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## Summary

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 [Mw 1][OEIS 1] 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 [Mw 2][OEIS 2] 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 [Mw 3][OEIS 3] Positive root of $x^{2}=3$  465 to 398 BCE $\mathbb {A}$
Square root of 5 ${\sqrt {5}}$  2.23606 79774 99789 69640 [OEIS 4] Positive root of $x^{2}=5$  $\mathbb {A}$
Phi, Golden ratio $\varphi$  1.61803 39887 49894 84820 [Mw 4][OEIS 5] ${\frac {1+{\sqrt {5}}}{2}}$  ~300 BCE $\mathbb {A}$
Silver ratio $\delta _{S}$  2.41421 35623 73095 04880 [Mw 5][OEIS 6] ${\sqrt {2}}+1$  ~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 [Mw 6][OEIS 7] Real root of $x^{3}=2$  46 to 120 CE $\mathbb {A}$
Cube root of 3 ${\sqrt[{3}]{3}}$  1.44224 95703 07408 38232 [OEIS 8] Real root of $x^{3}=3$  $\mathbb {A}$
Twelfth root of 2 ${\sqrt[{12}]{2}}$  1.05946 30943 59295 26456 [OEIS 9] Real root of $x^{12}=2$  $\mathbb {A}$
Supergolden ratio $\psi$  1.46557 12318 76768 02665 [OEIS 10] ${\frac {1+{\sqrt[{3}]{\frac {29+3{\sqrt {93}}}{2}}}+{\sqrt[{3}]{\frac {29-3{\sqrt {93}}}{2}}}}{3}}$

Real root of $x^{3}=x^{2}+1$

$\mathbb {A}$
Imaginary unit $i$  0 + 1i Either of the two roots of $x^{2}=-1$ [nb 1] 1501 to 1576 $\mathbb {C}$
Connective constant for the hexagonal lattice $\mu$  1.84775 90650 22573 51225 [Mw 7][OEIS 11] ${\sqrt {2+{\sqrt {2}}}}$ , as a root of the polynomial $x^{4}-4x^{2}+2=0$  1593[OEIS 11] $\mathbb {A}$
Kepler–Bouwkamp constant $K'$  0.11494 20448 53296 20070 [Mw 8][OEIS 12] $\prod _{n=3}^{\infty }\cos \left({\frac {\pi }{n}}\right)=\cos \left({\frac {\pi }{3}}\right)\cos \left({\frac {\pi }{4}}\right)\cos \left({\frac {\pi }{5}}\right)...$  1596[OEIS 12]
Wallis's constant 2.09455 14815 42326 59148 [Mw 9][OEIS 13] ${\sqrt[{3}]{\frac {45-{\sqrt {1929}}}{18}}}+{\sqrt[{3}]{\frac {45+{\sqrt {1929}}}{18}}}$

Real root of $x^{3}-2x-5=0$

1616 to 1703 $\mathbb {A}$
Euler's number $e$  2.71828 18284 59045 23536 [Mw 10][OEIS 14] $\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}=\sum _{n=0}^{\infty }{\frac {1}{n!}}=1+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}\cdots$  1618 $\mathbb {T}$
Natural logarithm of 2 $\ln 2$  0.69314 71805 59945 30941 [Mw 11][OEIS 15] 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 [Mw 12][OEIS 16] $\pi \,{G}=4{\sqrt {\tfrac {2}{\pi }}}\,\Gamma {\left({\tfrac {5}{4}}\right)^{2}}={\tfrac {1}{4}}{\sqrt {\tfrac {2}{\pi }}}\,\Gamma {\left({\tfrac {1}{4}}\right)^{2}}$

