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.

Rational

Main article: Rational number

Name	Symbol	Decimal expansion	Formula
1.96	1.96	1.96
Belphegor's prime		1000000000000066600000000000001	$10^{30}+666\times 10^{14}+1$
Googol	10100	1000000...	$10^{100}$
Googolplex	1010100	1000000...	$10^{10^{100}}$
Graham's number	$G_{64}$	...67018485186439059104575627262464195387	$g_{64}$ where $g_{n}=3{\uparrow }^{g_{n-1}}3$ and $g_{1}=3\uparrow \uparrow \uparrow \uparrow 3$ using Knuth's up-arrow notation
Hardy–Ramanujan number	1729	1729	$1^{3}+12^{3}=9^{3}+10^{3}$
Kaprekar's constant	6174	6174
Milü	355/113	3.14159292035398230088...	${\frac {355}{113}}$
Negative one	−1	−1	$1-2$
One	1	1	None
One half	1/2	0.5	$1/2$
Rayo's number	$\operatorname {Rayo} (10^{100})$	Too large to calculate	The smallest number bigger than any finite number named by an expression in the language of first-order set theory with a googol symbols or less.
Two	2	2	$1+1$
Zero	0	0	The additive identity: $x+0=x$

Irrational Algebraic

Main article: Algebraic number

Name	Symbol	Decimal expansion	Formula
Connective constant	${\mu }$	1.84775 90650 22573 51225	${\sqrt {2+{\sqrt {2}}}}\;=\lim _{n\rightarrow \infty }c_{n}^{1/n}$ as a root of the polynomial $:\;x^{4}-4x^{2}+2=0$
Conway constant	${\lambda }$	1.30357 72690 34296 39125	${\begin{smallmatrix}x^{71}-x^{69}-2x^{68}-x^{67}+2x^{66}+2x^{65}+x^{64}-x^{63}-x^{62}-x^{61}-x^{60}\\-x^{59}+2x^{58}+5x^{57}+3x^{56}-2x^{55}-10x^{54}-3x^{53}-2x^{52}+6x^{51}+6x^{50}\\+x^{49}+9x^{48}-3x^{47}-7x^{46}-8x^{45}-8x^{44}+10x^{43}+6x^{42}+8x^{41}-5x^{40}\\-12x^{39}+7x^{38}-7x^{37}+7x^{36}+x^{35}-3x^{34}+10x^{33}+x^{32}-6x^{31}-2x^{30}\\-10x^{29}-3x^{28}+2x^{27}+9x^{26}-3x^{25}+14x^{24}-8x^{23}-7x^{21}+9x^{20}\\+3x^{19}\!-4x^{18}\!-10x^{17}\!-7x^{16}\!+12x^{15}\!+7x^{14}\!+2x^{13}\!-12x^{12}\!-4x^{11}\!-2x^{10}\\+5x^{9}+x^{7}-7x^{6}+7x^{5}-4x^{4}+12x^{3}-6x^{2}+3x-6\ =\ 0\quad \quad \quad \end{smallmatrix}}$
Cube root of 2 (Delian Constant)	${\sqrt[{3}]{2}}$	1.25992 10498 94873 16476	Real root of $x^{3}=2$
Cube root of 3	${\sqrt[{3}]{3}}$	1.44224 95703 07408 38232	Real root of $x^{3}=3$
DeVicci's tesseract constant	${f_{(3,4)}}$	1.00743 47568 84279 37609	The largest cube that can pass through in an 4D hypercube. Positive root of $4x^{8}{-}28x^{6}{-}7x^{4}{+}16x^{2}{+}16=0$
Phi, Golden ratio	${\varphi }$	1.61803 39887 49894 84820	${\frac {1+{\sqrt {5}}}{2}}$
Golden ratio Conjugate	${\varphi }^{-1}$	0.61803 39887 49894 84820	${\frac {1-{\sqrt {5}}}{2}}$
Hermite constant	$\gamma _{_{2}}$	1.15470 05383 79251 52901	${\frac {2}{\sqrt {3}}}={\frac {1}{\cos \,({\frac {\pi }{6}})}}$
Lehmer's conjecture constant	${\sigma _{_{10}}}$	1.17628 08182 59917 50654	$x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1=0$
Lieb's square ice constant	${W}_{2D}$	1.53960 07178 39002 03869	$\lim _{n\to \infty }(f(n))^{n^{-2}}=\left({\frac {4}{3}}\right)^{\frac {3}{2}}={\frac {8}{3{\sqrt {3}}}}$
Plastic number	${\rho }$	1.32471 79572 44746 02596	${\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\cdots }}}}}}=\textstyle {\sqrt[{3}]{{\frac {1}{2}}+{\frac {\sqrt {69}}{18}}}}+{\sqrt[{3}]{{\frac {1}{2}}-{\frac {\sqrt {69}}{18}}}}$
Ramanujan nested radical	$R_{5}$	2.74723 82749 32304 33305	$\scriptstyle {\sqrt {5+{\sqrt {5+{\sqrt {5-{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5-\cdots }}}}}}}}}}}}}}\;=\textstyle {\frac {2+{\sqrt {5}}+{\sqrt {15-6{\sqrt {5}}}}}{2}}$
Silver ratio	$\delta _{S}$	2.41421 35623 73095 04880	${\sqrt {2}}+1$
Square root of 2, Pythagoras constant.	${\sqrt {2}}$	1.41421 35623 73095 04880	Positive root of $x^{2}=2$
Square root of 3, Theodorus' constant	${\sqrt {3}}$	1.73205 08075 68877 29352	Positive root of $x^{2}=3$
Square root of 5	${\sqrt {5}}$	2.23606 79774 99789 69640	Positive root of $x^{2}=5$
Supergolden ratio	$\psi$	1.46557123187676802665	${\frac {1+{\sqrt[{3}]{\frac {29+3{\sqrt {93}}}{2}}}+{\sqrt[{3}]{\frac {29-3{\sqrt {93}}}{2}}}}{3}}$
Tribonacci Constant	$t_{3}$	1.83928 67552 14161	${\frac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}={\frac {1+4\cosh \left({\frac {1}{3}}\cosh ^{-1}\left(2+{\frac {3}{8}}\right)\right)}{3}}$
Twelfth root of 2	${\sqrt[{12}]{2}}$	1.05946 30943 59295 26456	Real root of $x^{12}=2$
Wallis Constant	$W$	2.09455 14815 42326 59148	${\sqrt[{3}]{\frac {45-{\sqrt {1929}}}{18}}}+{\sqrt[{3}]{\frac {45+{\sqrt {1929}}}{18}}}$

