List of mathematical constants explained

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] 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		data-sort-value="-2000"	Prehistory	data-sort-value="1"	N
Two	2	2		data-sort-value="-2000"	Prehistory	data-sort-value="1"	N
One half	1/2	data-sort-value="0.50000"	0.5		data-sort-value="-2000"	Prehistory	data-sort-value="3"	Q
Pi	\pi	3.14159 26535 89793 23846	Ratio of a circle's circumference to its diameter.	data-sort-value="-1900"	1900 to 1600 BCE	data-sort-value="5"	R \setminus A
Tau	\tau	6.28318 53071 79586 47692^[2]	Ratio of a circle's circumference to its radius. Equivalent to 2 \pi	data-sort-value="-1900"	1900 to 1600 BCE	data-sort-value="5"	R \setminus A
Square root of 2,Pythagoras constant.^[3]	\sqrt{ 2 }	1.41421 35623 73095 04880	Positive root of x ^{2=2}	data-sort-value="-1800"	1800 to 1600 BCE^[4]	data-sort-value="4"	A
Square root of 3,Theodorus' constant^[5]	\sqrt{ 3 }	1.73205 08075 68877 29352	Positive root of x ^{2=3}	data-sort-value="-465"	465 to 398 BCE	data-sort-value="4"	A
Square root of 5^[6]	\sqrt{ 5 }	2.23606 79774 99789 69640	Positive root of x ^{2=5}	data-sort-value="-464"		data-sort-value="4"	A
Phi, Golden ratio^[7]	\varphi or \phi	1.61803 39887 49894 84820	(1+ \sqrt{ 5)/() }	data-sort-value="-301"	~300 BCE	data-sort-value="4"	A
Silver ratio^[8]	\delta _{S}	2.41421 35623 73095 04880	\sqrt{ 2 } +1	data-sort-value="-301"	~300 BCE	data-sort-value="4"	A
Zero	0	0		data-sort-value="-300"	300 to 100 BCE^[9]	data-sort-value="2"	Z
Negative one	−1	−1		data-sort-value="-300"	300 to 200 BCE	data-sort-value="2"	Z
Cube root of 2	\sqrt[ 3 ]{ 2 }	1.25992 10498 94873 16476	Real root of x ^{3=2}	46 to 120 CE^[10]	data-sort-value="4"	A
Cube root of 3	\sqrt[ 3 ]{ 3 }	1.44224 95703 07408 38232	Real root of x ^{3=3}	data-sort-value="47"		data-sort-value="4"	A
	\sqrt[ 12 ]{ 2 }	1.05946 30943 59295 26456	Real root of x ^{12} =2	data-sort-value="47"		data-sort-value="4"	A
Supergolden ratio^[11]	\psi	1.46557 12318 76768 02665	(1 + \sqrt[ 3 ]{ (29+3 \sqrt{ 93)/())/(2) } + \sqrt[3]}Real root of x ^{3} = x ^{2} + 1	data-sort-value="47"		data-sort-value="4"	A
Imaginary unit^[12]	i	data-sort-value="0"		Principal root of x ^{2=-1}	1501 to 1576	data-sort-value="8"	C
Connective constant for the hexagonal lattice^[13] ^[14]	\mu	1.84775 90650 22573 51225	\sqrt{ 2 + \sqrt{ 2 }} , as a root of the polynomial x ^{4-4} x ^{2} + 2=0	1593	data-sort-value="4"	A
Kepler–Bouwkamp constant^[15]	K'	0.11494 20448 53296 20070	\prod ^{inf ty} _{n =3} \cos\left( (\pi)/(n) \right) = \cos\left( (\pi)/(3) \right) \cos\left( (\pi)/(4) \right) \cos\left( (\pi)/(5) \right) ...	1596	data-sort-value="7"
Wallis's constant		2.09455 14815 42326 59148	\sqrt[ 3 ]{ (45- \sqrt{ 1929)/() }}+\sqrt[3]Real root of x ^{3} - 2 x - 5 = 0	1616 to 1703	data-sort-value="4"	A
Euler's number^[16]	e	2.71828 18284 59045 23536	\lim _{n} \left( 1 + (1)/(n) \right) ^{n} = \sum ^{inf ty} _{n =0} (1)/(n!) = 1 + (1)/(1 !) + (1)/(2 !) + (1)/(3 !) …	1618^[17]	data-sort-value="5"	R \setminus A
Natural logarithm of 2^[18]	ln 2	0.69314 71805 59945 30941	Real root of e ^{x} = 2 \sum ^{inf ty} _{n =1} (( -1 ) ^{n+1})/(n) = (1)/(1) - (1)/(2) + (1)/(3) - (1)/(4) + …	1619^[19] & 1668^[20]	data-sort-value="5"	R \setminus A