where $G$  is Gauss's constant

1718 to 1798 $\mathbb {T}$
Euler's constant $\gamma$  0.57721 56649 01532 86060 [Mw 13][OEIS 17] $\lim _{n\to \infty }\left(-\log n+\sum _{k=1}^{n}{\frac {1}{k}}\right)=\int _{1}^{\infty }\left(-{\frac {1}{x}}+{\frac {1}{\lfloor x\rfloor }}\right)\,dx$  1735
Erdős–Borwein constant $E$  1.60669 51524 15291 76378 [Mw 14][OEIS 18] $\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}={\frac {1}{1}}\!+\!{\frac {1}{3}}\!+\!{\frac {1}{7}}\!+\!{\frac {1}{15}}\!+\!\cdots$  1749 $\mathbb {R} \setminus \mathbb {Q}$
Omega constant $\Omega$  0.56714 32904 09783 87299 [Mw 15][OEIS 19] $W(1)={\frac {1}{\pi }}\int _{0}^{\pi }\log \left(1+{\frac {\sin t}{t}}e^{t\cot t}\right)dt$

where W is the Lambert W function

1758 & 1783 $\mathbb {T}$
Apéry's constant $\zeta (3)$  1.20205 69031 59594 28539 [Mw 16][OEIS 20] $\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}={\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{3}}}+{\frac {1}{5^{3}}}+\cdots$  1780[OEIS 20] $\mathbb {R} \setminus \mathbb {Q}$
Laplace limit 0.66274 34193 49181 58097 [Mw 17][OEIS 21] 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 [Mw 18][OEIS 22] $\mathrm {li} (x)=\int _{0}^{x}{\frac {dt}{\ln t}}=0$ ; root of the logarithmic integral function. 1792[OEIS 22]
Gauss's constant $G$  0.83462 68416 74073 18628 [Mw 19][OEIS 23] ${\frac {1}{\mathrm {agm} (1,{\sqrt {2}})}}={\frac {\Gamma ({\frac {1}{4}})^{2}}{2{\sqrt {2\pi ^{3}}}}}={\frac {2}{\pi }}\int _{0}^{1}{\frac {dx}{\sqrt {1-x^{4}}}}$

where agm is the arithmetic–geometric mean

1799 $\mathbb {T}$
Second Hermite constant $\gamma _{2}$  1.15470 05383 79251 52901 [Mw 20][OEIS 24] ${\frac {2}{\sqrt {3}}}$  1822 to 1901 $\mathbb {A}$
Liouville's constant $L$  0.11000 10000 00000 00000 0001 [Mw 21][OEIS 25] $\sum _{n=1}^{\infty }{\frac {1}{10^{n!}}}={\frac {1}{10^{1!}}}+{\frac {1}{10^{2!}}}+{\frac {1}{10^{3!}}}+{\frac {1}{10^{4!}}}+\cdots$  Before 1844 $\mathbb {T}$
First continued fraction constant $C_{1}$  0.69777 46579 64007 98201 [Mw 22][OEIS 26] ${\tfrac {1}{1+{\tfrac {1}{2+{\tfrac {1}{3+{\tfrac {1}{4+{\tfrac {1}{5+\cdots }}}}}}}}}}$

${\frac {I_{1}(2)}{I_{0}(2)}}$ , 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 [Mw 23][OEIS 27]
$e^{\pi {\sqrt {163}}}$  1859 $\mathbb {T}$
Glaisher–Kinkelin constant $A$  1.28242 71291 00622 63687[Mw 24][OEIS 28] $e^{{\frac {1}{12}}-\zeta ^{\prime }(-1)}=e^{{\frac {1}{8}}-{\frac {1}{2}}\sum \limits _{n=0}^{\infty }{\frac {1}{n+1}}\sum \limits _{k=0}^{n}\left(-1\right)^{k}{\binom {n}{k}}\left(k+1\right)^{2}\ln(k+1)}$  1860[OEIS 28]
Catalan's constant $G$  0.91596 55941 77219 01505 [Mw 25][OEIS 29] $\int _{0}^{1}\!\!\int _{0}^{1}\!\!{\frac {dx\,dy}{1{+}x^{2}y^{2}}}=\!\sum _{n=0}^{\infty }\!{\frac {(-1)^{n}}{(2n{+}1)^{2}}}\!=\!{\frac {1}{1^{2}}}{-}{\frac {1}{3^{2}}}{+}{\cdots }$  1864
Dottie number 0.73908 51332 15160 64165 [Mw 26][OEIS 30] Real root of $\cos x=x$  1865[Mw 26] $\mathbb {T}$
Meissel–Mertens constant $M$  0.26149 72128 47642 78375 [Mw 27][OEIS 31] $\lim _{n\to \infty }\left(\sum _{p\leq n}{\frac {1}{p}}\ln \ln n\right)=\gamma +\sum _{p}\left(\ln \left(1-{\frac {1}{p}}\right)+{\frac {1}{p}}\right)$

