68]}^{[OEIS 77]}	${\frac {4\!+\!17{\sqrt {2}}\!-6{\sqrt {3}}\!-7\pi }{105}}\!+\!{\frac {\ln(1\!+\!{\sqrt {2}})}{5}}\!+\!{\frac {2\ln(2\!+\!{\sqrt {3}})}{5}}$	1977^{[OEIS 77]}	✗	✗	✓
Weierstrass constant^[89]		0.47494 93799 87920 65033^{[Mw 69]}^{[OEIS 78]}	${\frac {2^{5/4}{\sqrt {\pi }}\,e^{\pi /8}}{\Gamma ({\frac {1}{4}})^{2}}}$	Before 1978^[90]	✗	✗	?
Fransén–Robinson constant^[91]	$F {\displaystyle F}$	2.80777 02420 28519 36522^{[Mw 70]}^{[OEIS 79]}	$\int _{0}^{\infty }{\frac {dx}{\Gamma (x)}}=e+\int _{0}^{\infty }{\frac {e^{-x}}{\pi ^{2}+\ln ^{2}x}}\,dx$	1978	?	?	?
Feigenbaum constant α^[92]	$\alpha$	2.50290 78750 95892 82228^{[Mw 66]}^{[OEIS 80]}	Ratio between the width of a tine and the width of one of its two subtines in abifurcation diagram	1979	?	?	?
Second du Bois-Reymond constant^[93]	$C_{2}$	0.19452 80494 65325 11361^{[Mw 71]}^{[OEIS 81]}	${\frac {e^{2}-7}{2}}=\int _{0}^{\infty }\left\|{{\frac {d}{dt}}\left({\frac {\sin t}{t}}\right)^{2}}\right\|\,dt-1$	1983^{[OEIS 81]}	✗	✗	?
Erdős–Tenenbaum–Ford constant	$\delta$	0.08607 13320 55934 20688^{[OEIS 82]}	$1-{\frac {1+\log \log 2}{\log 2}}$	1984	?	?	?
Conway's constant^[94]	$\lambda$	1.30357 72690 34296 39125^{[Mw 72]}^{[OEIS 83]}	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	✗	✓	✓
Hafner–Sarnak–McCurley constant^[95]	$\sigma$	0.35323 63718 54995 98454^{[Mw 73]}^{[OEIS 84]}	$\prod _{p{\text{ prime}}}{\left(1-\left(1-\prod _{n\geq 1}\left(1-{\frac {1}{p^{n}}}\right)\right)^{2}\right)}\!$	1991^{[OEIS 84]}	?	?	?
Backhouse's constant^[96]	$B {\displaystyle B}$	1.45607 49485 82689 67139^{[Mw 74]}^{[OEIS 85]}	$\lim _{k\to \infty }\left\|{\frac {q_{k+1}}{q_{k}}}\right\vert \quad \scriptstyle {\text{where:}}\displaystyle \;\;Q(x)={\frac {1}{P(x)}}=\!\sum _{k=1}^{\infty }q_{k}x^{k}$ $P(x)=1+\sum _{k=1}^{\infty }{p_{k}x^{k}}=1+2x+3x^{2}+5x^{3}+\cdots$ wherep_k is thek^th prime number	1995	?	?	?
Viswanath constant^[97]	$V {\displaystyle V}$	1.13198 82487 943^{[Mw 75]}^{[OEIS 86]}	$\lim _{n\to \infty }\|f_{n}\|^{\frac {1}{n}}$ wheref_n =f_n−1 ±f_n−2, where the signs + or − are chosenat random with equal probability 1/2	1997	?	?	?
Komornik–Loreti constant^[98]	$q {\displaystyle q}$	1.78723 16501 82965 93301^{[Mw 76]}^{[OEIS 87]}	Real number $q {\displaystyle q}$ such that $1=\sum _{k=1}^{\infty }{\frac {t_{k}}{q^{k}}}$ , or $\prod _{n=0}^{\infty }\left(1-{\frac {1}{q^{2^{n}}}}\right)+{\frac {q-2}{q-1}}=0$ wheret_k is thek^th term of theThue–Morse sequence	1998	✗	✗	?
Embree–Trefethen constant	$\beta ^{\star }$	0.70258		1999	?	?	?
Heath-Brown–Moroz constant^[99]	$C {\displaystyle C}$	0.00131 76411 54853 17810^{[Mw 77]}^{[OEIS 88]}	$\prod _{p{\text{ prime}}}\left(1-{\frac {1}{p}}\right)^{7}\left(1+{\frac {7p+1}{p^{2}}}\right)$	1999^{[OEIS 88]}	?	?	?
MRB constant^[100]^[101]^[102]	$S {\displaystyle S}$	0.18785 96424 62067 12024^{[Mw 78]}^{[Ow 1]}^{[OEIS 89]}	$\sum _{n=1}^{\infty }(-1)^{n}(n^{1/n}-1)=-{\sqrt[{1}]{1}}+{\sqrt[{2}]{2}}-{\sqrt[{3}]{3}}+\cdots$	1999	?	?	?
Prime constant^[103]	$\rho$	0.41468 25098 51111 66024^{[OEIS 90]}	$\sum _{p{\text{ prime}}}{\frac {1}{2^{p}}}={\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{32}}+\cdots$	1999^{[OEIS 90]}	✗	?	?
Somos' quadratic recurrence constant^[104]	$\sigma$	1.66168 79496 33594 12129^{[Mw 79]}^{[OEIS 91]}	$\prod _{n=1}^{\infty }n^{{1/2}^{n}}={\sqrt {1{\sqrt {2{\sqrt {3\cdots }}}}}}=1^{1/2}\;2^{1/4}\;3^{1/8}\cdots$	1999^{[Mw 79]}	?	?	?
Foias constant^[105]	$\alpha$	1.18745 23511 26501 05459^{[Mw 80]}^{[OEIS 92]}	$x_{n+1}=\left(1+{\frac {1}{x_{n}}}\right)^{n}{\text{ for }}n=1,2,3,\ldots$ Foias constant is the unique real number such that ifx₁ = α then the sequence diverges to infinity.	2000	?	?	?
Logarithmic capacity of the unit disk^[106]^[107]		0.59017 02995 08048 11302^{[Mw 81]}^{[OEIS 93]}	${\frac {\Gamma ({\tfrac {1}{4}})^{2}}{4\pi ^{3/2}}}={\frac {\varpi }{\pi {\sqrt {2}}}}$ where $\varpi$ is thelemniscate constant.	Before 2003^{[OEIS 93]}	✗	✗	?
Taniguchi constant^[82]	$C_{T}$	0.67823 44919 17391 97803^{[Mw 82]}^{[OEIS 94]}	$\prod _{p{\text{ prime}}}\left(1-{\frac {3}{p^{3}}}+{\frac {2}{p^{4}}}+{\frac {1}{p^{5}}}-{\frac {1}{p^{6}}}\right)$	Before 2005^[82]	?	?	?