Lemniscate constant^[21]	\varpi	2.62205 75542 92119 81046	\pi {G} = 4 \sqrt{\tfrac 2 \pi}\Gamma{\left(\tfrac 54 \right) ^{2}} = \tfrac 14 \sqrt{\tfrac{ 2 }{\pi}}\Gamma {\left(\tfrac 14 \right) ^{2}} where G is Gauss's constant	1718 to 1798	data-sort-value="5"	R \setminus A
Euler's constant	\gamma	0.57721 56649 01532 86060	\lim _{n\toinfty} \left( -log n + \sum ^{n} _{k =1} (1 {k}\right) = \int)/(_{1}) ^{inf ty\left( - (1 x + (1 {\lfloor)/(x\rfloor}\right)dx))/()}	1735	data-sort-value="7"
Erdős–Borwein constant^[22]	E	1.60669 51524 15291 76378	\sum ^{inf ty} _{n =1} (1)/(2 ^{n-1}) = (1)/(1) + \	\frac \! + \	\frac \! + \	\frac \! + \	\cdots	1749^[23]	data-sort-value="6"	R \setminus Q
Omega constant	\Omega	0.56714 32904 09783 87299	W( 1 ) = (1)/(\pi) \int ^{\pi log \left( 1+ (\sin t)/(t)} _{0} e ^{t\cot} \right)dt where W is the Lambert W function	1758 & 1783	data-sort-value="5"	R \setminus A
Apéry's constant^[24]	\zeta( 3 )	1.20205 69031 59594 28539	\sum ^{inf ty (1)/(n ^{3})} _{n =1} = (1)/(+ 1 ^{3}) (1)/(2 ^{3}) + (1)/(3 ^{3}) + (1)/(4 ^{3}) + (1)/(5 ^{3}) + …	1780	data-sort-value="6"	R \setminus A
First non-trivial zero of the zeta function	\gamma _{1}	14.13472 51417 34693 79045	\Im(x),\zeta(x) =0	1840
Laplace limit^[25]		0.66274 34193 49181 58097	Real root of (x e ^{\sqrt{x} ^{2+1})/() } = 1	data-sort-value="1782"	~1782	data-sort-value="5"	R \setminus A
Soldner constant^[26] ^[27]	\mu	1.45136 92348 83381 05028	li (x) = \int ^{x} _{0} (dt)/(ln t) = 0 root of the logarithmic integral function.	1792	data-sort-value="7"
Gauss's constant^[28]	G	0.83462 68416 74073 18628	(1)/(agm ( 1 , \sqrt{ 2) )} = (\Gamma( (1)/() ^{2} 4))/(2 \sqrt{ 2 \pi ^{3}) } = \frac\int_0^1\fracwhere agm is the arithmetic–geometric mean	1799^[29]	data-sort-value="5"	R \setminus A
Second Hermite constant^[30]	\gamma _{2}	1.15470 05383 79251 52901	(2)/(\sqrt{ 3) }	1822 to 1901	data-sort-value="4"	A
Liouville's constant^[31]	L	0.11000 10000 00000 00000 0001	\sum ^{inf ty} _{n =1} (1)/(10 ^{n!}) = (1)/(10 ^{1!}) + (1)/(10 ^{2!}) + (1)/(10 ^{3!}) + (1)/(10 ^{4!}) + …	data-sort-value="1844"	Before 1844	data-sort-value="5"	R \setminus A
First continued fraction constant	C _{1}	0.69777 46579 64007 98201	\tfrac 1 { 1+ \tfrac 1 { 2+ \tfrac 1 { 3+ \tfrac 1 { 4+ \tfrac 1 { 5+ … }}}}} (I _{1(2)})/(I _{0(2)}) , where I _{\alpha} (x) is the modified Bessel function	1855^[32]	data-sort-value="6"	R \setminus Q
Ramanujan's constant^[33]		262 53741 26407 68743 .99999 99999 99250 073	e ^{\pi\sqrt{163} }	1859	data-sort-value="5"	R \setminus A
Glaisher–Kinkelin constant	A	1.28242 71291 00622 63687	e ^{(1)/(- \zeta ^{\prime(-1)} 12)} = e ^{(1)/(- (1)/(2) \sum\limits ^{inf ty} _{n =0} (1)/(n +1) \sum\limits ^{n} _{k =0} \left( -1 \right) ^{k} \binom{n 8)} {k} \left(k +1 \right) ^{2} ln (k +1 )}	1860	data-sort-value="7"
Catalan's constant^[34] ^[35] ^[36]	G	0.91596 55941 77219 01505	\int ^{1} _{0} \	\int_0^1 \!\	\frac= \! \sum_^\infty \	\frac \!= \	\frac\frac	1864	data-sort-value="7"
Dottie number^[37]		0.73908 51332 15160 64165	Real root of \cos x = x	1865	data-sort-value="5"	R \setminus A
Meissel–Mertens constant^[38]	M	0.26149 72128 47642 78375	\lim _{n\toinfty} \left(\sum _{p\le} (1)/(p) -lnln n\right) = \gamma + \sum _{p} \left( ln \left( 1 - (1)/(p) \right) + (1)/(p) \right) where γ is the Euler–Mascheroni constant and p is prime	1866 & 1873	data-sort-value="7"