where γ is the Euler–Mascheroni constant and p is prime

1866 & 1873
Universal parabolic constant $P$  2.29558 71493 92638 07403 [Mw 28][OEIS 32] $\ln(1+{\sqrt {2}})+{\sqrt {2}}\;=\;\operatorname {arsinh} (1)+{\sqrt {2}}$  Before 1891 $\mathbb {T}$
Cahen's constant $C$  0.64341 05462 88338 02618 [Mw 29][OEIS 33] $\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{s_{k}-1}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{6}}-{\frac {1}{42}}+{\frac {1}{1806}}{\,\pm \cdots }$

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 [Mw 30][OEIS 34] $(-1)^{-i}=i^{-2i}=\sum _{n=0}^{\infty }{\frac {\pi ^{n}}{n!}}=1+{\frac {\pi ^{1}}{1}}+{\frac {\pi ^{2}}{2}}+{\frac {\pi ^{3}}{6}}+\cdots$  1900 $\mathbb {T}$
Gelfond–Schneider constant $2^{\sqrt {2}}$  2.66514 41426 90225 18865 [Mw 31][OEIS 35] Before 1902[OEIS 35] $\mathbb {T}$
Second Favard constant $K_{2}$  1.23370 05501 36169 82735 [Mw 32][OEIS 36] ${\frac {\pi ^{2}}{8}}=\sum _{n=0}^{\infty }{\frac {1}{(2n-1)^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\cdots$  1902 to 1965 $\mathbb {T}$
Golden angle $g$  2.39996 32297 28653 32223 [Mw 33][OEIS 37] ${\frac {2\pi }{\varphi ^{2}}}=\pi (3-{\sqrt {5}})$  or

$180(3-{\sqrt {5}})=137.50776\ldots$  in degrees

1907 $\mathbb {T}$
Sierpiński's constant $K$  2.58498 17595 79253 21706 [Mw 34][OEIS 38] {\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 [Mw 35][OEIS 39] ${\frac {1}{\sqrt {2}}}\prod _{{p\equiv 3{\text{ mod }}4} \atop p\;{\rm {prime}}}{\left(1-{\frac {1}{p^{2}}}\right)^{-{\frac {1}{2}}}}\!\!={\frac {\pi }{4}}\prod _{{p\equiv 1{\text{ mod }}4} \atop p\;{\rm {prime}}}{\left(1-{\frac {1}{p^{2}}}\right)^{\frac {1}{2}}}$  1908[OEIS 39]
First NielsenRamanujan constant $a_{1}$  0.82246 70334 24113 21823 [Mw 36][OEIS 40] ${\frac {{\zeta }(2)}{2}}={\frac {\pi ^{2}}{12}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{2}}}={\frac {1}{1^{2}}}{-}{\frac {1}{2^{2}}}{+}{\frac {1}{3^{2}}}{-}{\frac {1}{4^{2}}}{+}\cdots$  1909 $\mathbb {T}$
Gieseking constant $G$  1.01494 16064 09653 62502 [Mw 37][OEIS 41] ${\frac {3{\sqrt {3}}}{4}}\left(1-\sum _{n=0}^{\infty }{\frac {1}{(3n+2)^{2}}}+\sum _{n=1}^{\infty }{\frac {1}{(3n+1)^{2}}}\right)=$

$\textstyle {\frac {3{\sqrt {3}}}{4}}\left(1-{\frac {1}{2^{2}}}+{\frac {1}{4^{2}}}-{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}-{\frac {1}{8^{2}}}+{\frac {1}{10^{2}}}\pm \cdots \right)$ .