Irrational but not known to be transcendental

Main article: Rational number

Name	Symbol	Decimal expansion	Formula
Apéry's constant	$\zeta (3)$	1.20205 69031 59594 28539	$\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}={\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{3}}}+{\frac {1}{5^{3}}}+\cdots =$ ${\frac {1}{2}}\sum _{n=1}^{\infty }{\frac {H_{n}}{n^{2}}}={\frac {1}{2}}\sum _{i=1}^{\infty }\sum _{j=1}^{\infty }{\frac {1}{ij(i{+}j)}}=\!\!\int _{0}^{1}\!\!\int _{0}^{1}\!\!\int _{0}^{1}{\frac {dx\,dy\,dz}{1-xyz}}$
Copeland–Erdős constant	${{\mathcal {C}}_{CE}}$	0.23571 11317 19232 93137	$\sum _{n=1}^{\infty }{\frac {p_{n}}{10^{n+\sum \limits _{k=1}^{n}\lfloor \log _{10}{p_{k}}\rfloor }}}$
Erdős–Borwein constant	${E}_{\,B}$	1.60669 51524 15291 76378	$\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {1}{2^{mn}}}=\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}={\frac {1}{1}}\!+\!{\frac {1}{3}}\!+\!{\frac {1}{7}}\!+\!{\frac {1}{15}}\!+\!\cdots$
Prime constant	$\rho$	0.41468 25098 51111 66024	$\sum _{p}{\frac {1}{2^{p}}}={\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{32}}+\cdots$
Prévost constant, Reciprocal Fibonacci constant	$\Psi$	3.35988 56662 43177 55317	$\sum _{n=1}^{\infty }{\frac {1}{F_{n}}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{8}}+{\frac {1}{13}}+\cdots$ Fn: Fibonacci series
Van der Pauw constant	${\frac {\pi }{\ln 2}}$	4.53236 01418 27193 80962	${\frac {\sum \limits _{n=0}^{\infty }{\frac {4(-1)^{n}}{2n+1}}}{\sum \limits _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}}}={\frac {{\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-\cdots }{{\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots }}$

