# List of mathematical constants

A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems.[1][2] 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.

Explanations of the symbols in the right hand column can be found by clicking on them.

## Antiquity

Name Symbol Decimal Expansion Formula Year Set
One 1 1 None[nb 1] Prehistory ${\displaystyle \mathbb {N} }$
Two 2 2 ${\displaystyle 1+1}$  Prehistory ${\displaystyle \mathbb {N} }$
One half 1/2 0.5 ${\displaystyle 1/2}$  Prehistory ${\displaystyle \mathbb {Q} }$
Pi ${\displaystyle \pi }$  3.14159 26535 89793 23846 [Mw 1][OEIS 1] Ratio of a circle's circumference to its diameter. 1900 to 1600 BCE [3] ${\displaystyle \mathbb {T} }$
Square root of 2,

Pythagoras constant.[4]

${\displaystyle {\sqrt {2}}}$  1.41421 35623 73095 04880 [Mw 2][OEIS 2] Positive root of ${\displaystyle x^{2}=2}$  1800 to 1600 BCE[5] ${\displaystyle \mathbb {A} }$
Square root of 3,

Theodorus' constant[6]

${\displaystyle {\sqrt {3}}}$  1.73205 08075 68877 29352 [Mw 3][OEIS 3] Positive root of ${\displaystyle x^{2}=3}$  465 to 398 BCE ${\displaystyle \mathbb {A} }$
Square root of 5[7] ${\displaystyle {\sqrt {5}}}$  2.23606 79774 99789 69640[OEIS 4] Positive root of ${\displaystyle x^{2}=5}$  ${\displaystyle \mathbb {A} }$
Phi, Golden ratio[1][8] ${\displaystyle {\varphi }}$  1.61803 39887 49894 84820 [Mw 4][OEIS 5] Positive root of ${\displaystyle x^{2}-x-1=0}$  ~300 BCE ${\displaystyle \mathbb {A} }$
Zero 0 0 The additive identity: ${\displaystyle x+0=x}$  300-100 century BCE[9] ${\displaystyle \mathbb {Z} }$
Negative one -1 -1 ${\displaystyle 1-2}$  300-200 BCE ${\displaystyle \mathbb {Z} }$
Cube root of 2 (Delian Constant) ${\displaystyle {\sqrt[{3}]{2}}}$  1.25992 10498 94873 16476 [Mw 5][OEIS 6] Real root of ${\displaystyle x^{3}=2}$  46 -120 CE

[10]

${\displaystyle \mathbb {A} }$
Cube root of 3 ${\displaystyle {\sqrt[{3}]{3}}}$  1.44224 95703 07408 38232[OEIS 7] Real root of ${\displaystyle x^{3}=3}$  ${\displaystyle \mathbb {A} }$

## Medieval and Early Modern

Name Symbol Decimal Expansion Formula Year Set
Imaginary unit [1][11] ${\displaystyle {i}}$  0 + 1i Either of the two roots of ${\displaystyle x^{2}=-1}$ [nb 2] 1501 to 1576 ${\displaystyle \mathbb {C} }$
Wallis Constant ${\displaystyle W}$  2.09455 14815 42326 59148 [Mw 6][OEIS 8] ${\displaystyle {\sqrt[{3}]{\frac {45-{\sqrt {1929}}}{18}}}+{\sqrt[{3}]{\frac {45+{\sqrt {1929}}}{18}}}}$  1616
to
1703
${\displaystyle \mathbb {A} }$
Euler's number[1][12] ${\displaystyle {e}}$  2.71828 18284 59045 23536 [Mw 7][OEIS 9] ${\displaystyle \lim _{n\to \infty }\!\left(\!1\!+\!{\frac {1}{n}}\right)^{n}}$ [nb 3] 1618[13] ${\displaystyle \mathbb {T} }$
Natural logarithm of 2 [14] ${\displaystyle \ln 2}$  0.69314 71805 59945 30941 [Mw 8][OEIS 10] ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n2^{n}}}=\sum _{n=1}^{\infty }{\frac {({-}1)^{n+1}}{n}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\cdots }}$  1619,[15]1668[16] ${\displaystyle \mathbb {T} }$
Sophomore's dream1
J.Bernoulli [17]
${\displaystyle {I}_{1}}$  0.78343 05107 12134 40705 [OEIS 11] ${\displaystyle \int _{0}^{1}\!x^{x}\,dx=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{n}}}={\frac {1}{1^{1}}}-{\frac {1}{2^{2}}}+{\frac {1}{3^{3}}}-{\cdots }}$  1697
Sophomore's dream2
J.Bernoulli [18]
${\displaystyle {I}_{2}}$  1.29128 59970 62663 54040 [Mw 9][OEIS 12] ${\displaystyle \int _{0}^{1}\!{\frac {1}{x^{x}}}\,dx=\sum _{n=1}^{\infty }{\frac {1}{n^{n}}}={\frac {1}{1^{1}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{4}}}+\cdots }$  1697
Lemniscate constant[19] ${\displaystyle {\varpi }}$  2.62205 75542 92119 81046 [Mw 10][OEIS 13] ${\displaystyle \pi \,{G}=4{\sqrt {\tfrac {2}{\pi }}}\,\Gamma {\left({\tfrac {5}{4}}\right)^{2}}={\tfrac {1}{4}}{\sqrt {\tfrac {2}{\pi }}}\,\Gamma {\left({\tfrac {1}{4}}\right)^{2}}=4{\sqrt {\tfrac {2}{\pi }}}\left({\tfrac {1}{4}}!\right)^{2}}$  1718 to 1798 ${\displaystyle \mathbb {T} }$
Euler–Mascheroni constant[20] ${\displaystyle {\gamma }}$  0.57721 56649 01532 86060 [Mw 11][OEIS 14] ${\displaystyle \sum _{n=1}^{\infty }\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2^{n}+k}}=\sum _{n=1}^{\infty }\left({\frac {1}{n}}-\ln \left(1+{\frac {1}{n}}\right)\right)}$

${\displaystyle =\int _{0}^{1}-\ln \left(\ln {\frac {1}{x}}\right)\,dx=-\Gamma '(1)=-\Psi (1)}$

1735 ${\displaystyle \mathbb {R} \setminus \mathbb {Q} }$ ?
Erdős–Borwein constant[21] ${\displaystyle {E}_{\,B}}$  1.60669 51524 15291 76378 [Mw 12][OEIS 15] ${\displaystyle \sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {1}{2^{mn}}}=\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}={\frac {1}{1}}\!+\!{\frac {1}{3}}\!+\!{\frac {1}{7}}\!+\!{\frac {1}{15}}\!+\!...}$  1749[22] ${\displaystyle \mathbb {R} \setminus \mathbb {Q} }$
Laplace limit [23] ${\displaystyle {\lambda }}$  0.66274 34193 49181 58097 [Mw 13][OEIS 16] ${\displaystyle {\frac {x\;e^{\sqrt {x^{2}+1}}}{{\sqrt {x^{2}+1}}+1}}=1}$  ~1782 ${\displaystyle \mathbb {T} }$ ?
Gauss's constant [24] ${\displaystyle {G}}$  0.83462 68416 74073 18628 [Mw 14][OEIS 17] ${\displaystyle {\frac {1}{\mathrm {agm} (1,{\sqrt {2}})}}={\frac {4{\sqrt {2}}\,({\tfrac {1}{4}}!)^{2}}{\pi ^{3/2}}}={\frac {2}{\pi }}\int _{0}^{1}{\frac {dx}{\sqrt {1-x^{4}}}}}$

where agm = Arithmetic–geometric mean

1799[25] ${\displaystyle \mathbb {T} }$ ?