Universal parabolic constant^[39]	P	2.29558 71493 92638 07403	ln ( 1 + \sqrt 2 ) + \sqrt 2 = \operatorname{arsinh}( 1 ) + \sqrt{ 2 }	data-sort-value="1891"	Before 1891^[40]	data-sort-value="5"	R \setminus A
Cahen's constant^[41]	C	0.64341 05462 88338 02618	\sum ^{inf ty} _{k =1} (( -1 ) ^{k})/(s _{k-1}) = (1)/(1) - (1)/(2) + (1)/(6) - (1)/(42) + (1)/(1806) {\pm … } where sk is the kth term of Sylvester's sequence 2, 3, 7, 43, 1807, ...	1891	data-sort-value="5"	R \setminus A
Gelfond's constant^[42]	e ^{\pi}	23.14069 26327 79269 0057	( -1 ) ^{-i} = i ^{-2i} = \sum ^{inf ty} _{n =0} (\pi ^{n})/(n!) = 1 + (\pi ^{1})/(1) + (\pi ^{2})/(2) + (\pi ^{3})/(6) + …	1900^[43]	data-sort-value="5"	R \setminus A
Gelfond–Schneider constant^[44]	2 ^{\sqrt{2} }	2.66514 41426 90225 18865		data-sort-value="1902"	Before 1902	data-sort-value="5"	R \setminus A
Second Favard constant^[45]	K _{2}	1.23370 05501 36169 82735	(\pi ^{2})/(8) = \sum ^{inf ty} _{n =0} (1)/(( 2 n -1 ) ^{2}) = (1)/(+ 1 ^{2}) (1)/(+ 3 ^{2}) (1)/(+ 5 ^{2}) (1)/(7 ^{2}) + …	1902 to 1965	data-sort-value="5"	R \setminus A
Golden angle^[46]	g	2.39996 32297 28653 32223	(2 \pi)/(\varphi ^{2}) = \pi ( 3- \sqrt{ 5 }) or180 ( 3- \sqrt{ 5 }) =137 . 50776 \ldots in degrees	1907	data-sort-value="5"	R \setminus A
Sierpiński's constant^[47]	K	2.58498 17595 79253 21706	\begin{align} &\pi\left( 2 \gamma +ln (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{align}	1907	data-sort-value="7"
Landau–Ramanujan constant^[48]	K	0.76422 36535 89220 66299	(1 {\sqrt 2 }\prod)/(_{{p}) \atop p {\rm prime}} {\left( 1- (1 {p)/(^{2}\right)}) ^{- (1)/(2)} }\!\	=\frac\pi4\prod_	1908	data-sort-value="7"
First Nielsen–Ramanujan constant^[49]	a _{1}	0.82246 70334 24113 21823	({\zeta)/(( 2 )}{ 2 }) = (\pi ^{2})/(12) = \sum ^{inf ty (( -1 ) ^{n+1})/(n ^{2})} _{n =1} = (1)/(1 ^{2}) { - } (1)/(2 ^{2}) { + } (1)/(3 ^{2}) { - } (1)/(4 ^{2}) { + } …	1909	data-sort-value="5"	R \setminus A
Gieseking constant^[50]	G	1.01494 16064 09653 62502	(3 \sqrt{ 3)/() } \left(1- \sum_^\infty \frac+ \sum_^\infty\frac \right)= style (3 \sqrt{ 3)/() } \left(1 - \frac + \frac-\frac+\frac-\frac+\frac \pm \cdots \right).	1912	data-sort-value="7"
Bernstein's constant^[51]	\beta	0.28016 94990 23869 13303	\lim _{n\toinfty} 2 n E _{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	data-sort-value="7"
Tribonacci constant^[52]		1.83928 67552 14161 13255	$\backslash frac\; =\; \backslash frac$Real root of x ^{3} - x ^{2} - x - 1 = 0	1914 to 1963	data-sort-value="4"	A
Brun's constant^[53]	B _{2}	1.90216 05831 04	style {\sum\limits _{p( (1 {p} + (1 {p +2 })})/(=))/(( (1 { 3 })/())} + \	\frac1) + (\tfrac1 \! + \	\tfrac1) + (\tfrac1 \! + \	\tfrac1) + \cdots where the sum ranges over all primes p such that p + 2 is also a prime	1919	data-sort-value="7"
Twin primes constant	C _{2}	0.66016 18158 46869 57392	\prod _{style{p {\rmprime} \atop p \ge 3 }} \left( 1 - (1)/((p -1 ) ^{2}) \right)	1922	data-sort-value="7"
Plastic ratio^[54]	\rho	1.32471 79572 44746 02596	\sqrt[ 3 ]{ 1 + \sqrt[ 3 ]{ 1 + \sqrt[ 3 ]{ 1 + … }}} =style \sqrt[ 3 ]{ (1)/(+ 2) (\sqrt{ 69)/() }}+\sqrt[3]Real root of x ^{3} = x + 1	1924	data-sort-value="4"	A
Bloch's constant^[55]	B	data-sort-value=0.43320	0 . 4332 \leq B\leq 0 . 4719	The best known bounds are (\sqrt{ 3)/() }+2\times10^\leq B\leq \sqrt\cdot\frac	1925	data-sort-value="7"