Transcendental

Main article: Transcendental number

Name	Symbol	Decimal expansion	Formula
Pi	$\pi$	3.14159 26535 89793 23846	Ratio of a circle's circumference to its diameter.
Euler's number	${e}$	2.71828 18284 59045 23536	$\lim _{n\to \infty }\!\left(\!1\!+\!{\frac {1}{n}}\right)^{n}$
Natural logarithm of 2	$\ln 2$	0.69314 71805 59945 30941	$\sum _{n=1}^{\infty }{\frac {1}{n2^{n}}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+\cdots$
Lemniscate constant	${\varpi }$	2.62205 75542 92119 81046	$\pi \,{G}=4{\sqrt {\tfrac {2}{\pi }}}\,\Gamma {\left({\tfrac {5}{4}}\right)^{2}}={\tfrac {1}{4}}{\sqrt {\tfrac {2}{\pi }}}\,\Gamma {\left({\tfrac {1}{4}}\right)^{2}}=4{\sqrt {\tfrac {2}{\pi }}}\left({\tfrac {1}{4}}!\right)^{2}$
Omega constant	$\Omega$	0.56714329040978387299	$\Omega =W(1)={\frac {1}{\pi }}\int _{0}^{\pi }\log \left(1+{\frac {\sin t}{t}}e^{t\cot t}\right)dt$ Where W is the Lambert W function
Laplace limit	${\lambda }$	0.66274 34193 49181 58097	${\frac {x\;e^{\sqrt {x^{2}+1}}}{{\sqrt {x^{2}+1}}+1}}=1$
Gauss's constant	${G}$	0.83462 68416 74073 18628	${\frac {1}{\mathrm {agm} (1,{\sqrt {2}})}}={\frac {4{\sqrt {2}}\,\left({\tfrac {1}{4}}!\right)^{2}}{\pi ^{3}{2}}}={\frac {2}{\pi }}\int _{0}^{1}{\frac {dx}{\sqrt {1-x^{4}}}}$ where agm is the arithmetic–geometric mean
Liouville number	${\text{£}}_{Li}$	0.11000 10000 00000 00000 0001	$\sum _{n=1}^{\infty }{\frac {1}{10^{n!}}}={\frac {1}{10^{1!}}}+{\frac {1}{10^{2!}}}+{\frac {1}{10^{3!}}}+{\frac {1}{10^{4!}}}+\cdots$
Hermite–Ramanujan constant	${R}$	262 53741 26407 68743 .99999 99999 99250 073	$e^{\pi {\sqrt {163}}}$
Dottie number	$d$	0.73908 51332 15160 64165	$\lim _{x\to \infty }\cos ^{[x]}(c)=\lim _{x\to \infty }\underbrace {\cos(\cos(\cos(\cdots (\cos(c)))))} _{x}$
Hafner–Sarnak–McCurley constant (2)	${\frac {1}{\zeta (2)}}$	0.60792 71018 54026 62866	${\frac {6}{\pi ^{2}}}=\prod _{n=0}^{\infty }{\!\left(\!1-{\frac {1}{{p_{n}}^{2}}}\!\right)}\!=\!\textstyle \left(1\!-\!{\frac {1}{2^{2}}}\right)\!\left(1\!-\!{\frac {1}{3^{2}}}\right)\!\left(1\!-\!{\frac {1}{5^{2}}}\right)\cdots$where pn is a prime
Universal parabolic constant	${P}_{\,2}$	2.29558 71493 92638 07403	$\ln(1+{\sqrt {2}})+{\sqrt {2}}\;=\;\operatorname {arsinh} (1)+{\sqrt {2}}$
Cahen's constant	$\xi _{2}$	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, ... Defined as: $S_{0}=\,2,\,\,S_{k}=\,1+\prod _{n=0}^{k-1}S_{n}{\text{ for }}k>0$
Gelfond's constant	${e}^{\pi }$	23.14069 26327 79269 0057	$(-1)^{-i}=i^{-2i}=\sum _{n=0}^{\infty }{\frac {\pi ^{n}}{n!}}=1+{\frac {\pi ^{1}}{1}}+{\frac {\pi ^{2}}{2}}+{\frac {\pi ^{3}}{6}}+\cdots$
Favard constant	${\tfrac {3}{4}}\zeta (2)$	1.23370 05501 36169 82735	${\frac {\pi ^{2}}{8}}=\sum _{n=0}^{\infty }{\frac {1}{(2n-1)^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\cdots$
Golden angle	${b}$	2.39996 32297 28653 32223	$(4-2\,\Phi )\,\pi =(3-{\sqrt {5}})\,\pi$ = 137.5077640500378546 ...°
Nielsen–Ramanujan constant	${\frac {{\zeta }(2)}{2}}$	0.82246 70334 24113 21823	${\frac {\pi ^{2}}{12}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{2}}}={\frac {1}{1^{2}}}{-}{\frac {1}{2^{2}}}{+}{\frac {1}{3^{2}}}{-}{\frac {1}{4^{2}}}{+}{\frac {1}{5^{2}}}{-}\cdots$
Base 10 Champernowne constant	$C_{10}$	0.12345 67891 01112 13141	$\sum _{n=1}^{\infty }\;\sum _{k=10^{n-1}}^{10^{n}-1}{\frac {k}{10^{kn-9\sum _{j=0}^{n-1}10^{j}(n-j-1)}}}$
Gelfond–Schneider constant	$G_{\,GS}$	2.66514 41426 90225 18865	$2^{\sqrt {2}}$
Magic angle	${\theta _{m}}$	0.95531 66181 245092 78163	$\arctan {\sqrt {2}}=\arccos {\tfrac {1}{\sqrt {3}}}\approx \textstyle {54.7356}^{\circ }$
Baker constant	$\beta _{3}$	0.83564 88482 64721 05333	$\int _{0}^{1}{\frac {\mathrm {d} t}{1+t^{3}}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{3n+1}}={\frac {1}{3}}\left(\ln 2+{\frac {\pi }{\sqrt {3}}}\right)$
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
Robbins constant	$\Delta (3)$	0.66170 71822 67176 23515	${\frac {4\!+\!17{\sqrt {2}}\!-6{\sqrt {3}}\!-7\pi }{105}}\!+\!{\frac {\ln(1\!+\!{\sqrt {2}})}{5}}\!+\!{\frac {2\ln(2\!+\!{\sqrt {3}})}{5}}$
Fractal dimension of the Cantor set	$d_{f}(k)$	0.63092 97535 71457 43709	$\lim _{\varepsilon \to 0}{\frac {\log N(\varepsilon )}{\log(1/\varepsilon )}}={\frac {\log 2}{\log 3}}$
Komornik–Loreti constant	${q}$	1.78723 16501 82965 93301	$1=\!\sum _{n=1}^{\infty }{\frac {t_{k}}{q^{k}}}\qquad \scriptstyle {\text{Raiz real de}}\displaystyle \prod _{n=0}^{\infty }\!\left(\!1{-}{\frac {1}{q^{2^{n}}}}\!\right)\!{+}{\frac {q{-}2}{q{-}1}}=0$ tk = Thue–Morse sequence
Hausdorff dimension, Sierpinski triangle	${\log _{2}3}$	1.58496 25007 21156 18145	${\frac {\log 3}{\log 2}}={\frac {\sum _{n=0}^{\infty }{\frac {1}{2^{2n+1}(2n+1)}}}{\sum _{n=0}^{\infty }{\frac {1}{3^{2n+1}(2n+1)}}}}={\frac {{\frac {1}{2}}+{\frac {1}{24}}+{\frac {1}{160}}+\cdots }{{\frac {1}{3}}+{\frac {1}{81}}+{\frac {1}{1215}}+\cdots }}$
2nd du Bois-Reymond constant	${C_{2}}$	0.19452 80494 65325 11361	${\frac {e^{2}-7}{2}}=\int _{0}^{\infty }\left|{{\frac {d}{dt}}\left({\frac {\sin t}{t}}\right)^{n}}\right|\,dt-1$
Prouhet–Thue–Morse constant	$\tau$	0.41245 40336 40107 59778	$\sum _{n=0}^{\infty }{\frac {t_{n}}{2^{n+1}}}$    where ${t_{n}}$ is the Thue–Morse sequence  and Where $\tau (x)=\sum _{n=0}^{\infty }(-1)^{t_{n}}\,x^{n}=\prod _{n=0}^{\infty }(1-x^{2^{n}})$