## 19th century

Name Symbol Decimal Expansion Formula Year Set
Ramanujan–Soldner constant[26][27] ${\displaystyle {\mu }}$  1.45136 92348 83381 05028 [Mw 15][OEIS 18] ${\displaystyle \mathrm {li} (x)=\int \limits _{0}^{x}{\frac {dt}{\ln t}}=0}$ ; root of the logarithmic integral function. 1812[Mw 16]
Hermite constant [28] ${\displaystyle \gamma _{_{2}}}$  1.15470 05383 79251 52901 [Mw 17] ${\displaystyle {\frac {2}{\sqrt {3}}}={\frac {1}{\cos \,({\frac {\pi }{6}})}}}$  1822 to 1901 ${\displaystyle \mathbb {A} }$
Liouville number [29] ${\displaystyle {\text{£}}_{Li}}$  0.11000 10000 00000 00000 0001 [Mw 18][OEIS 19] ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{10^{n!}}}={\frac {1}{10^{1!}}}+{\frac {1}{10^{2!}}}+{\frac {1}{10^{3!}}}+{\frac {1}{10^{4!}}}+\cdots }$  Before 1844 ${\displaystyle \mathbb {T} }$
Hermite–Ramanujan constant[30] ${\displaystyle {R}}$  262 53741 26407 68743
.99999 99999 99250 073 [Mw 19][OEIS 20]
${\displaystyle e^{\pi {\sqrt {163}}}}$  1859 ${\displaystyle \mathbb {T} }$
Catalan's constant[31][32][33] ${\displaystyle {C}}$  0.91596 55941 77219 01505 [Mw 20][OEIS 21] ${\displaystyle \int _{0}^{1}\!\!\int _{0}^{1}\!\!{\frac {1}{1{+}x^{2}y^{2}}}\,dx\,dy=\!\sum _{n=0}^{\infty }\!{\frac {(-1)^{n}}{(2n{+}1)^{2}}}\!=\!{\frac {1}{1^{2}}}{-}{\frac {1}{3^{2}}}{+}{\cdots }}$  1864 ${\displaystyle \mathbb {T} }$ ?
Dottie number [34] ${\displaystyle d}$  0.73908 51332 15160 64165 [Mw 21][OEIS 22] ${\displaystyle \lim _{x\to \infty }\cos ^{[x]}(c)=\lim _{x\to \infty }\underbrace {\cos(\cos(\cos(\cdots (\cos(c)))))} _{x}}$  1865[Mw 21] ${\displaystyle \mathbb {T} }$
Meissel–Mertens constant [35] ${\displaystyle {M}}$  0.26149 72128 47642 78375 [Mw 22][OEIS 23] ${\displaystyle \lim _{n\rightarrow \infty }\!\!\left(\sum _{p\leq n}{\frac {1}{p}}\!-\ln(\ln(n))\!\right)\!\!={\underset {\!\!\!\!\gamma :\,{\text{Euler constant}},\,\,p:\,{\text{prime}}}{\!\gamma \!+\!\!\sum _{p}\!\left(\!\ln \!\left(\!1\!-\!{\frac {1}{p}}\!\right)\!\!+\!{\frac {1}{p}}\!\right)}}}$  1866
&
1873
${\displaystyle \mathbb {T} }$ ?
Weierstrass constant [36] ${\displaystyle \sigma ({\tfrac {1}{2}})}$  0.47494 93799 87920 65033 [Mw 23][OEIS 24] ${\displaystyle {\frac {e^{\frac {\pi }{8}}{\sqrt {\pi }}}{4\cdot 2^{3/4}{({\frac {1}{4}}!)^{2}}}}}$  1872 ?
Hafner–Sarnak–McCurley constant (2) [37] ${\displaystyle {\frac {1}{\zeta (2)}}}$  0.60792 71018 54026 62866 [Mw 24][OEIS 25] ${\displaystyle {\frac {6}{\pi ^{2}}}=\prod _{n=0}^{\infty }{\underset {p_{n}:{\text{ prime}}}{\!\left(\!1-{\frac {1}{{p_{n}}^{2}}}\!\right)}}\!=\!\textstyle \left(1\!-\!{\frac {1}{2^{2}}}\right)\!\left(1\!-\!{\frac {1}{3^{2}}}\right)\!\left(1\!-\!{\frac {1}{5^{2}}}\right)\cdots }$  1883[Mw 24] ${\displaystyle \mathbb {T} }$
Cahen's constant [38] ${\displaystyle \xi _{2}}$  0.64341 05462 88338 02618 [Mw 25][OEIS 26] ${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k}}{s_{k}-1}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{6}}-{\frac {1}{42}}+{\frac {1}{1806}}{\,\pm \cdots }}$

Where sk is the kth term of Sylvester's sequence 2, 3, 7, 43, 1807, ...
Defined as: ${\displaystyle \,\,S_{0}=\,2,\,\,S_{k}=\,1+\prod \limits _{n=0}^{k-1}S_{n}{\text{ for}}\;k>0}$

1891 ${\displaystyle \mathbb {T} }$
Universal parabolic constant [39] ${\displaystyle {P}_{\,2}}$  2.29558 71493 92638 07403 [Mw 26][OEIS 27] ${\displaystyle \ln(1+{\sqrt {2}})+{\sqrt {2}}\;=\;\operatorname {arcsinh} (1)+{\sqrt {2}}}$  Before 1891[40] ${\displaystyle \mathbb {T} }$
Apéry's constant [41] ${\displaystyle \zeta (3)}$  1.20205 69031 59594 28539 [Mw 27][OEIS 28] ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{3}}}={\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{3}}}+{\frac {1}{5^{3}}}+\cdots =}$

${\displaystyle {\frac {1}{2}}\sum _{n=1}^{\infty }{\frac {H_{n}}{n^{2}}}={\frac {1}{2}}\sum _{i=1}^{\infty }\sum _{j=1}^{\infty }{\frac {1}{ij(i{+}j)}}=\!\!\int \limits _{0}^{1}\!\!\int \limits _{0}^{1}\!\!\int \limits _{0}^{1}{\frac {\mathrm {d} x\mathrm {d} y\mathrm {d} z}{1-xyz}}}$

1895[42] ${\displaystyle \mathbb {R} \setminus \mathbb {Q} }$
Gelfond's constant [43] ${\displaystyle {e}^{\pi }}$  23.14069 26327 79269 0057 [Mw 28][OEIS 29] ${\displaystyle (-1)^{-i}=i^{-2i}=\sum _{n=0}^{\infty }{\frac {\pi ^{n}}{n!}}=1+{\frac {\pi ^{1}}{1}}+{\frac {\pi ^{2}}{2}}+{\frac {\pi ^{3}}{6}}+\cdots }$  1900[44] ${\displaystyle \mathbb {T} }$

## 1900–1949

Name Symbol Decimal Expansion Formula Year Set
Favard constant [45] ${\displaystyle {\tfrac {3}{4}}\zeta (2)}$  1.23370 05501 36169 82735 [Mw 29][OEIS 30] ${\displaystyle {\frac {\pi ^{2}}{8}}=\sum _{n=0}^{\infty }{\frac {1}{(2n-1)^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\cdots }$  1902
to
1965
${\displaystyle \mathbb {T} }$
Golden angle [46] ${\displaystyle {b}}$  2.39996 32297 28653 32223 [Mw 30][OEIS 31] ${\displaystyle (4-2\,\Phi )\,\pi =(3-{\sqrt {5}})\,\pi }$  = 137.5077640500378546 ...° 1907 ${\displaystyle \mathbb {T} }$
Sierpiński's constant [47] ${\displaystyle {K}}$  2.58498 17595 79253 21706 [Mw 31][OEIS 32] ${\displaystyle \pi \left(2\gamma +\ln {\frac {4\pi ^{3}}{\Gamma ({\tfrac {1}{4}})^{4}}}\right)=\pi (2\gamma +4\ln \Gamma ({\tfrac {3}{4}})-\ln \pi )}$

${\displaystyle =\pi \left(2\ln 2+3\ln \pi +2\gamma -4\ln \Gamma ({\tfrac {1}{4}})\right)}$