Z score for the 97.5 percentile point^[56] ^[57]	z _{.975}	1.95996 39845 40054 23552	\sqrt{ 2 }\operatorname{erf} ^{-1} ( 0 . 95 ) where is the inverse error functionReal number z such that (1)/(\sqrt{ 2 \pi) }\int_^ e^ \, \mathrmx = 0.975	1925	data-sort-value="7"
Landau's constant	L	data-sort-value=0.50000	0 . 5 < L \le 0 . 54326	The best known bounds are 0 . 5 < L \le (\Gamma( (1)/()\Gamma( (5)/(6) ) 3))/(\Gamma( (1)/() 6))	1929	data-sort-value="7"
Landau's third constant	A	data-sort-value=0.50000	0 . 5 < A \le 0 . 7853		1929	data-sort-value="7"
Prouhet–Thue–Morse constant^[58]	\tau	0.41245 40336 40107 59778	\sum ^{inf ty} _{n =0} (t _{n})/(2 ^{n+1}) = (1)/(4) \left[ 2- \prod ^{inf ty} _{n =0} \left( 1- (1)/(2 ^{2 ^{n}}) \right)\right] where {t _{n}} is the n^th term of the Thue–Morse sequence	1929	data-sort-value="5"	R \setminus A
Golomb–Dickman constant^[59]	λ	0.62432 99885 43550 87099	\int ^{1} _{0} e ^{Li(t)} dt = \int ^{inf ty} _{0} (\rho(t))/(t +2) dt where Li(t) is the logarithmic integral, and ρ(t) is the Dickman function	1930 & 1964	data-sort-value="7"
Constant related to the asymptotic behavior of Lebesgue constants^[60]	c	0.98943 12738 31146 95174	\lim _{n\toinfty} \	\left(\!\	\!\right)\	\frac\!\left(\frac\	\!\right)	1930	data-sort-value="7"
Feller–Tornier constant^[61]	l{C} _{FT}	0.66131 70494 69622 33528	{ (1)/(2) \prod _{pprime} \left( 1- (2)/(p ^{2}) \right) + (1)/(2) } =\frac\prod_ \left(1-\frac\right) + \frac	1932	data-sort-value="7"
Base 10 Champernowne constant^[62]	C _{10}	0.12345 67891 01112 13141	Defined by concatenating representations of successive integers:0.1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...	1933	data-sort-value="5"	R \setminus A
Salem constant^[63]	\sigma _{10}	1.17628 08182 59917 50654	Largest real root of x ^{10} + x ^{9-x} ^{7-x} ^{6-x} ^{5-x} ^{4-x} ^{3+x+1=0}	1933	data-sort-value="4"	A
Khinchin's constant^[64]	K _{0}		\prod ^{inf ty} _{n =1} \left[{ 1+ { 1 \over n(n +2 )}}\right] ^{log _{2(n)}}	1934	data-sort-value="7"
Lévy's constant (1)^[65]	\beta	1.18656 91104 15625 45282	(\pi ^{2})/(12ln2)	1935	data-sort-value="7"
Lévy's constant (2)^[66]	e ^{\beta}	3.27582 29187 21811 15978	e ^{\pi ^{2/(12ln2)}}	1936	data-sort-value="7"
Copeland–Erdős constant^[67]	l{C} _{CE}	0.23571 11317 19232 93137	Defined by concatenating representations of successive prime numbers:0.2 3 5 7 11 13 17 19 23 29 31 37 ...	1946	data-sort-value="6"	R \setminus Q
Mills' constant^[68]	A	1.30637 78838 63080 69046	Smallest positive real number A such that \lfloor A ^{3 ^{n}} \rfloor is prime for all positive integers n	1947	data-sort-value="7"
Gompertz constant^[69]	\delta	0.59634 73623 23194 07434	\int ^{inf ty} _{0} \	\frac \, dx = \!\	\int_0^1 \!\	\frac =	data-sort-value="1948"	Before 1948	data-sort-value="7"
de Bruijn–Newman constant	Λ	data-sort-value="0"	0 \le Λ \le 0 . 2	The number Λ such that H( λ ,z) = \int ^{λ u ^{2}} _{0 e} \Phi(u)\cos(zu)du has real zeros if and only if λ ≥ Λ. where \Phi(u) = \sum ^{inf ty} _{n =1} ( 2 \pi ^{2n} ^{4e} ^{9u} -3 \pi n ^{2e} ^{5u} )e ^{- \pi n ^{2e} ^{4u}} .	1950	data-sort-value="7"
Van der Pauw constant	(\pi)/(ln2)	4.53236 01418 27193 80962		data-sort-value="1958"	Before 1958	data-sort-value="6"	R \setminus Q
Magic angle^[70]	\theta _{m}	0.95531 66181 245092 78163	\arctan \sqrt{ 2 } = \arccos \tfrac{ 1 }{\sqrt 3 } ≈ style { 54 . 7356 }	data-sort-value="1959"	Before 1959^[71]	data-sort-value="5"	R \setminus A