1912
Bernstein's constant $\beta$  0.28016 94990 23869 13303 [Mw 38][OEIS 42] $\lim _{n\to \infty }2nE_{2n}(f)$ , 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 [Mw 39][OEIS 43] ${\textstyle {\frac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}={\frac {1+4\cosh \left({\frac {1}{3}}\cosh ^{-1}\left(2+{\frac {3}{8}}\right)\right)}{3}}}$

Real root of $x^{3}-x^{2}-x-1=0$

1914 to 1963 $\mathbb {A}$
Brun's constant $B_{2}$  1.90216 05831 04 [Mw 40][OEIS 44] $\textstyle {\sum \limits _{p}({\frac {1}{p}}+{\frac {1}{p+2}})}=({\frac {1}{3}}\!+\!{\frac {1}{5}})+({\tfrac {1}{5}}\!+\!{\tfrac {1}{7}})+({\tfrac {1}{11}}\!+\!{\tfrac {1}{13}})+\cdots$

where the sum ranges over all primes p such that p + 2 is also a prime

1919[OEIS 44]
Twin primes constant $C_{2}$  0.66016 18158 46869 57392 [Mw 41][OEIS 45] $\prod _{\textstyle {p\;{\rm {prime}} \atop p\geq 3}}\left(1-{\frac {1}{(p-1)^{2}}}\right)$  1922
Plastic number $\rho$  1.32471 79572 44746 02596 [Mw 42][OEIS 46] ${\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\cdots }}}}}}=\textstyle {\sqrt[{3}]{{\frac {1}{2}}+{\frac {\sqrt {69}}{18}}}}+{\sqrt[{3}]{{\frac {1}{2}}-{\frac {\sqrt {69}}{18}}}}$

Real root of $x^{3}=x+1$

1924[OEIS 46] $\mathbb {A}$
Bloch's constant $B$  $0.4332\leq B\leq 0.4719$  [Mw 43][OEIS 47] 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 47]
Z score for the 97.5 percentile point $z_{.975}$  1.95996 39845 40054 23552 [Mw 44][OEIS 48] ${\sqrt {2}}\operatorname {erf} ^{-1}(0.95)$  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  [Mw 45][OEIS 49] The best known bounds are $0.5  1929
Landau's third constant $A$  $0.5  1929
Prouhet–Thue–Morse constant $\tau$  0.41245 40336 40107 59778 [Mw 46][OEIS 50] $\sum _{n=0}^{\infty }{\frac {t_{n}}{2^{n+1}}}={\frac {1}{4}}\left[2-\prod _{n=0}^{\infty }\left(1-{\frac {1}{2^{2^{n}}}}\right)\right]$

where ${t_{n}}$  is the nth term of the Thue–Morse sequence

1929[OEIS 50] $\mathbb {T}$
Golomb–Dickman constant $\lambda$  0.62432 99885 43550 87099 [Mw 47][OEIS 51] $\int _{0}^{1}e^{\mathrm {Li} (t)}dt=\int _{0}^{\infty }{\frac {\rho (t)}{t+2}}dt$