1907
NielsenRamanujan constant [48] ${\displaystyle {\frac {{\zeta }(2)}{2}}}$  0.82246 70334 24113 21823 [Mw 32][OEIS 33] ${\displaystyle {\frac {\pi ^{2}}{12}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{2}}}={\frac {1}{1^{2}}}{-}{\frac {1}{2^{2}}}{+}{\frac {1}{3^{2}}}{-}{\frac {1}{4^{2}}}{+}{\frac {1}{5^{2}}}{-}\cdots }$  1909 ${\displaystyle \mathbb {T} }$
Area of the Mandelbrot fractal [49] ${\displaystyle \gamma }$  1.5065918849 ± 0.0000000028 [Mw 33][OEIS 34] 1912
Gieseking constant [50] ${\displaystyle {\pi \ln \beta }}$  1.01494 16064 09653 62502 [Mw 34][OEIS 35] ${\displaystyle {\frac {3{\sqrt {3}}}{4}}\left(1-\sum _{n=0}^{\infty }{\frac {1}{(3n+2)^{2}}}+\sum _{n=1}^{\infty }{\frac {1}{(3n+1)^{2}}}\right)=}$

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

1912
Bernstein's constant [51] ${\displaystyle {\beta }}$  0.28016 94990 23869 13303 [Mw 35][OEIS 36] ${\displaystyle \approx {\frac {1}{2{\sqrt {\pi }}}}}$  1913
Twin Primes Constant [52] ${\displaystyle {C}_{2}}$  0.66016 18158 46869 57392 [Mw 36][OEIS 37] ${\displaystyle \prod _{p=3}^{\infty }{\frac {p(p-2)}{(p-1)^{2}}}}$  1922
Plastic number [53] ${\displaystyle {\rho }}$  1.32471 79572 44746 02596 [Mw 37][OEIS 38] ${\displaystyle {\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\cdots }}}}}}=\textstyle {\sqrt[{3}]{{\frac {1}{2}}+{\frac {\sqrt {69}}{18}}}}+{\sqrt[{3}]{{\frac {1}{2}}-{\frac {\sqrt {69}}{18}}}}}$  1929 ${\displaystyle \mathbb {A} }$
Bloch–Landau constant [54] ${\displaystyle {L}}$  0.54325 89653 42976 70695 [Mw 38][OEIS 39] ${\displaystyle ={\frac {\Gamma ({\tfrac {1}{3}})\;\Gamma ({\tfrac {5}{6}})}{\Gamma ({\tfrac {1}{6}})}}={\frac {(-{\tfrac {2}{3}})!\;(-1+{\tfrac {5}{6}})!}{(-1+{\tfrac {1}{6}})!}}}$  1929
Golomb–Dickman constant [55] ${\displaystyle {\lambda }}$  0.62432 99885 43550 87099 [Mw 39][OEIS 40] ${\displaystyle \int \limits _{0}^{\infty }{\underset {{\text{Para }}x>2}{{\frac {f(x)}{x^{2}}}\,dx}}=\int \limits _{0}^{1}e^{\operatorname {Li} (n)}dn\quad \scriptstyle {\text{Li: Logarithmic integral}}}$  1930
&
1964
Feller–Tornier constant [56] ${\displaystyle {{\mathcal {C}}_{_{FT}}}}$  0.66131 70494 69622 33528 [Mw 40][OEIS 41] ${\displaystyle {\underset {p_{n}:\,{prime}}{{\frac {1}{2}}\prod _{n=1}^{\infty }\left(1-{\frac {2}{p_{n}^{2}}}\right){+}{\frac {1}{2}}}}={\frac {3}{\pi ^{2}}}\prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}^{2}-1}}\right){+}{\frac {1}{2}}}$  1932 ${\displaystyle \mathbb {T} }$ ?
Base 10 Champernowne constant [57] ${\displaystyle C_{10}}$  0.12345 67891 01112 13141 [Mw 41][OEIS 42] ${\displaystyle \sum _{n=1}^{\infty }\;\sum _{k=10^{n-1}}^{10^{n}-1}{\frac {k}{10^{kn-9\sum _{j=0}^{n-1}10^{j}(n-j-1)}}}}$  1933 ${\displaystyle \mathbb {T} }$
Gelfond–Schneider constant [58] ${\displaystyle G_{\,GS}}$  2.66514 41426 90225 18865 [Mw 42][OEIS 43] ${\displaystyle 2^{\sqrt {2}}}$  1934 ${\displaystyle \mathbb {T} }$
Khinchin's constant [59] ${\displaystyle K_{\,0}}$  2.68545 20010 65306 44530 [Mw 43][OEIS 44] ${\displaystyle \prod _{n=1}^{\infty }\left[{1+{1 \over n(n+2)}}\right]^{\ln n/\ln 2}}$  1934 ${\displaystyle \mathbb {T} }$ ?
Khinchin–Lévy constant[60] ${\displaystyle {\beta }}$  1.18656 91104 15625 45282 [Mw 44][OEIS 45] ${\displaystyle {\frac {\pi ^{2}}{12\,\ln 2}}}$  1935
Khinchin-Lévy constant [61] ${\displaystyle \gamma }$  3.27582 29187 21811 15978 [Mw 45][OEIS 46] ${\displaystyle e^{\pi ^{2}/(12\ln 2)}}$  1936
Mills' constant [62] ${\displaystyle {\theta }}$  1.30637 78838 63080 69046 [Mw 46][OEIS 47] ${\displaystyle \lfloor \theta ^{3^{n}}\rfloor }$  is prime 1947
Euler–Gompertz constant [63] ${\displaystyle {G}}$  0.59634 73623 23194 07434 [Mw 47][OEIS 48] ${\displaystyle \!\int \limits _{0}^{\infty }\!\!{\frac {e^{-n}}{1{+}n}}\,dn=\!\!\int \limits _{0}^{1}\!\!{\frac {1}{1{-}\ln n}}\,dn=\textstyle {\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {2}{1+{\tfrac {2}{1+{\tfrac {3}{1+3{/\cdots }}}}}}}}}}}}}}$  Before 1948[OEIS 48]

## 1950–1999

Name Symbol Decimal Expansion Formula Year Set
Van der Pauw constant ${\displaystyle {\alpha }}$  4.53236 01418 27193 80962[OEIS 49] ${\displaystyle {\frac {\pi }{\ln(2)}}={\frac {\sum \limits _{n=0}^{\infty }{\frac {4(-1)^{n}}{2n+1}}}{\sum \limits _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}}}={\frac {{\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-\cdots }{{\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots }}}$  Before 1958[OEIS 50]
Magic angle [64] ${\displaystyle {\theta _{m}}}$  0.95531 66181 245092 78163[OEIS 51] ${\displaystyle \arctan \left({\sqrt {2}}\right)=\arccos \left({\sqrt {\tfrac {1}{3}}}\right)\approx \textstyle {54.7356}^{\circ }}$  Before 1959[65][64] ${\displaystyle \mathbb {T} }$
Lochs constant [66] ${\displaystyle {{\text{£}}_{_{Lo}}}}$  0.97027 01143 92033 92574 [Mw 48][OEIS 52] ${\displaystyle {\frac {6\ln 2\ln 10}{\pi ^{2}}}}$  1964
Lieb's square ice constant [67] ${\displaystyle {W}_{2D}}$  1.53960 07178 39002 03869 [Mw 49][OEIS 53] ${\displaystyle \lim _{n\to \infty }(f(n))^{n^{-2}}=\left({\frac {4}{3}}\right)^{\frac {3}{2}}={\frac {8}{3{\sqrt {3}}}}}$  1967 ${\displaystyle \mathbb {A} }$
Niven's constant [68] ${\displaystyle {C}}$  1.70521 11401 05367 76428 [Mw 50][OEIS 54] ${\displaystyle 1+\sum _{n=2}^{\infty }\left(1-{\frac {1}{\zeta (n)}}\right)}$  1969
Baker constant [69] ${\displaystyle \beta _{3}}$  0.83564 88482 64721 05333[OEIS 55] ${\displaystyle \int _{0}^{1}{\frac {\mathrm {d} t}{1+t^{3}}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{3n+1}}={\frac {1}{3}}\left(\ln 2+{\frac {\pi }{\sqrt {3}}}\right)}$  Before 1969[69]
Porter's constant[70] ${\displaystyle {C}}$  1.46707 80794 33975 47289 [Mw 51][OEIS 56] ${\displaystyle {\frac {6\ln 2}{\pi ^{2}}}\left(3\ln 2+4\,\gamma -{\frac {24}{\pi ^{2}}}\,\zeta '(2)-2\right)-{\frac {1}{2}}}$