Artin's constant^[72]	C _{Artin}	0.37395 58136 19202 28805	\prod _{pprime} \left( 1- (1)/(p(p -1 )) \right)	data-sort-value="1961"	Before 1961	data-sort-value="7"
Porter's constant^[73]	C	1.46707 80794 33975 47289	(6ln2)/(\pi ^{2}) \left( 3 ln 2 + 4 \gamma - (24)/(\pi ^{2}) \zeta '( 2 ) -2 \right) - (1)/(2) where γ is the Euler–Mascheroni constant and is the derivative of the Riemann zeta function evaluated at s = 2	1961	data-sort-value="7"
Lochs constant^[74]	L	0.97027 01143 92033 92574	(6ln2ln10)/(\pi ^{2})	1964	data-sort-value="7"
DeVicci's tesseract constant		1.00743 47568 84279 37609	The largest cube that can pass through in an 4D hypercube.Positive root of 4 x ^{8{-}28x} ^{6{-}7x} ^{4{+}16x} ^{2{+}16=0}	1966	data-sort-value="4"	A
Lieb's square ice constant^[75]		1.53960 07178 39002 03869	\left( (4)/(3) \right) ^{(3)/(2)} = (8)/(3 \sqrt 3)	1967	data-sort-value="4"	A
Niven's constant^[76]	C	1.70521 11401 05367 76428	1+ \sum ^{inf ty} _{n =2} \left( 1- (1)/(\zeta(n)) \right)	1969	data-sort-value="7"
Stephens' constant^[77]		0.57595 99688 92945 43964	\prod _{pprime} \left( 1 - (p)/(p ^{3-1}) \right)	1969	data-sort-value="7"
Regular paperfolding sequence^[78] ^[79]	P	0.85073 61882 01867 26036	\sum ^{inf ty} _{n =0} (8 ^{2 ^{n}})/(2 ^{2 ^{n+2}} -1) = \sum ^{inf ty} _{n =0} \cfrac {\tfrac { 1 }{ 2 ^{2 ^{n}} }}	1970	data-sort-value="5"	R \setminus A
Reciprocal Fibonacci constant^[80]	\psi	3.35988 56662 43177 55317	\sum ^{inf ty} _{n =1} (1)/(F _{n}) = (1)/(1) + (1)/(1) + (1)/(2) + (1)/(3) + (1)/(5) + (1)/(8) + (1)/(13) + … where Fn is the n^th Fibonacci number	1974	data-sort-value="6"	R \setminus Q
Chvátal–Sankoff constant for the binary alphabet	\gamma _{2}	data-sort-value=0.78807 10000	0 . 788071 \le \gamma _{2} \le 0 . 826280	\lim _{n\toinfty} (\operatorname{E)/([ λ) _{n,2} ]}{n} where is the expected longest common subsequence of two random length-n binary strings	1975	data-sort-value="7"
Feigenbaum constant δ^[81]	\delta	4.66920 16091 02990 67185	\lim _{n} (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	data-sort-value="7"
Chaitin's constants^[82]	\Omega	data-sort-value=0.00787 49969 97812 3844	In general they are uncomputable numbers. But one such number is 0.00787 49969 97812 3844.	\sum _{p} 2 ^{-|p|} Halted program Size in bits of program Domain of all programs that stop. See also: Halting problem.	1975	data-sort-value="5"	R \setminus A
Robbins constant^[83]	\Delta( 3 )	0.66170 71822 67176 23515	(4+17 \sqrt 2-6 \sqrt 3-7 \pi)/(105) + \	\frac \! + \	\frac	1977	data-sort-value="5"	R \setminus A
Weierstrass constant^[84]		0.47494 93799 87920 65033	(2 ^{5/4} \sqrt{\pi)/() e ^{\pi/8} }	data-sort-value="1978"	Before 1978^[85]	data-sort-value="5"	R \setminus A
Fransén–Robinson constant^[86]	F	2.80777 02420 28519 36522	\int ^{inf ty} _{0} (dx)/(\Gamma(x)) = e + \int ^{inf ty} _{0} (e ^{-x})/(\pi ^{2} +ln ^{2} x) dx	1978	data-sort-value="7"
Feigenbaum constant α^[87]	\alpha	2.50290 78750 95892 82228	Ratio between the width of a tine and the width of one of its two subtines in a bifurcation diagram	1979	data-sort-value="7"
Second du Bois-Reymond constant^[88]	C _{2}	0.19452 80494 65325 11361	(e ^{2-7})/(2) = \int ^{inf ty} _{0} \left	\right	\,dt-1	1983	data-sort-value="5"	R \setminus A