Real unknown rationality

Main article: Real number

Name	Symbol	Decimal expansion	Formula
Sophomore's dream1 J.Bernoulli	${I}_{1}$	0.78343 05107 12134 40705	$\int _{0}^{1}\!x^{x}\,dx=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{n}}}={\frac {1}{1^{1}}}-{\frac {1}{2^{2}}}+{\frac {1}{3^{3}}}-\cdots$
Sophomore's dream2 J.Bernoulli	${I}_{2}$	1.29128 59970 62663 54040	$\int _{0}^{1}\!{\frac {dx}{x^{x}}}=\sum _{n=1}^{\infty }{\frac {1}{n^{n}}}={\frac {1}{1^{1}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{4}}}+\cdots$
Euler–Mascheroni constant	${\gamma }$	0.57721 56649 01532 86060	${\begin{aligned}&\sum _{n=1}^{\infty }\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2^{n}+k}}=\sum _{n=1}^{\infty }\left({\frac {1}{n}}-\ln \left(1+{\frac {1}{n}}\right)\right)\\&=\int _{0}^{1}-\ln \left(\ln {\frac {1}{x}}\right)\,dx=-\Gamma '(1)=-\Psi (1)\end{aligned}}$
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.
Glaisher–Kinkelin constant	${A}$	1.28242 71291 00622 63687	$e^{{\frac {1}{12}}-\zeta ^{\prime }(-1)}=e^{{\frac {1}{8}}-{\frac {1}{2}}\sum \limits _{n=0}^{\infty }{\frac {1}{n+1}}\sum \limits _{k=0}^{n}\left(-1\right)^{k}{\binom {n}{k}}\left(k+1\right)^{2}\ln(k+1)}$
Catalan's constant	${C}$	0.91596 55941 77219 01505	$\int _{0}^{1}\!\!\int _{0}^{1}\!\!{\frac {dx\,dy}{1{+}x^{2}y^{2}}}=\!\sum _{n=0}^{\infty }\!{\frac {(-1)^{n}}{(2n{+}1)^{2}}}\!=\!{\frac {1}{1^{2}}}{-}{\frac {1}{3^{2}}}{+}{\cdots }$
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 Euler's constant and p is a prime
Weierstrass constant	$\sigma ({\tfrac {1}{2}})$	0.47494 93799 87920 65033	${\frac {e^{{\pi }/{8}}{\sqrt {\pi }}}{4\cdot 2^{3/4}{\left({\frac {1}{4}}!\right)^{2}}}}$
Sierpiński's constant	${K}$	2.58498 17595 79253 21706	${\begin{aligned}&\pi \left(2\gamma +\ln {\frac {4\pi ^{3}}{\Gamma ({\tfrac {1}{4}})^{4}}}\right)=\pi (2\gamma +4\ln \Gamma ({\tfrac {3}{4}})-\ln \pi )\\&=\pi \left(2\ln 2+3\ln \pi +2\gamma -4\ln \Gamma ({\tfrac {1}{4}})\right)\end{aligned}}$
Gieseking constant	${\pi \ln \beta }$	1.01494 16064 09653 62502	${\frac {3{\sqrt {3}}}{4}}\left(1-\sum _{n=0}^{\infty }{\frac {1}{(3n+2)^{2}}}+\sum _{n=1}^{\infty }{\frac {1}{(3n+1)^{2}}}\right)=$ $\textstyle {\frac {3{\sqrt {3}}}{4}}\left(1-{\frac {1}{2^{2}}}+{\frac {1}{4^{2}}}-{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}-{\frac {1}{8^{2}}}+{\frac {1}{10^{2}}}\pm \cdots \right)$.
Twin Primes Constant	${C}_{2}$	0.66016 18158 46869 57392	$\prod _{p=3}^{\infty }{\frac {p(p-2)}{(p-1)^{2}}}$
Golomb–Dickman constant	${\lambda }$	0.62432 99885 43550 87099	$\int _{0}^{\infty }{\underset {{\text{Para }}x>2}{{\frac {f(x)}{x^{2}}}\,dx}}=\int _{0}^{1}e^{\operatorname {Li} (n)}dn$where Li is the logarithmic integral