Mathematical constants sorted by their representations as continued fractions

[edit]

The following list includes thecontinued fractions of some constants and is sorted by their representations. Continued fractions with more than 20 known terms have been truncated, with anellipsis to show that they continue. Rational numbers have two continued fractions; the version in this list is the shorter one. Decimal representations arerounded 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, …]^{[OEIS 95]}	E. Weisstein noted that the continued fraction has an unusually large number of 1s.^{[Mw 83]}
Cahen's constant	$C_{2}$	$\mathbb {R} \setminus \mathbb {A}$	0.64341 05463	[0; 1, 1, 1, 2², 3², 13², 129², 25298², 420984147², 269425140741515486², …]^{[OEIS 96]}	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^[108]	[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, 1, …]^[108]^{[OEIS 97]}	Using the continued fraction expansion, it was shown that ifγ is rational, its denominator must exceed 10²⁴⁴⁶⁶³.
Firstcontinued fraction constant	$C_{1}$	$\mathbb {R} \setminus \mathbb {A}$	0.69777 46579	[0; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, …]	Equal to the ratio $I_{1}(2)/I_{0}(2)$ ofmodified Bessel functions of the first kind evaluated at 2.
Catalan's constant	$G {\displaystyle G}$		0.91596 55942^[109]	[0; 1, 10, 1, 8, 1, 88, 4, 1, 1, 7, 22, 1, 2, 3, 26, 1, 11, 1, 10, 1, …]^[109]^{[OEIS 98]}	Computed up to4851389025 terms by E. Weisstein.^{[Mw 84]}
One half	⁠1/2⁠	$\mathbb {Q}$	0.50000 00000	[0; 2]
Prouhet–Thue–Morse constant	$\tau$	$\mathbb {R} \setminus \mathbb {A}$	0.41245 40336	[0; 2, 2, 2, 1, 4, 3, 5, 2, 1, 4, 2, 1, 5, 44, 1, 4, 1, 2, 4, 1, …]^{[OEIS 99]}	Infinitely many partial quotients are 4 or 5, and infinitely many partial quotients are greater than or equal to 50.^[110]
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, …]^{[OEIS 100]}	Computed up to1011597392 terms by E. Weisstein. He also noted that while theChampernowne constant continued fraction contains sporadic large terms, the continued fraction of the Copeland–Erdős Constant do not exhibit this property.^{[Mw 85]}
Base 10Champernowne constant	$C_{10}$	$\mathbb {R} \setminus \mathbb {A}$	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, …]^{[OEIS 101]}	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^[111]	[1; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …]^[112]	The convergents are ratios of successiveFibonacci 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, …]	Then^th roots of the denominators of then^th convergents are close toKhinchin's constant, suggesting that $B_{2}$ is irrational. If true, this will prove thetwin prime conjecture.^[113]
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 successivePell numbers.
Two	2	$\mathbb {N}$	2.00000 00000	[2; ]
Euler's number	$e {\displaystyle e}$	$\mathbb {R} \setminus \mathbb {A}$	2.71828 18285^[114]	[2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, …]^[115]^{[OEIS 102]}	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^[116]	[2; 1, 2, 5, 1, 1, 2, 1, 1, 3, 10, 2, 1, 3, 2, 24, 1, 3, 2, 3, 1, …]^[117]^{[OEIS 103]}	Foralmost all real numbersx, the coefficients of the continued fraction ofx have a finitegeometric mean known as Khinchin's constant.
Three	3	$\mathbb {N}$	3.00000 00000	[3; ]
Pi	$\pi$	$\mathbb {R} \setminus \mathbb {A}$	3.14159 26536	[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, …]^{[OEIS 104]}	The first few convergents (3, 22/7, 333/106, 355/113, ...) are among the best-known and most widely used historical approximations ofπ.