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 [Mw 48][OEIS 52] $\lim _{n\to \infty }\!\!\left(\!{L_{n}{-}{\frac {4}{\pi ^{2}}}\ln(2n{+}1)}\!\!\right)\!{=}{\frac {4}{\pi ^{2}}}\!\left({\sum _{k=1}^{\infty }\!{\frac {2\ln k}{4k^{2}{-}1}}}{-}{\frac {\Gamma '({\tfrac {1}{2}})}{\Gamma ({\tfrac {1}{2}})}}\!\!\right)$  1930[Mw 48]
Feller–Tornier constant ${\mathcal {C}}_{\mathrm {FT} }$  0.66131 70494 69622 33528 [Mw 49][OEIS 53] ${{\frac {1}{2}}\prod _{p{\text{ prime}}}\left(1-{\frac {2}{p^{2}}}\right)+{\frac {1}{2}}}={\frac {3}{\pi ^{2}}}\prod _{p{\text{ prime}}}\left(1-{\frac {1}{p^{2}-1}}\right)+{\frac {1}{2}}$  1932
Base 10 Champernowne constant $C_{10}$  0.12345 67891 01112 13141 [Mw 50][OEIS 54] 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 [Mw 51][OEIS 55] Largest real root of $x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1=0$  1933[OEIS 55] $\mathbb {A}$
Khinchin's constant $K_{0}$  2.68545 20010 65306 44530 [Mw 52][OEIS 56] $\prod _{n=1}^{\infty }\left[{1+{1 \over n(n+2)}}\right]^{\log _{2}(n)}$  1934
Lévy's constant (1) $\beta$  1.18656 91104 15625 45282 [Mw 53][OEIS 57] ${\frac {\pi ^{2}}{12\,\ln 2}}$  1935
Lévy's constant (2) $e^{\beta }$  3.27582 29187 21811 15978 [Mw 54][OEIS 58] $e^{\pi ^{2}/(12\ln 2)}$  1936
Copeland–Erdős constant ${\mathcal {C}}_{CE}$  0.23571 11317 19232 93137 [Mw 55][OEIS 59] Defined by concatenating representations of successive prime numbers:

0.2 3 5 7 11 13 17 19 23 29 31 37 ...

1946[OEIS 59] $\mathbb {R} \setminus \mathbb {Q}$
Mills' constant $A$  1.30637 78838 63080 69046 [Mw 56][OEIS 60] 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 [Mw 57][OEIS 61] $\int _{0}^{\infty }\!\!{\frac {e^{-x}}{1+x}}\,dn=\!\!\int _{0}^{1}\!\!{\frac {dx}{1-\ln x}}={\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {2}{1+{\tfrac {2}{1+{\tfrac {3}{1+3{/\cdots }}}}}}}}}}}}}$  Before 1948[OEIS 61]
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 [OEIS 62] Before 1958[OEIS 63] $\mathbb {R} \setminus \mathbb {Q}$
Magic angle $\theta _{\mathrm {m} }$  0.95531 66181 245092 78163 [OEIS 64] $\arctan {\sqrt {2}}=\arccos {\tfrac {1}{\sqrt {3}}}\approx \textstyle {54.7356}^{\circ }$  Before 1959 $\mathbb {T}$
Artin's constant $C_{\mathrm {Artin} }$  0.37395 58136 19202 28805 [Mw 58][OEIS 65] $\prod _{p{\text{ prime}}}\left(1-{\frac {1}{p(p-1)}}\right)$  Before 1961[OEIS 65]
Porter's constant $C$  1.46707 80794 33975 47289 [Mw 59][OEIS 66] ${\frac {6\ln 2}{\pi ^{2}}}\left(3\ln 2+4\,\gamma -{\frac {24}{\pi ^{2}}}\,\zeta '(2)-2\right)-{\frac {1}{2}}$

where γ is the Euler–Mascheroni constant and ζ '(2) is the derivative of the Riemann zeta function evaluated at s = 2