Feller–Tornier constant	${{\mathcal {C}}_{_{FT}}}$	0.66131 70494 69622 33528	${\underset {p_{n}:\,{prime}}{{\frac {1}{2}}\prod _{n=1}^{\infty }\left(1-{\frac {2}{p_{n}^{2}}}\right){+}{\frac {1}{2}}}}={\frac {3}{\pi ^{2}}}\prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}^{2}-1}}\right){+}{\frac {1}{2}}$where pn is a prime
Khinchin's constant	$K_{\,0}$	2.68545 20010 65306 44530	$\prod _{n=1}^{\infty }\left[{1+{1 \over n(n+2)}}\right]^{\ln n/\ln 2}$
Khinchin–Lévy constant (1)	${\beta }$	1.18656 91104 15625 45282	${\frac {\pi ^{2}}{12\,\ln 2}}$
Khinchin-Lévy constant (2)	$e^{\beta }$	3.27582 29187 21811 15978	$e^{\pi ^{2}/(12\ln 2)}$
Mills' constant	${\theta }$	1.30637 78838 63080 69046	$\lfloor \theta ^{3^{n}}\rfloor$ is prime
Euler–Gompertz constant	${G}$	0.59634 73623 23194 07434	$\int _{0}^{\infty }\!\!{\frac {e^{-n}}{1{+}n}}\,dn=\!\!\int _{0}^{1}\!\!{\frac {dn}{1{-}\ln n}}={\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {2}{1+{\tfrac {2}{1+{\tfrac {3}{1+3{/\cdots }}}}}}}}}}}}}$
Lochs constant	${{\text{£}}_{_{Lo}}}$	0.97027 01143 92033 92574	${\frac {6\ln 2\ln 10}{\pi ^{2}}}$
Niven's constant	${C}$	1.70521 11401 05367 76428	$1+\sum _{n=2}^{\infty }\left(1-{\frac {1}{\zeta (n)}}\right)$
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 (= 0.5772156649...) is the Euler–Mascheroni Constant $\scriptstyle \zeta '(2)\,{\text{= Derivative of }}\zeta (2)=-\sum \limits _{n=2}^{\infty }{\frac {\ln n}{n^{2}}}=-0.9375482543\ldots$
Feigenbaum constant δ	${\delta }$	4.66920 16091 02990 67185	$\lim _{n\to \infty }{\frac {x_{n+1}-x_{n}}{x_{n+2}-x_{n+1}}}\qquad \scriptstyle x\in (3.8284;\,3.8495)$ $\scriptstyle x_{n+1}=\,ax_{n}(1-x_{n})\quad {\text{or}}\quad x_{n+1}=\,a\sin(x_{n})$
Fransén–Robinson constant	${F}$	2.80777 02420 28519 36522	$\int _{0}^{\infty }{\frac {dx}{\Gamma (x)}}=e+\int _{0}^{\infty }{\frac {e^{-x}}{\pi ^{2}+\ln ^{2}x}}\,dx$
Feigenbaum constant α	$\alpha$	2.50290 78750 95892 82228	$\lim _{n\to \infty }{\frac {d_{n}}{d_{n+1}}}$
Chebyshev constant ·	$\lambda _{\text{Ch}}$	0.59017 02995 08048 11302	${\frac {\Gamma ({\tfrac {1}{4}})^{2}}{4\pi ^{3/2}}}={\frac {4({\tfrac {1}{4}}!)^{2}}{\pi ^{3/2}}}$
Brun 2 constant = Σ inverse of Twin primes	${B}_{\,2}$	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 p is a prime such that p + 2 is also a prime
Hafner–Sarnak–McCurley constant (1)	${\sigma }$	0.35323 63718 54995 98454	$\prod _{k=1}^{\infty }\left\{1-\left[1-\prod _{j=1}^{n}\left(1-p_{k}^{-j}\right)\right]^{2}\right\}$where pk is a prime
Fractal dimension of the Apollonian packing of circles	$\varepsilon$	1.30568 6729 ≈ by Thomas & Dhar 1.30568 8 ≈ by McMullen
Backhouse's constant	${B}$	1.45607 49485 82689 67139	$\lim _{k\to \infty }\left|{\frac {q_{k+1}}{q_{k}}}\right\vert \quad \scriptstyle {\text{where:}}\displaystyle \;\;Q(x)={\frac {1}{P(x)}}=\!\sum _{k=1}^{\infty }q_{k}x^{k}$ $P(x)=\sum _{k=1}^{\infty }{\underset {p_{k}{\text{ prime}}}{p_{k}x^{k}}}=1+2x+3x^{2}+5x^{3}+\cdots$
Viswanath constant	${C}_{Vi}$	1.13198 82487 943	$\lim _{n\to \infty }|a_{n}|^{\frac {1}{n}}$      where an = Fibonacci sequence