Sequences of constants

[edit]

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^{[Mw 86]}	$\gamma _{n}$	For a lattice L in Euclidean spaceRⁿ with unit covolume, i.e. vol(Rⁿ/L) = 1, let λ₁(L) denote the least length of a nonzero element of L. Then √γ_nn is the maximum of λ₁(L) over all such lattices L.	1822 to 1901	$\mathbb {R}$
Hafner–Sarnak–McCurley constant^[118]	$D(n)$	$D(n)=\prod _{k=1}^{\infty }\left\{1-\left[1-\prod _{j=1}^{n}(1-p_{k}^{-j})\right]^{2}\right\}$	1883^{[Mw 87]}	$\mathbb {R}$
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^[48]^{[Mw 88]}	$K_{r}$	${\frac {4}{\pi }}\sum _{n=0}^{\infty }\left({\frac {(-1)^{n}}{2n+1}}\right)^{\!r+1}={\frac {4}{\pi }}\left({\frac {(-1)^{0(r+1)}}{1^{r}}}+{\frac {(-1)^{1(r+1)}}{3^{r}}}+{\frac {(-1)^{2(r+1)}}{5^{r}}}+{\frac {(-1)^{3(r+1)}}{7^{r}}}+\cdots \right)$	1902 to 1965	$\mathbb {R}$
Generalized Brun's Constant^[56]	$B_{n}$	${\sum \limits _{p}\left({\frac {1}{p}}+{\frac {1}{p+n}}\right)}$ where the sum ranges over all primesp such thatp + n is also a prime	1919^{[OEIS 45]}	$\mathbb {R}$
Champernowne constants^[67]	$C_{b}$	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	$\mathbb {R} \setminus \mathbb {A}$
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	$\mathbb {A}$
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	$\mathbb {R} \setminus \mathbb {A}$
Beraha constants	$B(n)$	$2+2\cos \left({\frac {2\pi }{n}}\right)$	1974	$\mathbb {A}$
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	$G_{x}$	$\sum _{n=0}^{\infty }(-1)^{n}{\frac {1}{(2n+1)x^{2n+1}}}=\operatorname {arccot}(x)$ for rational x greater than or equal to one.	before 1996	$\mathbb {R} \setminus \mathbb {A}$
Metallic mean		${\frac {n+{\sqrt {n^{2}+4}}}{2}}$	before 1998	$\mathbb {A}$

Notes

[edit]

^Bothi and−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 ofi and−i is in some ways arbitrary, but a useful notational device. Seeimaginary unit for more information.

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[edit]

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^Rees, DG (1987),Foundations of Statistics, CRC Press, p. 246,ISBN 0-412-28560-6,Why 95% confidence? Why not some otherconfidence 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 fromthe original on 5 February 2008. Retrieved4 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.
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Bibliography

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Arndt, Jörg; Haenel, Christoph (2006).Pi Unleashed. Springer-Verlag.ISBN 978-3-540-66572-4. Retrieved2013-06-05. 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.

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