1961[OEIS 66]
Lochs constant $L$  0.97027 01143 92033 92574 [Mw 60][OEIS 67] ${\frac {6\ln 2\ln 10}{\pi ^{2}}}$  1964
DeVicci's tesseract constant 1.00743 47568 84279 37609 [OEIS 68] 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[OEIS 68] $\mathbb {A}$
Lieb's square ice constant 1.53960 07178 39002 03869 [Mw 61][OEIS 69] $\left({\frac {4}{3}}\right)^{\frac {3}{2}}={\frac {8}{3{\sqrt {3}}}}$  1967 $\mathbb {A}$
Niven's constant $C$  1.70521 11401 05367 76428 [Mw 62][OEIS 70] $1+\sum _{n=2}^{\infty }\left(1-{\frac {1}{\zeta (n)}}\right)$  1969
Stephens' constant 0.57595 99688 92945 43964 [Mw 63][OEIS 71] $\prod _{p{\text{ prime}}}\left(1-{\frac {p}{p^{3}-1}}\right)$  1969[OEIS 71]
Regular paperfolding sequence $P$  0.85073 61882 01867 26036 [Mw 64][OEIS 72] $\sum _{n=0}^{\infty }{\frac {8^{2^{n}}}{2^{2^{n+2}}-1}}=\sum _{n=0}^{\infty }{\cfrac {\tfrac {1}{2^{2^{n}}}}{1-{\tfrac {1}{2^{2^{n+2}}}}}}$  1970[OEIS 72] $\mathbb {T}$
Reciprocal Fibonacci constant $\psi$  3.35988 56662 43177 55317 [Mw 65][OEIS 73] $\sum _{n=1}^{\infty }{\frac {1}{F_{n}}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{8}}+{\frac {1}{13}}+\cdots$

where Fn is the nth Fibonacci number

1974[OEIS 73] $\mathbb {R} \setminus \mathbb {Q}$
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 [Mw 66][OEIS 74] $\lim _{n\to \infty }{\frac {x_{n+1}-x_{n}}{x_{n+2}-x_{n+1}}}$

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.
[Mw 67][OEIS 75]
$\sum _{p\in P}2^{-|p|}$
• p: Halted program
• |p|: Size in bits of program p
• P: Domain of all programs that stop.
1975 $\mathbb {T}$
Robbins constant $\Delta (3)$  0.66170 71822 67176 23515 [Mw 68][OEIS 76] ${\frac {4\!+\!17{\sqrt {2}}\!-6{\sqrt {3}}\!-7\pi }{105}}\!+\!{\frac {\ln(1\!+\!{\sqrt {2}})}{5}}\!+\!{\frac {2\ln(2\!+\!{\sqrt {3}})}{5}}$  1977[OEIS 76] $\mathbb {T}$
Weierstrass constant  0.47494 93799 87920 65033 [Mw 69][OEIS 77] ${\frac {2^{5/4}{\sqrt {\pi }}\,e^{\pi /8}}{\Gamma ({\frac {1}{4}})^{2}}}$  Before 1978 $\mathbb {T}$
Fransén–Robinson constant $F$  2.80777 02420 28519 36522 [Mw 70][OEIS 78] $\int _{0}^{\infty }{\frac {dx}{\Gamma (x)}}=e+\int _{0}^{\infty }{\frac {e^{-x}}{\pi ^{2}+\ln ^{2}x}}\,dx$  1978
Fractal dimension of the Cantor set $\log _{3}2$  0.63092 97535 71457 43709 [Mw 71][OEIS 79] ${\frac {\log 2}{\log 3}}$  Before 1979[OEIS 79] $\mathbb {T}$
Feigenbaum constant α $\alpha$  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 a bifurcation diagram 1979
Second du Bois-Reymond constant $C_{2}$  0.19452 80494 65325 11361 [Mw 72][OEIS 81] ${\frac {e^{2}-7}{2}}=\int _{0}^{\infty }\left|{{\frac {d}{dt}}\left({\frac {\sin t}{t}}\right)^{n}}\right|\,dt-1$  1983[OEIS 81] $\mathbb {T}$
Erdős–Tenenbaum–Ford constant $\delta$  0.86071 33205 59342 06887 [OEIS 82] $1-{\frac {1+\log \log 2}{\log 2}}$  1984
Conway's constant $\lambda$  1.30357 72690 34296 39125 [Mw 73][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 $\mathbb {A}$
Fractal dimension of the Apollonian packing of circles