Regular paperfolding sequence	${P_{f}}$	0.85073 61882 01867 26036	$\sum _{n=0}^{\infty }{\frac {8^{2^{n}}}{2^{2^{n+2}}-1}}=\sum _{n=0}^{\infty }{\cfrac {\tfrac {1}{2^{2^{n}}}}{1-{\tfrac {1}{2^{2^{n+2}}}}}}$
Artin constant	${C}_{Artin}$	0.37395 58136 19202 28805	$\prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}(p_{n}-1)}}\right)\quad p_{n}\scriptstyle {\text{ = prime}}$
MRB constant	$C_{{}_{MRB}}$	0.18785 96424 62067 12024	$\sum _{n=1}^{\infty }(-1)^{n}(n^{1/n}-1)=-{\sqrt[{1}]{1}}+{\sqrt[{2}]{2}}-{\sqrt[{3}]{3}}+\cdots$
Somos' quadratic recurrence constant	${\sigma }$	1.66168 79496 33594 12129	$\prod _{n=1}^{\infty }n^{{1/2}^{n}}={\sqrt {1{\sqrt {2{\sqrt {3\cdots }}}}}}=1^{1/2}\;2^{1/4}\;3^{1/8}\cdots$
Foias constant α	$F_{\alpha }$	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 ∞. When x1 = α, $\,\lim _{n\to \infty }x_{n}{\tfrac {\log n}{n}}=1$
Foias constant β	$F_{\beta }$	2.29316 62874 11861 03150	$x^{x+1}=(x+1)^{x}$
Brun 4 constant = Σ inv.prime quadruplets	${B}_{\,4}$	0.87058 83799 75	$\textstyle {\sum ({\frac {1}{p}}+{\frac {1}{p+2}}+{\frac {1}{p+6}}+{\frac {1}{p+8}})}\scriptstyle \quad {p,\;p+2,\;p+6,\;p+8:{\text{ prime}}}$ $\textstyle {\left({\tfrac {1}{5}}+{\tfrac {1}{7}}+{\tfrac {1}{11}}+{\tfrac {1}{13}}\right)}+\left({\tfrac {1}{11}}+{\tfrac {1}{13}}+{\tfrac {1}{17}}+{\tfrac {1}{19}}\right)+\dots$
Heath-Brown–Moroz constant	${C_{_{HBM}}}$	0.00131 76411 54853 17810	${\underset {p_{n}:\,{prime}}{\prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}}}\right)^{7}\left(1+{\frac {7p_{n}+1}{p_{n}^{2}}}\right)}}$
Lebesgue constant	${C_{1}}$	0.98943 12738 31146 95174	$\lim _{n\to \infty }\!\!\left(\!{L_{n}{-}{\frac {4}{\pi ^{2}}}\ln(2n{+}1)}\!\!\right)\!{=}{\frac {4}{\pi ^{2}}}\!\left({\sum _{k=1}^{\infty }\!{\frac {2\ln k}{4k^{2}{-}1}}}{-}{\frac {\Gamma '({\tfrac {1}{2}})}{\Gamma ({\tfrac {1}{2}})}}\!\!\right)$
Landau–Ramanujan constant	$K$	0.76422 36535 89220 66299	${\frac {1}{\sqrt {2}}}\prod _{p\equiv 3\!\!\!\!\!\mod \!4}\!\!{\underset {\!\!\!\!\!\!\!\!p:{\text{ prime}}}{\left(1-{\frac {1}{p^{2}}}\right)^{-{\frac {1}{2}}}}}\!\!={\frac {\pi }{4}}\prod _{p\equiv 1\!\!\!\!\!\mod \!4}\!\!{\underset {\!\!\!\!p:{\text{ prime}}}{\left(1-{\frac {1}{p^{2}}}\right)^{\frac {1}{2}}}}$
Stephens constant	$C_{S}$	0.57595 99688 92945 43964	$\prod _{n=1}^{\infty }\left(1-{\frac {p}{p^{3}-1}}\right)$
Taniguchi constant	$C_{T}$	0.67823 44919 17391 97803	$\prod _{n=1}^{\infty }\left(1-{\frac {3}{{p_{n}}^{3}}}+{\frac {2}{{p_{n}}^{4}}}+{\frac {1}{{p_{n}}^{5}}}-{\frac {1}{{p_{n}}^{6}}}\right)$ $\scriptstyle p_{n}=\,{\text{prime}}$
Kepler–Bouwkamp constant	${\rho }$	0.11494 20448 53296 20070	$\prod _{n=3}^{\infty }\cos \left({\frac {\pi }{n}}\right)=\cos \left({\frac {\pi }{3}}\right)\cos \left({\frac {\pi }{4}}\right)\cos \left({\frac {\pi }{5}}\right)...$