${\displaystyle \scriptstyle \gamma \,{\text{= Euler–Mascheroni Constant}}=0.5772156649\ldots }$  ${\displaystyle \scriptstyle \zeta '(2)\,{\text{= Derivative of }}\zeta (2)=-\sum \limits _{n=2}^{\infty }{\frac {\ln n}{n^{2}}}=-0.9375482543\ldots }$

1974
Feigenbaum constant δ [71] ${\displaystyle {\delta }}$  4.66920 16091 02990 67185 [Mw 52][OEIS 57] ${\displaystyle \lim _{n\to \infty }{\frac {x_{n+1}-x_{n}}{x_{n+2}-x_{n+1}}}\qquad \scriptstyle x\in (3.8284;\,3.8495)}$

${\displaystyle \scriptstyle x_{n+1}=\,ax_{n}(1-x_{n})\quad {\text{or}}\quad x_{n+1}=\,a\sin(x_{n})}$

1975
Chaitin's constants [72] ${\displaystyle \Omega }$  In general they are uncomputable numbers.
But one such number is 0.00787 49969 97812 3844
[Mw 53][OEIS 58]
${\displaystyle \sum _{p\in P}2^{-|p|}}$
• p: Halted program
• |p|: Size in bits of program p
• P: Domain of all programs that stop.
1975 ${\displaystyle \mathbb {T} }$
Fransén–Robinson constant [73] ${\displaystyle {F}}$  2.80777 02420 28519 36522 [Mw 54][OEIS 59] ${\displaystyle \int _{0}^{\infty }{\frac {1}{\Gamma (x)}}\,dx=e+\int _{0}^{\infty }{\frac {e^{-x}}{\pi ^{2}+\ln ^{2}x}}\,dx}$  1978
Robbins constant [74] ${\displaystyle \Delta (3)}$  0.66170 71822 67176 23515 [Mw 55][OEIS 60] ${\displaystyle {\frac {4\!+\!17{\sqrt {2}}\!-6{\sqrt {3}}\!-7\pi }{105}}\!+\!{\frac {\ln(1\!+\!{\sqrt {2}})}{5}}\!+\!{\frac {2\ln(2\!+\!{\sqrt {3}})}{5}}}$  1978
Feigenbaum constant α[75] ${\displaystyle \alpha }$  2.50290 78750 95892 82228 [Mw 52][OEIS 61] ${\displaystyle \lim _{n\to \infty }{\frac {d_{n}}{d_{n+1}}}}$  1979 ${\displaystyle \mathbb {T} }$ ?
Fractal dimension of the Cantor set [76] ${\displaystyle d_{f}(k)}$  0.63092 97535 71457 43709 [Mw 56][OEIS 62] ${\displaystyle \lim _{\varepsilon \to 0}{\frac {\log N(\varepsilon )}{\log(1/\varepsilon )}}={\frac {\log 2}{\log 3}}}$  Before 1979[OEIS 62] ${\displaystyle \mathbb {T} }$
Connective constant [77][78] ${\displaystyle {\mu }}$  1.84775 90650 22573 51225 [Mw 57][OEIS 63] ${\displaystyle {\sqrt {2+{\sqrt {2}}}}\;=\lim _{n\rightarrow \infty }c_{n}^{1/n}}$

as a root of the polynomial ${\displaystyle :\;x^{4}-4x^{2}+2=0}$

1982[79] ${\displaystyle \mathbb {A} }$
Salem number,[80] ${\displaystyle {\sigma _{_{10}}}}$  1.17628 08182 59917 50654 [Mw 58][OEIS 64] ${\displaystyle x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1}$  1983? ${\displaystyle \mathbb {A} }$
Chebyshev constant [81] · [82] ${\displaystyle \lambda _{\text{Ch}}}$  0.59017 02995 08048 11302 [Mw 59][OEIS 65] ${\displaystyle {\frac {\Gamma ({\tfrac {1}{4}})^{2}}{4\pi ^{3/2}}}={\frac {4({\tfrac {1}{4}}!)^{2}}{\pi ^{3/2}}}}$  Before 1987[Mw 59]
Conway constant [83] ${\displaystyle {\lambda }}$  1.30357 72690 34296 39125 [Mw 60][OEIS 66] ${\displaystyle {\begin{smallmatrix}x^{71}\quad \ -x^{69}-2x^{68}-x^{67}+2x^{66}+2x^{65}+x^{64}-x^{63}-x^{62}-x^{61}-x^{60}\\-x^{59}+2x^{58}+5x^{57}+3x^{56}-2x^{55}-10x^{54}-3x^{53}-2x^{52}+6x^{51}+6x^{50}\\+x^{49}+9x^{48}-3x^{47}-7x^{46}-8x^{45}-8x^{44}+10x^{43}+6x^{42}+8x^{41}-5x^{40}\\-12x^{39}+7x^{38}-7x^{37}+7x^{36}+x^{35}-3x^{34}+10x^{33}+x^{32}-6x^{31}-2x^{30}\\-10x^{29}-3x^{28}+2x^{27}+9x^{26}-3x^{25}+14x^{24}-8x^{23}\quad \ -7x^{21}+9x^{20}\\+3x^{19}\!-4x^{18}\!-10x^{17}\!-7x^{16}\!+12x^{15}\!+7x^{14}\!+2x^{13}\!-12x^{12}\!-4x^{11}\!-2x^{10}\\+5x^{9}+x^{7}\quad \ -7x^{6}+7x^{5}-4x^{4}+12x^{3}-6x^{2}+3x-6\ =\ 0\quad \quad \quad \end{smallmatrix}}}$  1987 ${\displaystyle \mathbb {A} }$
Prévost constant, Reciprocal Fibonacci constant[84] ${\displaystyle \Psi }$  3.35988 56662 43177 55317 [Mw 61][OEIS 67] ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{n}}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{8}}+{\frac {1}{13}}+\cdots }$  Before 1988[OEIS 67] ${\displaystyle \mathbb {R} \setminus \mathbb {Q} }$
Brun 2 constant = Σ inverse of Twin primes [85] ${\displaystyle {B}_{\,2}}$  1.90216 05831 04 [Mw 62][OEIS 68] ${\displaystyle \textstyle {\underset {p,\,p+2:{\text{ prime}}}{\sum ({\frac {1}{p}}+{\frac {1}{p+2}})}}=({\frac {1}{3}}\!+\!{\frac {1}{5}})+({\tfrac {1}{5}}\!+\!{\tfrac {1}{7}})+({\tfrac {1}{11}}\!+\!{\tfrac {1}{13}})+\cdots }$  1989[OEIS 68]
Hafner–Sarnak–McCurley constant (1) [86] ${\displaystyle {\sigma }}$  0.35323 63718 54995 98454 [Mw 63][OEIS 69] ${\displaystyle \prod _{k=1}^{\infty }\left\{1-[1-\prod _{j=1}^{n}{\underset {p_{k}:{\text{ prime}}}{(1-p_{k}^{-j})]^{2}}}\right\}}$  1993
Fractal dimension of the Apollonian packing of circles
[87][88]

${\displaystyle \varepsilon }$
1.30568 6729 ≈ by Thomas & Dhar
1.30568 8 ≈ by McMullen [Mw 64][OEIS 70]
1994
1998
Backhouse's constant [89] ${\displaystyle {B}}$  1.45607 49485 82689 67139 [Mw 65][OEIS 71] ${\displaystyle \lim _{k\to \infty }\left|{\frac {q_{k+1}}{q_{k}}}\right\vert \quad \scriptstyle {\text{where:}}\displaystyle \;\;Q(x)={\frac {1}{P(x)}}=\!\sum _{k=1}^{\infty }q_{k}x^{k}}$