Erdős–Tenenbaum–Ford constant	\delta	0.08607 13320 55934 20688	1- (1+loglog2)/(log2)	1984	data-sort-value="7"
Conway's constant^[89]	λ	1.30357 72690 34296 39125	Real root of the polynomial:\begin{smallmatrix} x ^{71} - x ^{69} -2 x ^{68} - x ^{67} +2 x ^{66} +2 x ^{65} + x ^{64} - x ^{63} - x ^{62} - x ^{61} - x ^{60} \\ - x ^{59} +2 x ^{58} +5 x ^{57} +3 x ^{56} -2 x ^{55} -10 x ^{54} -3 x ^{53} -2 x ^{52} +6 x ^{51} +6 x ^{50} \\ + x ^{49} +9 x ^{48} -3 x ^{47} -7 x ^{46} -8 x ^{45} -8 x ^{44} +10 x ^{43} +6 x ^{42} +8 x ^{41} -5 x ^{40} \\ -12 x ^{39} +7 x ^{38} -7 x ^{37} +7 x ^{36} + x ^{35} -3 x ^{34} +10 x ^{33} + x ^{32} -6 x ^{31} -2 x ^{30} \\ -10 x ^{29} -3 x ^{28} +2 x ^{27} +9 x ^{26} -3 x ^{25} +14 x ^{24} -8 x ^{23} -7 x ^{21} +9 x ^{20} \\ +3 x ^{19} -4 x ^{18} \	-10x^\!-7x^\	+12x^\!+7x^\	+2x^\!-12x^\	-4x^\!-2x^\\+5x^+x^-7x^+7x^-4x^+12x^-6x^+3x-6\ =\ 0 \quad\quad\quad\end	1987	data-sort-value="4"	A
Hafner–Sarnak–McCurley constant^[90]	\sigma	0.35323 63718 54995 98454	\prod _{pprime} {\left( 1- \left( 1- \prod _{n\ge1} \left( 1- (1)/(p ^{n}) \right)\right) ^{2} \right)}	1991	data-sort-value="7"
Backhouse's constant^[91]	B	1.45607 49485 82689 67139	\lim _{k} \left	\frac \right \vert \quad \scriptstyle \text \displaystyle \;\; Q(x)=\frac= \! \sum_^\infty q_k x^k P(x) = 1+ \sum ^{inf ty} _{k =1} {p _{k} x ^{k}} = 1+2 x +3 x ^{2+5x} ^{3+ …} where pk is the k^th prime number	1995	data-sort-value="7"
Viswanath constant^[92]		1.13198 82487 943	\lim _{n}	f_n	^\frac      where fn = fn−1 ± fn−2, where the signs + or − are chosen at random with equal probability 1/2	1997	data-sort-value="7"
Komornik–Loreti constant^[93]	q	1.78723 16501 82965 93301	Real number q such that 1 = \sum ^{inf ty} _{k =1} (t _{k})/(q ^{k}) , or \prod ^{inf ty\left} _{n =0} ( 1- (1)/(q ^{2 ^{n}}) \right ) + (q -2)/(q -1) =0 where tk is the k^th term of the Thue–Morse sequence	1998	data-sort-value="5"	R \setminus A
Embree–Trefethen constant	\beta ^{\star}	0.70258		1999	data-sort-value="7"
Heath-Brown–Moroz constant^[94]	C	0.00131 76411 54853 17810	\prod _{pprime} \left( 1- (1)/(p) \right) ^{7 \left( 1+ (7 p +1)/(p ^{2})} \right)	1999	data-sort-value="7"
MRB constant^[95] ^[96] ^[97]	S	0.18785 96424 62067 12024 ^[98]	\sum ^{inf ty} _{n =1} ( -1 ) ^{n} (n ^{1/n} -1 ) = - \sqrt[ 1 ]{ 1 } + \sqrt[ 2 ]{ 2 } - \sqrt[ 3 ]{ 3 } + …	1999	data-sort-value="7"
Prime constant^[99]	\rho	0.41468 25098 51111 66024	\sum _{pprime} (1)/(2 ^{p}) = (1)/(4) + (1)/(8) + (1)/(32) + …	1999	data-sort-value="6"	R \setminus Q
Somos' quadratic recurrence constant^[100]	\sigma	1.66168 79496 33594 12129	\prod ^{inf ty} _{n =1} n ^{{1/2} ^{n}} = \sqrt { 1 \sqrt { 2 \sqrt{ 3 … }}} = 1 ^{1/2} 2 ^{1/4} 3 ^{1/8} …	1999	data-sort-value="7"
Foias constant^[101]	\alpha	1.18745 23511 26501 05459	x _{n+1} = \left( 1 + (1)/(x _{n}) \right) ^{nforn=1,2,3,\ldots} Foias constant is the unique real number such that if x1 = α then the sequence diverges to infinity.	2000	data-sort-value="7"
Logarithmic capacity of the unit disk^[102] ^[103]		0.59017 02995 08048 11302	(\Gamma(\tfrac 14 ) ^{2})/(4 \pi ^{3/2})	data-sort-value="2003"	Before 2003	data-sort-value="5"	R \setminus A
Taniguchi constant		0.67823 44919 17391 97803	\prod _{pprime} \left( 1 - (3)/(+ p ^{3}) (2)/(+ p ^{4}) (1)/(- p ^{5}) (1)/(p ^{6}) \right)	data-sort-value="2005"	Before 2005	data-sort-value="7"