1.30568 6729 ≈ by Thomas & Dhar
1.30568 8 ≈ by McMullen [Mw 74][OEIS 84]
1991[OEIS 84]
Hafner–Sarnak–McCurley constant $\sigma$  0.35323 63718 54995 98454 [Mw 75][OEIS 85] $\prod _{p{\text{ prime}}}{\left(1-\left(1-\prod _{n\geq 1}\left(1-{\frac {1}{p^{n}}}\right)\right)^{2}\right)}\!$  1991[OEIS 85]
Hausdorff dimension of the Sierpinski triangle $\log _{2}3$  1.58496 25007 21156 18145 [Mw 76][OEIS 86] ${\frac {\log 3}{\log 2}}$  1993[OEIS 86] $\mathbb {T}$
Backhouse's constant $B$  1.45607 49485 82689 67139 [Mw 77][OEIS 87] $\lim _{k\to \infty }\left|{\frac {q_{k+1}}{q_{k}}}\right\vert \quad {\text{where:}}\displaystyle \;\;Q(x)={\frac {1}{P(x)}}=\!\sum _{k=1}^{\infty }q_{k}x^{k}$

$P(x)=1+\sum _{k=1}^{\infty }{p_{k}x^{k}}=1+2x+3x^{2}+5x^{3}+\cdots$ where pk is the kth prime number

1995
Viswanath constant 1.13198 82487 943 [Mw 78][OEIS 88] $\lim _{n\to \infty }|f_{n}|^{\frac {1}{n}}$       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 [Mw 79][OEIS 89] 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 [Mw 80][OEIS 90] $\prod _{p{\text{ prime}}}\left(1-{\frac {1}{p}}\right)^{7}\left(1+{\frac {7p+1}{p^{2}}}\right)$  1999[OEIS 90]
MRB constant $S$  0.18785 96424 62067 12024 [Mw 81][Ow 1][OEIS 91] $\sum _{n=1}^{\infty }(-1)^{n}(n^{1/n}-1)=-{\sqrt[{1}]{1}}+{\sqrt[{2}]{2}}-{\sqrt[{3}]{3}}+\cdots$  1999
Prime constant $\rho$  0.41468 25098 51111 66024 [OEIS 92] $\sum _{p{\text{ prime}}}{\frac {1}{2^{p}}}={\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{32}}+\cdots$  1999[OEIS 92] $\mathbb {R} \setminus \mathbb {Q}$
Somos' quadratic recurrence constant $\sigma$  1.66168 79496 33594 12129 [Mw 82][OEIS 93] $\prod _{n=1}^{\infty }n^{{1/2}^{n}}={\sqrt {1{\sqrt {2{\sqrt {3\cdots }}}}}}=1^{1/2}\;2^{1/4}\;3^{1/8}\cdots$  1999[Mw 82]
Foias constant $\alpha$  1.18745 23511 26501 05459 [Mw 83][OEIS 94] $x_{n+1}=\left(1+{\frac {1}{x_{n}}}\right)^{n}{\text{ for }}n=1,2,3,\ldots$

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 [Mw 84][OEIS 95] ${\frac {\Gamma ({\tfrac {1}{4}})^{2}}{4\pi ^{3/2}}}$  Before 2003[OEIS 95] $\mathbb {T}$
Taniguchi constant 0.67823 44919 17391 97803 [Mw 85][OEIS 96] $\prod _{p{\text{ prime}}}\left(1-{\frac {3}{p^{3}}}+{\frac {2}{p^{4}}}+{\frac {1}{p^{5}}}-{\frac {1}{p^{6}}}\right)$  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, …] [OEIS 97] E. Weisstein noted that the continued fraction has an unusually large number of 1s.[Mw 86]
Cahen's constant $C_{2}$  $\mathbb {T}$  0.64341 05463 [0; 1, 1, 1, 22, 32, 132, 1292, 252982, 4209841472, 2694251407415154862, …] [OEIS 98] 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, …] [OEIS 99] 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, …] [OEIS 100] Computed up to 4851389025 terms by E. Weisstein.[Mw 87]
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, …] [OEIS 101] 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, …] [OEIS 102] 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.[Mw 88]
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, …] [OEIS 103] 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, …] [OEIS 104] 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, …] [OEIS 105] 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, …] [OEIS 106] The first few convergents (3, 22/7, 333/106, 355/113, ...) are among the best-known and most widely used historical approximations of π.