${\displaystyle P(x)=\sum _{k=1}^{\infty }{\underset {p_{k}{\text{ prime}}}{p_{k}x^{k}}}=1+2x+3x^{2}+5x^{3}+\cdots }$

1995
Viswanath constant[90] ${\displaystyle {C}_{Vi}}$  1.13198 82487 943 [Mw 66][OEIS 72] ${\displaystyle \lim _{n\to \infty }|a_{n}|^{\frac {1}{n}}}$       where an = Fibonacci sequence 1997 ${\displaystyle \mathbb {T} }$ ?
Time constant [91] ${\displaystyle {\tau }}$  0.63212 05588 28557 67840 [Mw 67][OEIS 73] ${\displaystyle \lim _{n\to \infty }1-{\frac {!n}{n!}}=\lim _{n\to \infty }P(n)=\int _{0}^{1}e^{-x}dx=1{-}{\frac {1}{e}}=}$

${\displaystyle \sum \limits _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n!}}={\frac {1}{1!}}{-}{\frac {1}{2!}}{+}{\frac {1}{3!}}{-}{\frac {1}{4!}}{+}{\frac {1}{5!}}{-}{\frac {1}{6!}}{+}\cdots }$

Before 1997[91] ${\displaystyle \mathbb {T} }$
Komornik–Loreti constant [92] ${\displaystyle {q}}$  1.78723 16501 82965 93301 [Mw 68][OEIS 74] ${\displaystyle 1=\!\sum _{n=1}^{\infty }{\frac {t_{k}}{q^{k}}}\qquad \scriptstyle {\text{Raiz real de}}\displaystyle \prod _{n=0}^{\infty }\!\left(\!1{-}{\frac {1}{q^{2^{n}}}}\!\right)\!{+}{\frac {q{-}2}{q{-}1}}=0}$  1998 ${\displaystyle \mathbb {T} }$
Regular paperfolding sequence [93][94] ${\displaystyle {P_{f}}}$  0.85073 61882 01867 26036 [Mw 69][OEIS 75] ${\displaystyle \sum _{n=0}^{\infty }{\frac {8^{2^{n}}}{2^{2^{n+2}}-1}}=\sum _{n=0}^{\infty }{\cfrac {\tfrac {1}{2^{2^{n}}}}{1-{\tfrac {1}{2^{2^{n+2}}}}}}}$  Before 1998[94]
Artin constant [95] ${\displaystyle {C}_{Artin}}$  0.37395 58136 19202 28805 [Mw 70][OEIS 76] ${\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}(p_{n}-1)}}\right)\quad p_{n}\scriptstyle {\text{ = prime}}}$  1999
MRB constant[96][97][98] ${\displaystyle C_{{}_{MRB}}}$  0.18785 96424 62067 12024 [Mw 71][Ow 1][OEIS 77] ${\displaystyle \sum _{n=1}^{\infty }(-1)^{n}(n^{1/n}-1)=-{\sqrt[{1}]{1}}+{\sqrt[{2}]{2}}-{\sqrt[{3}]{3}}+\cdots }$  1999
Somos' quadratic recurrence constant [99] ${\displaystyle {\sigma }}$  1.66168 79496 33594 12129 [Mw 72][OEIS 78] ${\displaystyle \prod _{n=1}^{\infty }n^{{1/2}^{n}}={\sqrt {1{\sqrt {2{\sqrt {3\cdots }}}}}}=1^{1/2}\;2^{1/4}\;3^{1/8}\cdots }$  1999[Mw 72] ${\displaystyle \mathbb {T} }$ ?

## 2000 onwards

Name Symbol Decimal Expansion Formula Year Set
Foias constant α [100] ${\displaystyle F_{\alpha }}$  1.18745 23511 26501 05459 [Mw 73][OEIS 79] ${\displaystyle x_{n+1}=\left(1+{\frac {1}{x_{n}}}\right)^{n}{\text{ for }}n=1,2,3,\ldots }$

Foias constant is the unique real number

such that if x1 = α then the sequence diverges to ∞. When x1 = α, ${\displaystyle \,\lim _{n\to \infty }x_{n}{\tfrac {\log n}{n}}=1}$

2000
Foias constant β ${\displaystyle F_{\beta }}$  2.29316 62874 11861 03150 [Mw 73][OEIS 80] ${\displaystyle x^{x+1}=(x+1)^{x}}$  2000
Raabe's formula [101] ${\displaystyle {\zeta '(0)}}$  0.91893 85332 04672 74178 [Mw 74][OEIS 81] ${\displaystyle \int \limits _{a}^{a+1}\log \Gamma (t)\,\mathrm {d} t={\tfrac {1}{2}}\log 2\pi +a\log a-a,\quad a\geq 0}$  Before 2011[101]
Kepler–Bouwkamp constant [102] ${\displaystyle {\rho }}$  0.11494 20448 53296 20070 [Mw 75][OEIS 82] ${\displaystyle \prod _{n=3}^{\infty }\cos \left({\frac {\pi }{n}}\right)=\cos \left({\frac {\pi }{3}}\right)\cos \left({\frac {\pi }{4}}\right)\cos \left({\frac {\pi }{5}}\right)...}$  Before 2013[102]