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	Z	0.00000 00000	[0; ]
Golomb–Dickman constant	λ		0.62432 99885	[0; 1, 1, 1, 1, 1, 22, 1, 2, 3, 1, 1, 11, 1, 1, 2, 22, 2, 6, 1, 1, …]	E. Weisstein noted that the continued fraction has an unusually large number of 1s.
Cahen's constant	C _{2}	R \setminus A	0.64341 05463	[0; 1, 1, 1, 2<sup>2</sup>, 3<sup>2</sup>, 13<sup>2</sup>, 129<sup>2</sup>, 25298<sup>2</sup>, 420984147<sup>2</sup>, 269425140741515486<sup>2</sup>, …]	All terms are squares and truncated at 10 terms due to large size. Davison and Shallit used the continued fraction expansion to prove that the constant is transcendental.
Euler–Mascheroni constant	\gamma		0.57721 56649	[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, 1, …]	Using the continued fraction expansion, it was shown that if is rational, its denominator must exceed 10^244663.
First continued fraction constant	C _{1}	R \setminus Q	0.69777 46579	[0; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, …]	Equal to the ratio I _{1} ( 2 )/I _{0} ( 2 ) of modified Bessel functions of the first kind evaluated at 2.
Catalan's constant	G		0.91596 55942	[0; 1, 10, 1, 8, 1, 88, 4, 1, 1, 7, 22, 1, 2, 3, 26, 1, 11, 1, 10, 1, …]	Computed up to terms by E. Weisstein.
One half	1/2	Q	0.50000 00000	[0; 2]
Prouhet–Thue–Morse constant	\tau	R \setminus A	0.41245 40336	[0; 2, 2, 2, 1, 4, 3, 5, 2, 1, 4, 2, 1, 5, 44, 1, 4, 1, 2, 4, 1, …]	Infinitely many partial quotients are 4 or 5, and infinitely many partial quotients are greater than or equal to 50.^[104]
Copeland–Erdős constant	l{C} _{CE}	R \setminus Q	0.23571 11317	[0; 4, 4, 8, 16, 18, 5, 1, 1, 1, 1, 7, 1, 1, 6, 2, 9, 58, 1, 3, 4, …]	Computed up to terms by E. Weisstein. He also noted that while the Champernowne constant continued fraction contains sporadic large terms, the continued fraction of the Copeland–Erdős Constant do not exhibit this property.
Base 10 Champernowne constant	C _{10}	R \setminus A	0.12345 67891	[0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, {{val|4.57540e165|fmt=none}}, 6, 1, …]	Champernowne constants in any base exhibit sporadic large numbers; the 40th term in C _{10} has 2504 digits.
One	1	N	1.00000 00000	[1; ]
Phi, Golden ratio	\varphi	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 n^th roots of the denominators of the n^th convergents are close to Khinchin's constant, suggesting that B _{2} is irrational. If true, this will prove the twin prime conjecture.^[105]
Square root of 2	\sqrt 2	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	N	2.00000 00000	[2; ]
Euler's number	e	R \setminus 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, …]	The continued fraction expansion has the pattern [2; 1, 2, 1, 1, 4, 1, ..., 1, 2''n'', 1, ...].
Khinchin's constant	K _{0}		2.68545 20011	[2; 1, 2, 5, 1, 1, 2, 1, 1, 3, 10, 2, 1, 3, 2, 24, 1, 3, 2, 3, 1, …]	For almost all real numbers x, the coefficients of the continued fraction of x have a finite geometric mean known as Khinchin's constant.
Three	3	N	3.00000 00000	[3; ]
Pi	\pi	R \setminus 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, …]	The first few convergents (3, 22/7, 333/106, 355/113, ...) are among the best-known and most widely used historical approximations of .