No Explicit formula known

Name	Symbol	Decimal expansion	Formula
Bernstein's constant	${\beta }$	0.28016 94990 23869 13303	$\approx {\frac {1}{2{\sqrt {\pi }}}}$
Eddington number	$N_{Edd}$	~1080	The number of protons in the universe
de Bruijn–Newman constant	$\Lambda$	$0\leq \Lambda \leq 0.2$	The number 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 and $\Phi (u)=\sum _{n=1}^{\infty }(2\pi ^{2}n^{4}e^{9u}-3\pi n^{2}e^{5u})e^{-\pi n^{2}e^{4u}}$

Complex

Name	Symbol	Decimal expansion	Formula
Imaginary unit	${i}$	0 + 1i	Either of the two roots of $x^{2}=-1$

Mathematical constants sorted by their representations as continued fractions

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	Member of	decimal	Continued fraction	Notes
	$0$	$\mathbb {Z}$	0.00000 00000	[0; ]
Golden ratio conjugate	$\phi ^{-1}$	$\mathbb {A} \setminus \mathbb {Q}$	0.61803 39887	[0; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …]	irrational
Cahen's constant	$C$	$\mathbb {R}$	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.
First Hardy–Littlewood conjecture	$C_{2}$	$\mathbb {R}$	0.66016 18158	[0; 1, 1, 1, 16, 2, 2, 2, 2, 1, 18, 2, 2, 11, 1, 1, 2, 4, 1, 16, 3, …]	Hardy–Littlewood's twin prime constant. Presumed irrational, but not proved.
Euler-Mascheroni constant	$\gamma$	$\mathbb {R}$	0.57721 56649	[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, 1, …]	Presumed irrational, but not proved.
Omega constant	$\Omega$	$\mathbb {R} \setminus \mathbb {A}$	0.56714 32904	[0; 1, 1, 3, 4, 2, 10, 4, 1, 1, 1, 1, 2, 7, 306, 1, 5, 1, 2, 1, 5, …]
Embree–Trefethen constant	$\beta ^{\star }$	$\mathbb {R}$	0.70258	[0; 1, 2, 2, 1, 3, 5, 1, 2, 6, 1, 1, 5, …]	Value only known to 5 decimal places.
Continued fraction constant	Continued fraction constant	$\mathbb {R} \setminus \mathbb {A}$	0.69777 46579	[0; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, …]	Equal to a ratio $I_{1}(2)/I_{2}(2)$ of modified Bessel functions of the first kind evaluated at 2
Landau–Ramanujan constant	$K$	$\mathbb {R} (\setminus \mathbb {Q} ?)$	0.76422 36535	[0; 1, 3, 4, 6, 1, 15, 1, 2, 2, 3, 1, 23, 3, 1, 1, 3, 1, 1, 7, 2, …]	May have been proven irrational.
Gauss's constant	${\frac {1}{M(1,{\sqrt {2}})}}$	$\mathbb {R} \setminus \mathbb {A}$	0.83462 68417	[0; 1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, 3, 8, 36, 1, 2, …]	Gauss's constant
Brun's theorem	$B_{4}$	$\mathbb {R}$	0.87058 83800	[0; 1, 6, 1, 2, 1, 2, 956, 8, 1, 1, 1, 23, …]	Brun's prime quadruplet constant. Estimated value; 99% confidence interval ± 0.00000 00005.
Champernowne constant	$C_{2}$	$\mathbb {R} \setminus \mathbb {A}$	0.86224 01259	[0; 1, 6, 3, 1, 6, 5, 3, 3, 1, 6, 4, 1, 3, 298, 1, 6, 1, 1, 3, 285, …]	Base 2 Champernowne constant. The binary expansion is $C_{2}=0.110111001011101111000\ldots _{2}$
Catalan's constant	$G$	$\mathbb {R}$	0.91596 55942	[0; 1, 10, 1, 8, 1, 88, 4, 1, 1, 7, 22, 1, 2, 3, 26, 1, 11, 1, 10, 1, …]	Presumed irrational, but not proved.
One half	${\frac {1}{2}}$	$\mathbb {Q}$	0.50000 00000	[0; 2]
Bernstein's constant	$\beta$	$\mathbb {R}$	0.28016 94990	[0; 3, 1, 1, 3, 9, 6, 3, 1, 3, 13, 1, 16, 3, 3, 4, …]	Presumed irrational, but not proved.
Meissel–Mertens constant	$M$	$\mathbb {R}$	0.26149 72128	[0; 3, 1, 4, 1, 2, 5, 2, 1, 1, 1, 1, 13, 4, 2, 4, 2, 1, 33, 296, 2, …]	Presumed irrational, but not proved.
MRB constant	$C_{{}_{MRB}}$	$\mathbb {R}$	0.18785 96424	[0; 5, 3, 10, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 12, 2, 17, 2, 2, 1, 1, …]
Champernowne 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, $4.57540\times 10^{165}$, 6, 1, …]	Base 10 Champernowne constant. Champernowne constants in any base exhibit sporadic large numbers; the 40th term in $C_{10}$ has 2504 digits.