Prouhet–Thue–Morse constant [103] ${\displaystyle \tau }$  0.41245 40336 40107 59778 [Mw 76][OEIS 83] ${\displaystyle \sum _{n=0}^{\infty }{\frac {t_{n}}{2^{n+1}}}}$     where ${\displaystyle {t_{n}}}$  is the Thue–Morse sequence  and
Where ${\displaystyle \tau (x)=\sum _{n=0}^{\infty }(-1)^{t_{n}}\,x^{n}=\prod _{n=0}^{\infty }(1-x^{2^{n}})}$
Before 2014[103] ${\displaystyle \mathbb {T} }$
Heath-Brown–Moroz constant[104] ${\displaystyle {C_{_{HBM}}}}$  0.00131 76411 54853 17810 [Mw 77][OEIS 84] ${\displaystyle {\underset {p_{n}:\,{prime}}{\prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}}}\right)^{7}\left(1+{\frac {7p_{n}+1}{p_{n}^{2}}}\right)}}}$  Before 2002[104] ${\displaystyle \mathbb {T} }$ ?
Lebesgue constant [105] ${\displaystyle {C_{1}}}$  0.98943 12738 31146 95174 [Mw 78][OEIS 85] ${\displaystyle \lim _{n\to \infty }\!\!\left(\!{L_{n}{-}{\frac {4}{\pi ^{2}}}\ln(2n{+}1)}\!\!\right)\!{=}{\frac {4}{\pi ^{2}}}\!\left({\sum _{k=1}^{\infty }\!{\frac {2\ln k}{4k^{2}{-}1}}}{-}{\frac {\Gamma '({\tfrac {1}{2}})}{\Gamma ({\tfrac {1}{2}})}}\!\!\right)}$  Before 2002[105]
2nd du Bois-Reymond constant [106] ${\displaystyle {C_{2}}}$  0.19452 80494 65325 11361 [Mw 79][OEIS 86] ${\displaystyle {\frac {e^{2}-7}{2}}=\int _{0}^{\infty }\left|{{\frac {d}{dt}}\left({\frac {\sin t}{t}}\right)^{n}}\right|\,dt-1}$  Before 2003[106] ${\displaystyle \mathbb {T} }$
Stephens constant [107] ${\displaystyle C_{S}}$  0.57595 99688 92945 43964 [Mw 80][OEIS 87] ${\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {p}{p^{3}-1}}\right)}$  Before 2005[107] ${\displaystyle \mathbb {T} }$ ?
Taniguchi constant [107] ${\displaystyle C_{T}}$  0.67823 44919 17391 97803 [Mw 81][OEIS 88] ${\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {3}{{p_{n}}^{3}}}+{\frac {2}{{p_{n}}^{4}}}+{\frac {1}{{p_{n}}^{5}}}-{\frac {1}{{p_{n}}^{6}}}\right)}$
${\displaystyle \scriptstyle p_{n}=\,{\text{prime}}}$
Before 2005[107] ${\displaystyle \mathbb {T} }$ ?
Copeland–Erdős constant [108] ${\displaystyle {{\mathcal {C}}_{CE}}}$  0.23571 11317 19232 93137 [Mw 82][OEIS 89] ${\displaystyle \sum _{n=1}^{\infty }{\frac {p_{n}}{10^{n+\sum \limits _{k=1}^{n}\lfloor \log _{10}{p_{k}}\rfloor }}}}$  Before 2012[108] ${\displaystyle \mathbb {R} \setminus \mathbb {Q} }$
Hausdorff dimension, Sierpinski triangle [109] ${\displaystyle {\log _{2}3}}$  1.58496 25007 21156 18145 [Mw 83][OEIS 90] ${\displaystyle {\frac {\log 3}{\log 2}}={\frac {\sum _{n=0}^{\infty }{\frac {1}{2^{2n+1}(2n+1)}}}{\sum _{n=0}^{\infty }{\frac {1}{3^{2n+1}(2n+1)}}}}={\frac {{\frac {1}{2}}+{\frac {1}{24}}+{\frac {1}{160}}+\cdots }{{\frac {1}{3}}+{\frac {1}{81}}+{\frac {1}{1215}}+\cdots }}}$  Before 2002[109] ${\displaystyle \mathbb {T} }$
Landau–Ramanujan constant [110] ${\displaystyle K}$  0.76422 36535 89220 66299 [Mw 84][OEIS 91] ${\displaystyle {\frac {1}{\sqrt {2}}}\prod _{p\equiv 3\!\!\!\!\!\mod \!4}\!\!{\underset {\!\!\!\!\!\!\!\!p:{\text{ prime}}}{\left(1-{\frac {1}{p^{2}}}\right)^{-{\frac {1}{2}}}}}\!\!={\frac {\pi }{4}}\prod _{p\equiv 1\!\!\!\!\!\mod \!4}\!\!{\underset {\!\!\!\!p:{\text{ prime}}}{\left(1-{\frac {1}{p^{2}}}\right)^{\frac {1}{2}}}}}$  Before 2005[110] ${\displaystyle \mathbb {T} }$ ?
Brun 4 constant = Σ inv.prime quadruplets [111] ${\displaystyle {B}_{\,4}}$  0.87058 83799 75 [Mw 62][OEIS 92] ${\displaystyle \textstyle {\sum ({\frac {1}{p}}+{\frac {1}{p+2}}+{\frac {1}{p+6}}+{\frac {1}{p+8}})}\scriptstyle \quad {p,\;p+2,\;p+6,\;p+8:{\text{ prime}}}}$

${\displaystyle \textstyle {\left({\tfrac {1}{5}}+{\tfrac {1}{7}}+{\tfrac {1}{11}}+{\tfrac {1}{13}}\right)}+\left({\tfrac {1}{11}}+{\tfrac {1}{13}}+{\tfrac {1}{17}}+{\tfrac {1}{19}}\right)+\dots }$

Before 2002[111]
Ramanujan nested radical [112] ${\displaystyle R_{5}}$  2.74723 82749 32304 33305 ${\displaystyle \scriptstyle {\sqrt {5+{\sqrt {5+{\sqrt {5-{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5-\cdots }}}}}}}}}}}}}}\;=\textstyle {\frac {2+{\sqrt {5}}+{\sqrt {15-6{\sqrt {5}}}}}{2}}}$  Before 2001[112] ${\displaystyle \mathbb {A} }$

## Other constants

Name Symbol Decimal Expansion Formula Year Set
DeVicci's tesseract constant ${\displaystyle {f_{(3,4)}}}$  1.00743 47568 84279 37609[Mw 85][OEIS 93] The largest cube that can pass through in an 4D hypercube.

Positive root of ${\displaystyle :\;\;4x^{4}{-}28x^{3}{-}7x^{2}{+}16x{+}16=0}$

${\displaystyle \mathbb {A} }$
Glaisher–Kinkelin constant ${\displaystyle {A}}$  1.28242 71291 00622 63687[Mw 86][OEIS 94] ${\displaystyle e^{{\frac {1}{12}}-\zeta ^{\prime }(-1)}=e^{{\frac {1}{8}}-{\frac {1}{2}}\sum \limits _{n=0}^{\infty }{\frac {1}{n+1}}\sum \limits _{k=0}^{n}\left(-1\right)^{k}{\binom {n}{k}}\left(k+1\right)^{2}\ln(k+1)}}$

## Notes

1. ^ 1 can be given as a primitive notion within Peano arithmetic. Alternatively, 0 can be a primitive notion in Peano arithmetic and 1 defined as the successor to 0. This article uses the former definition for pedagogical and chronological simplicity.
2. ^ Both i 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 of i and -i is in some ways arbitrary, but a useful notational device. See imaginary unit for more information.
3. ^ Can also be defined by the infinite series ${\displaystyle \!\sum _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{0!}}+{\frac {1}{1}}+{\frac {1}{2!}}+{\frac {1}{3!}}+\textstyle \cdots }$

## References

1. ^ a b c d "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-08.
2. ^ Weisstein, Eric W. "Constant". mathworld.wolfram.com. Retrieved 2020-08-08.
3. ^ Arndt & Haenel 2006, p. 167
4. ^ Calvin C Clawson (2001). Mathematical sorcery: revealing the secrets of numbers. p. IV. ISBN 978 0 7382 0496-3.
5. ^
6. ^ Vijaya AV (2007). Figuring Out Mathematics. Dorling Kindcrsley (India) Pvt. Lid. p. 15. ISBN 978-81-317-0359-5.
7. ^ P A J Lewis (2008). Essential Mathematics 9. Ratna Sagar. p. 24. ISBN 9788183323673.
8. ^ Timothy Gowers; June Barrow-Green; Imre Leade (2007). The Princeton Companion to Mathematics. Princeton University Press. p. 316. ISBN 978-0-691-11880-2.
9. ^ Kim Plofker (2009), Mathematics in India, Princeton University Press, ISBN 978-0-691-12067-6, pp. 54–56. Quote – "In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, [ ...] Pingala's use of a zero symbol [śūnya] as a marker seems to be the first known explicit reference to zero." Kim Plofker (2009), Mathematics in India, Princeton University Press, ISBN 978-0-691-12067-6, 55–56. "In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, there are five questions concerning the possible meters for any value "n". [ ...] The answer is (2)7 = 128, as expected, but instead of seven doublings, the process (explained by the sutra) required only three doublings and two squarings – a handy time saver where "n" is large. Pingala's use of a zero symbol as a marker seems to be the first known explicit reference to zero.
10. ^ Plutarch. "718ef". Quaestiones convivales VIII.ii. And therefore Plato himself dislikes Eudoxus, Archytas, and Menaechmus for endeavoring to bring down the doubling the cube to mechanical operations
11. ^ Keith J. Devlin (1999). Mathematics: The New Golden Age. Columbia University Press. p. 66. ISBN 978-0-231-11638-1.
12. ^ E.Kasner y J.Newman. (2007). Mathematics and the Imagination. Conaculta. p. 77. ISBN 978-968-5374-20-0.
13. ^ O'Connor, J J; Robertson, E F. "The number e". MacTutor History of Mathematics.
14. ^ Annie Cuyt; Vigdis Brevik Petersen; Brigitte Verdonk; Haakon Waadeland; William B. Jones (2008). Handbook of Continued Fractions for Special Functions. Springer. p. 182. ISBN 978-1-4020-6948-2.
15. ^ Cajori, Florian (1991). A History of Mathematics (5th ed.). AMS Bookstore. p. 152. ISBN 0-8218-2102-4.
16. ^ O'Connor, J. J.; Robertson, E. F. (September 2001). "The number e". The MacTutor History of Mathematics archive. Retrieved 2009-02-02.
17. ^ William Dunham (2005). The Calculus Gallery: Masterpieces from Newton to Lebesgue. Princeton University Press. p. 51. ISBN 978-0-691-09565-3.
18. ^ Jean Jacquelin (2010). SOPHOMORE'S DREAM FUNCTION.
19. ^ J. Coates; Martin J. Taylor (1991). L-Functions and Arithmetic. Cambridge University Press. p. 333. ISBN 978-0-521-38619-7.
20. ^ "Greek/Hebrew/Latin-based Symbols in Mathematics". Math Vault. 2020-03-20. Retrieved 2020-08-08.
21. ^ Robert Baillie (2013). "Summing The Curious Series of Kempner and Irwin". arXiv:0806.4410 [math.CA].
22. ^ Leonhard Euler (1749). Consideratio quarumdam serierum, quae singularibus proprietatibus sunt praeditae. p. 108.
23. ^ Howard Curtis (2014). Orbital Mechanics for Engineering Students. Elsevier. p. 159. ISBN 978-0-08-097747-8.
24. ^ Keith B. Oldham; Jan C. Myland; Jerome Spanier (2009). An Atlas of Functions: With Equator, the Atlas Function Calculator. Springer. p. 15. ISBN 978-0-387-48806-6.
25. ^ Nielsen, Mikkel Slot. (July 2016). Undergraduate convexity : problems and solutions. p. 162. ISBN 9789813146211. OCLC 951172848.
26. ^ Johann Georg Soldner (1809). Théorie et tables d'une nouvelle fonction transcendante (in French). J. Lindauer, München. p. 42.
27. ^ Lorenzo Mascheroni (1792). Adnotationes ad calculum integralem Euleri (in Latin). Petrus Galeatius, Ticini. p. 17.
28. ^ Steven Finch (2014). Errata and Addenda to Mathematical Constants (PDF). Harvard.edu. Archived from the original (PDF) on 2016-03-16. Retrieved 2013-12-17.
29. ^ Calvin C. Clawson (2003). Mathematical Traveler: Exploring the Grand History of Numbers. Perseus. p. 187. ISBN 978-0-7382-0835-0.
30. ^ L. J. Lloyd James Peter Kilford (2008). Modular Forms: A Classical and Computational Introduction. Imperial College Press. p. 107. ISBN 978-1-84816-213-6.
31. ^ Henri Cohen (2000). Number Theory: Volume II: Analytic and Modern Tools. Springer. p. 127. ISBN 978-0-387-49893-5.
32. ^ H. M. Srivastava; Choi Junesang (2001). Series Associated With the Zeta and Related Functions. Kluwer Academic Publishers. p. 30. ISBN 978-0-7923-7054-3.
33. ^ E. Catalan (1864). Mémoire sur la transformation des séries, et sur quelques intégrales définies, Comptes rendus hebdomadaires des séances de l'Académie des sciences 59. Kluwer Academic éditeurs. p. 618.
34. ^ James Stewart (2010). Single Variable Calculus: Concepts and Contexts. Brooks/Cole. p. 314. ISBN 978-0-495-55972-6.
35. ^ Julian Havil (2003). Gamma: Exploring Euler's Constant. Princeton University Press. p. 64. ISBN 9780691141336.
36. ^ Eric W. Weisstein (2003). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. p. 151. ISBN 978-1-58488-347-0.
37. ^ Holger Hermanns; Roberto Segala (2000). Process Algebra and Probabilistic Methods. Springer-Verlag. p. 270. ISBN 978-3-540-67695-9.
38. ^ Yann Bugeaud (2004). Series representations for some mathematical constants. p. 72. ISBN 978-0-521-82329-6.
39. ^ Steven Finch (2014). Errata and Addenda to Mathematical Constants (PDF). Harvard.edu. p. 59. Archived from the original (PDF) on 2016-03-16. Retrieved 2013-12-17.
40. ^ Osborne, George Abbott (1891). An Elementary Treatise on the Differential and Integral Calculus. Leach, Shewell, and Sanborn. pp. 250.
41. ^ Annie Cuyt; Vigdis Brevik Petersen; Brigitte Verdonk; Haakon Waadelantl; William B. Jones. (2008). Handbook of Continued Fractions for Special Functions. Springer. p. 188. ISBN 978-1-4020-6948-2.
42. ^ See Jensen 1895.
43. ^ David Wells (1997). The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books Ltd. p. 4. ISBN 9780141929408.
44. ^ Tijdeman, Robert (1976). "On the Gel'fond–Baker method and its applications". In Felix E. Browder (ed.). Mathematical Developments Arising from Hilbert Problems. Proceedings of Symposia in Pure Mathematics. XXVIII.1. American Mathematical Society. pp. 241–268. ISBN 0-8218-1428-1. Zbl 0341.10026.
45. ^ Helmut Brass; Knut Petras (2010). Quadrature Theory: The Theory of Numerical Integration on a Compact Interval. AMS. p. 274. ISBN 978-0-8218-5361-0.
46. ^
47. ^ Eric W. Weisstein (2002). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. p. 1356. ISBN 9781420035223.
48. ^ Mauro Fiorentini. Nielsen – Ramanujan (costanti di).
49. ^ Robert P. Munafo (2012). Pixel Counting.
50. ^ Steven Finch. Volumes of Hyperbolic 3-Manifolds (PDF). Harvard University. Archived from the original (PDF) on 2015-09-19.
51. ^ Lloyd N. Trefethen (2013). Approximation Theory and Approximation Practice. SIAM. p. 211. ISBN 978-1-611972-39-9.
52. ^ R. M. ABRAROV AND S. M. ABRAROV (2011). "PROPERTIES AND APPLICATIONS OF THE PRIME DETECTING FUNCTION". arXiv:1109.6557 [math.GM].
53. ^ Ian Stewart (1996). Professor Stewart's Cabinet of Mathematical Curiosities. Birkhäuser Verlag. ISBN 978-1-84765-128-0.
54. ^ Eric W. Weisstein (2003). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. p. 1688. ISBN 978-1-58488-347-0.
55. ^ Eric W. Weisstein (2002). CRC Concise Encyclopedia of Mathematics. Crc Press. p. 1212. ISBN 9781420035223.
56. ^ ECKFORD COHEN (1962). SOME ASYMPTOTIC FORMULAS IN THE THEORY OF NUMBERS (PDF). University of Tennessee. p. 220.