	$1$	$\mathbb {N}$	1.00000 00000	[1; ]
Golden ratio	$\phi$	$\mathbb {A} \setminus \mathbb {Q}$	1.61803 39887	[1; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …]
Erdős–Borwein constant	$E$	$\mathbb {R} \setminus \mathbb {Q}$	1.60669 51524	[1; 1, 1, 1, 1, 5, 2, 1, 2, 29, 4, 1, 2, 2, 2, 2, 6, 1, 7, 1, 6, …]	Not known whether algebraic or transcendental.
Brun's constant	$B_{2}$	$\mathbb {R}$	1.90216 05831	[1; 1, 9, 4, 1, 1, 8, 3, 4, 7, 1, 3, 3, 1, 2, 1, 1, 12, 4, 2, 1, …]	Brun's twin prime constant. Estimated value; best bounds $1.8304<B_{2}<2.347$.
Square root of 2	${\sqrt {2}}$	$\mathbb {A} \setminus \mathbb {Q}$	1.41421 35624	[1; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, …]
Ramanujan-Soldner constant	$\mu$	$\mathbb {R}$	1.45136 92349	[1; 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, 4, 1, 12, 1, 1, 2, 2, 1, …]	Presumed irrational, but not proved.
Backhouse's constant	$B$	$\mathbb {R}$	1.45607 49485	[1; 2, 5, 5, 4, 1, 1, 18, 1, 1, 1, 1, 1, 2, 13, 3, 1, 2, 4, 16, 4, …]
Plastic number	$\rho$	$\mathbb {A} \setminus \mathbb {Q}$	1.32471 95724	[1; 3, 12, 1, 1, 3, 2, 3, 2, 4, 2, 141, 80, 2, 5, 1, 2, 8, 2, 1, 1, …]
Apéry's constant	$\zeta (3)$	$\mathbb {R} \setminus \mathbb {Q}$	1.20205 69032	[1; 4, 1, 18, 1, 1, 1, 4, 1, 9, 9, 2, 1, 1, 1, 2, 7, 1, 1, 7, 11, …]
Random Fibonacci sequence	$V$	$\mathbb {R}$	1.13198 82488	[1; 7, 1, 1, 2, 1, 3, 2, 1, 2, 1, 17, 1, 1, 2, 1, 2, 4, 1, 2, …]	Viswanath's constant. Apparently, Eric Weisstein calculated this constant to be approximately 1.13215 06911 with Mathematica.
	$2$	$\mathbb {N}$	2.00000 00000	[2; ]
Gelfond–Schneider constant	$2^{\sqrt {2}}$	$\mathbb {R} \setminus \mathbb {A}$	2.66514 41426	[2; 1, 1, 1, 72, 3, 4, 1, 3, 2, 1, 1, 1, 14, 1, 2, 1, 1, 3, 1, 3, …]
Second Feigenbaum constant	$\alpha$	$\mathbb {R}$	2.50290 78751	[2; 1, 1, 85, 2, 8, 1, 10, 16, 3, 8, 9, 2, 1, 40, 1, 2, 3, 2, 2, 1, …]
Base of the natural logarithm	$e$	$\mathbb {R} \setminus \mathbb {A}$	2.71828 18285	[2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, …]
Khinchin's constant	$K_{0}$	$\mathbb {R}$	2.68545 20011	[2; 1, 2, 5, 1, 1, 2, 1, 1, 3, 10, 2, 1, 3, 2, 24, 1, 3, 2, 3, 1, …]
Fransén–Robinson constant	$F$	$\mathbb {R}$	2.80777 02420	[2; 1, 4, 4, 1, 18, 5, 1, 3, 4, 1, 5, 3, 6, 1, 1, 1, 5, 1, 1, 1, …]
Universal parabolic constant	$P_{2}$	$\mathbb {R} \setminus \mathbb {A}$	2.29558 71494	[2; 3, 2, 1, 1, 1, 1, 3, 3, 1, 1, 4, 2, 3, 2, 7, 1, 6, 1, 8, 7, …]
	$3$	$\mathbb {N}$	3.00000 00000	[3; ]
Reciprocal Fibonacci constant	$\psi$	$\mathbb {R} \setminus \mathbb {Q}$	3.35988 56662	[3; 2, 1, 3, 1, 1, 13, 2, 3, 3, 2, 1, 1, 6, 3, 2, 4, 362, 2, 4, 8, …]
	$\pi$	$\mathbb {R} \setminus \mathbb {A}$	3.14159 26536	[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, …]
	$4$	$\mathbb {N}$	4.00000 00000	[4; ]
First Feigenbaum constant	$\delta$	$\mathbb {R}$	4.66920 16091	[4; 1, 2, 43, 2, 163, 2, 3, 1, 1, 2, 5, 1, 2, 3, 80, 2, 5, 2, 1, 1, …]
	$5$	$\mathbb {N}$	5.00000 00000	[5; ]
Gelfond's constant	$e^{\pi }$	$\mathbb {R} \setminus \mathbb {A}$	23.14069 26328	[23; 7, 9, 3, 1, 1, 591, 2, 9, 1, 2, 34, 1, 16, 1, 30, 1, 1, 4, 1, 2, …]	Gelfond's constant. Can also be expressed as $(-1)^{-i}$; from this form, it is transcendental due to the Gelfond–Schneider theorem.

See also

• Invariant (mathematics)
• List of mathematical symbols
• List of mathematical symbols by subject
• List of numbers
• List of physical constants
• Mathematical constants and functions
• 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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