57. ^ Michael J. Dinneen; Bakhadyr Khoussainov; Prof. Andre Nies (2012). Computation, Physics and Beyond. Springer. p. 110. ISBN 978-3-642-27653-8.
58. ^ David Cohen (2006). Precalculus: With Unit Circle Trigonometry. Thomson Learning Inc. p. 328. ISBN 978-0-534-40230-3.
59. ^ Julian Havil (2003). Gamma: Exploring Euler's Constant. Princeton University Press. p. 161. ISBN 9780691141336.
60. ^ Aleksandr I͡Akovlevich Khinchin (1997). Continued Fractions. Courier Dover Publications. p. 66. ISBN 978-0-486-69630-0.
61. ^ Marek Wolf (2018). "Two arguments that the nontrivial zeros of the Riemann zeta function are irrational". Computational Methods in Science and Technology. 24 (4): 215–220. arXiv:1002.4171. doi:10.12921/cmst.2018.0000049. S2CID 115174293.
62. ^ Laith Saadi (2004). Stealth Ciphers. Trafford Publishing. p. 160. ISBN 978-1-4120-2409-9.
63. ^ Annie Cuyt; Viadis Brevik Petersen; Brigitte Verdonk; William B. Jones (2008). Handbook of continued fractions for special functions. Springer Science. p. 190. ISBN 978-1-4020-6948-2.
64. ^ a b Andras Bezdek (2003). Discrete Geometry. Marcel Dekkcr, Inc. p. 150. ISBN 978-0-8247-0968-6.
65. ^ Lowe, I. J. (1959-04-01). "Free Induction Decays of Rotating Solids". Physical Review Letters. 2 (7): 285–287. doi:10.1103/PhysRevLett.2.285. ISSN 0031-9007.
66. ^ Steven Finch (2007). Continued Fraction Transformation (PDF). Harvard University. p. 7. Archived from the original (PDF) on 2016-04-19. Retrieved 2015-02-28.
67. ^ Robin Whitty. Lieb's Square Ice Theorem (PDF).
68. ^ Ivan Niven. Averages of exponents in factoring integers (PDF).
69. ^ a b Jean-Pierre Serre (1969–1970). Travaux de Baker (PDF). NUMDAM, Séminaire N. Bourbaki. p. 74.
70. ^ Michel A. Théra (2002). Constructive, Experimental, and Nonlinear Analysis. CMS-AMS. p. 77. ISBN 978-0-8218-2167-1.
71. ^ Kathleen T. Alligood (1996). Chaos: An Introduction to Dynamical Systems. Springer. ISBN 978-0-387-94677-1.
72. ^ David Darling (2004). The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes. Wiley & Sons inc. p. 63. ISBN 978-0-471-27047-8.
73. ^ Dusko Letic; Nenad Cakic; Branko Davidovic; Ivana Berkovic. Orthogonal and diagonal dimension fluxes of hyperspherical function (PDF). Springer.
74. ^ Steven R. Finch (2003). Mathematical Constants. Cambridge University Press. p. 479. ISBN 978-3-540-67695-9. Schmutz.
75. ^ K. T. Chau; Zheng Wang (201). Chaos in Electric Drive Systems: Analysis, Control and Application. John Wiley & Son. p. 7. ISBN 978-0-470-82633-1.
76. ^ Paul Manneville (2010). Instabilities, Chaos and Turbulence. Imperial College Press. p. 176. ISBN 978-1-84816-392-8.
77. ^ Mireille Bousquet-Mélou. Two-dimensional self-avoiding walks (PDF). CNRS, LaBRI, Bordeaux, France.
78. ^ Hugo Duminil-Copin & Stanislav Smirnov (2011). The connective constant of the honeycomb lattice √ (2 + √ 2) (PDF). Université de Geneve.
79. ^ B. Nienhuis (1982). "Exact critical point and critical exponents of O(n) models in two dimensions". Phys. Rev. Lett. 49 (15): 1062–1065. Bibcode:1982PhRvL..49.1062N. doi:10.1103/PhysRevLett.49.1062.
80. ^ Pei-Chu Hu,Chung-Chun (2008). Distribution Theory of Algebraic Numbers. Hong Kong University. p. 246. ISBN 978-3-11-020536-7.
81. ^ Steven Finch (2014). Electrical Capacitance (PDF). Harvard.edu. p. 1. Archived from the original (PDF) on 2016-04-19. Retrieved 2015-10-12.
82. ^ Thomas Ransford. Computation of Logarithmic Capacity (PDF). Université Laval, Quebec (QC), Canada. p. 557.
83. ^ Facts On File, Incorporated (1997). Mathematics Frontiers. p. 46. ISBN 978-0-8160-5427-5.
84. ^ Gérard P. Michon (2005). Numerical Constants. Numericana.
85. ^ Thomas Koshy (2007). Elementary Number Theory with Applications. Elsevier. p. 119. ISBN 978-0-12-372-487-8.
86. ^ Steven R. Finch (2003). Mathematical Constants. p. 110. ISBN 978-3-540-67695-9.
87. ^ Benoit Mandelbrot (2004). Fractals and Chaos: The Mandelbrot Set and Beyond. ISBN 978-1-4419-1897-0.
88. ^ Curtis T. McMullen (1997). Hausdorff dimension and conformal dynamics III: Computation of dimension (PDF).
89. ^ Eric W. Weisstein (2003). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. p. 151. ISBN 978-1-58488-347-0.
90. ^ DIVAKAR VISWANATH (1999). RANDOM FIBONACCI SEQUENCES AND THE NUMBER 1.13198824... (PDF). MATHEMATICS OF COMPUTATION.
91. ^ a b Kunihiko Kaneko; Ichiro Tsuda (1997). Complex Systems: Chaos and Beyond. p. 211. ISBN 978-3-540-67202-9.
92. ^ Christoph Lanz. k-Automatic Reals (PDF). Technischen Universität Wien.
93. ^ Francisco J. Aragón Artacho; David H. Baileyy; Jonathan M. Borweinz; Peter B. Borwein (2012). Tools for visualizing real numbers (PDF). p. 33.
94. ^ a b Papierfalten (PDF). 1998.
95. ^ Paulo Ribenboim (2000). My Numbers, My Friends: Popular Lectures on Number Theory. Springer. p. 66. ISBN 978-0-387-98911-2.
96. ^ Richard E. Crandall (2012). Unified algorithms for polylogarithm, L-series, and zeta variants (PDF). perfscipress.com. Archived from the original on 2013-04-30.CS1 maint: BOT: original-url status unknown (link)
97. ^ RICHARD J. MATHAR (2010). "NUMERICAL EVALUATION OF THE OSCILLATORY INTEGRAL OVER exp(I pi x)x^1/x BETWEEN 1 AND INFINITY". arXiv:0912.3844 [math.CA].
98. ^ M.R.Burns (1999). Root constant. Marvin Ray Burns.
99. ^ Jesus Guillera; Jonathan Sondow (2008). "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent". The Ramanujan Journal. 16 (3): 247–270. arXiv:math/0506319. doi:10.1007/s11139-007-9102-0. S2CID 119131640.
100. ^ Andrei Vernescu (2007). Gazeta Matemetica Seria a revista de cultur Matemetica Anul XXV(CIV)Nr. 1, Constante de tip Euler generalízate (PDF). p. 14.
101. ^ a b István Mezö (2011). "On the integral of the fourth Jacobi theta function". arXiv:1106.1042 [math.NT].
102. ^ a b Richard J. Mathar (2013). "Circumscribed Regular Polygons". arXiv:1301.6293 [math.MG].
103. ^ a b Steven Finch (2014). Errata and Addenda to Mathematical Constants (PDF). Harvard.edu. p. 53. Archived from the original (PDF) on 2016-03-16. Retrieved 2013-12-17.
104. ^ a b J. B. Friedlander; A. Perelli; C. Viola; D.R. Heath-Brown; H.Iwaniec; J. Kaczorowski (2002). Analytic Number Theory. Springer. p. 29. ISBN 978-3-540-36363-7.
105. ^ a b Horst Alzer (2002). "Journal of Computational and Applied Mathematics, Volume 139, Issue 2" (PDF). Journal of Computational and Applied Mathematics. 139 (2): 215–230. doi:10.1016/S0377-0427(01)00426-5.
106. ^ a b Steven R. Finch (2003). Mathematical Constants. Cambridge University Press. p. 238. ISBN 978-3-540-67695-9.
107. ^ a b c d Steven Finch (2005). Class Number Theory (PDF). Harvard University. p. 8. Archived from the original (PDF) on 2016-04-19. Retrieved 2014-04-15.
108. ^ a b Yann Bugeaud (2012). Distribution Modulo One and Diophantine Approximation. Cambridge University Press. p. 87. ISBN 978-0-521-11169-0.
109. ^ a b Eric W. Weisstein (2002). CRC Concise Encyclopedia of Mathematics (Second ed.). CRC Press. p. 1356. ISBN 978-1-58488-347-0.
110. ^ a b Richard E. Crandall; Carl B. Pomerance (2005). Prime Numbers: A Computational Perspective. Springer. p. 80. ISBN 978-0387-25282-7.
111. ^ a b Pascal Sebah & Xavier Gourdon (2002). Introduction to twin primes and Brun's constant computation (PDF).
112. ^ a b Bruce C. Berndt; Robert Alexander Rankin (2001). Ramanujan: essays and surveys. American Mathematical Society, London Mathematical Society. p. 219. ISBN 978-0-8218-2624-9.

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## Bibliography

• Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 2013-06-05.CS1 maint: ref=harv (link) 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