Sequences of constants

Name	Symbol	Formula	Year	Set
Harmonic number	H _{n}	\sum ^{n} _{k =1} (1)/(k)	data-sort-value="-400"	Antiquity	data-sort-value="3"	Q
Gregory coefficients	G _{n}	(1)/(n!) \int ^{1} _{0} x(x -1 )(x -2 ) … (x - n +1 ) dx = \int ^{1} _{0} \binom x n dx	1670	data-sort-value="3"	Q
Bernoulli number	B ^{\pm} _{n}	(t)/(2) \left( \operatorname{coth} (t)/(2) \pm 1 \right) = \sum ^{inf ty} _{m =0} (B ^{\pm{})/(_{m}) t ^{m}{m!}}	1689	data-sort-value="3"	Q
Hermite constants^[106]	\gamma _{n}	For a lattice L in Euclidean space R^n with unit covolume, i.e. vol(R^n/L) = 1, let λ(L) denote the least length of a nonzero element of L. Then √γn is the maximum of λ(L) over all such lattices L.	data-sort-value="1822"	1822 to 1901	data-sort-value="7"	R
Hafner–Sarnak–McCurley constant^[107]	D(n)	D(n) = \prod ^{inf ty} _{k =1} \left\{ 1- \left[ 1- \prod ^{n} _{j =1} ( 1- p ^{- j} _{k} ) \right] ^{2} \right\}	data-sort-value="1883"	1883	data-sort-value="7"	R
Stieltjes constants	\gamma _{n}	{ (( -1 ) ^{n} n!)/(2 \pi) }\int _^e^\zeta \left(e^+1\right)dx.	data-sort-value="1894"	before 1894	data-sort-value="7"	R
Favard constants^[108]	K _{r}	(4)/(\pi) \sum ^{inf ty} _{n =0} \left( (( -1 ) ^{n})/(2 n +1) \right) ^{r+1} = (4)/(\pi) \left( (( -1 ) ^{0(r+1)})/(+ 1 ^{r}) (( -1 ) ^{1(r+1)})/(+ 3 ^{r}) (( -1 ) ^{2(r+1)})/(+ 5 ^{r}) (( -1 ) ^{3(r+1)})/(7 ^{r}) + … \right)	data-sort-value="1902"	1902 to 1965	data-sort-value="7"	R
Generalized Brun's Constant	B _{n}	{\sum\limits _{p( (1 {p} + (1 {p + n})})/())/()} where the sum ranges over all primes p such that p + n is also a prime	data-sort-value="1919"	1919	data-sort-value="7"	R
Champernowne constants	C _{b}	Defined by concatenating representations of successive integers in base b.C ^{inf ty} _{n =1} (n)/(b ^{\left(\sum ^{n} _{k =1} \lceil log _{b(k+1)\rceil\right)}})	1933	data-sort-value="5"	R \setminus A
Lagrange number	L(n)	\sqrt{ 9- (4)/({m _{n}) ^{2}}} where m _{n} is the nth smallest number such that m ^{2+x} ^{2+y} ^{2=3mxy} has positive (x,y).	data-sort-value="1957"	before 1957	data-sort-value="4"	A
Feller's coin-tossing constants	\alpha _{k,\beta} _{k}	\alpha _{k} is the smallest positive real root of x ^{k+1} =2 ^{k+1} (x -1 ),\beta _{k = (2- \alpha _{k})/(k +1- k\alpha _{k})}	1968	data-sort-value="4"	A
Stoneham number	\alpha _{b,c}	\sum _{n = c ^{k>1}} (1)/(b ^{nn}) = \sum ^{inf ty} _{k =1} (1)/(b ^{c ^{k}} c ^{k}) where b,c are coprime integers.	1973	data-sort-value="6"	R \setminus Q
Beraha constants	B(n)	2+2 \cos\left( (2 \pi)/(n) \right)	1974	data-sort-value="7"	A
Chvátal–Sankoff constants	\gamma _{k}	\lim _{n\toinfty} (E[ λ _{n,k} ])/(n)	1975	data-sort-value="7"	R
Hyperharmonic number	H ^{(r)} _{n}	\sum ^{n} _{k =1} H ^{(r -1 )} _{k} and H ^{( 0 )} _{n = (1)/(n)}	1995	data-sort-value="3"	Q
Gregory number	G _{x}	\sum ^{inf ty} ^{n =0} ( -1 ) ^{n} { (1)/(( 2 n +1 )x ^{2n+1}) } for rational x greater than one.	data-sort-value="1996"	before 1996	data-sort-value="7"	R
Metallic mean		(n + \sqrt{n ^{2+4})/() }	data-sort-value="1998"	before 1998	data-sort-value="4"	A

See also

• Invariant (mathematics)
• Glossary of mathematical symbols
• List of mathematical symbols by subject
• List of numbers
• List of physical constants
• Particular values of the Riemann zeta function
• Physical constant

References

Site OEIS Wiki

Bibliography

• Book: Arndt. Jörg. Haenel. Christoph. Pi Unleashed. Springer-Verlag. 2006. 978-3-540-66572-4 . 2013-06-05. English translation by Catriona and David Lischka.
  • Book: Handbook of Continued Fractions for Special Functions. Annie A.M.. Cuyt. Annie Cuyt. Vigdis. Petersen. Brigitte. Verdonk. Haakon. Waadeland. William B.. Jones. Springer Science + Business Media. 2008. 9781402069499. Mathematical constants . Dordrecht, Netherlands.
• Book: Neverending Fractions: An Introduction to Continued Fractions. 23. Australian Mathematical Society Lecture Series. 0950-2815. Jonathan. Borwein. Alf. van der Poorten. Jeffrey. Shallit. Wadim. Zudilin. Cambridge, United Kingdom. Cambridge University Press. 2014. 9780521186490.

Further reading

• Book: https://www.wolframscience.com/nks/notes-4-5--continued-fractions/ . 4: Systems Based on Numbers . Section 5: Mathematical Constants Continued fractions . Stephen . Wolfram . A New Kind of Science. A New Kind of Science.

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

Notes and References

1. Web site: Weisstein. Eric W.. Constant. 2020-08-08. mathworld.wolfram.com. en.
2. Web site: Hartl . Michael . 100,000 digits of Tau . Tau Day . 22 January 2023.
3. Book: Mathematical sorcery: revealing the secrets of numbers. Calvin C Clawson. 2001. 978 0 7382 0496-3. IV. Basic Books.
4. Fowler and Robson, p. 368.

   Photograph, illustration, and description of the root(2) tablet from the Yale Babylonian Collection

   High resolution photographs, descriptions, and analysis of the root(2) tablet (YBC 7289) from the Yale Babylonian Collection

5. Book: Figuring Out Mathematics. Vijaya AV. Dorling Kindcrsley (India) Pvt. Lid.. 2007. 978-81-317-0359-5. 15.
6. Book: Essential Mathematics 9. P A J Lewis. Ratna Sagar. 2008. 9788183323673. 24.
7. Book: The Princeton Companion to Mathematics. Timothy Gowers. June Barrow-Green. Imre Leade. Princeton University Press. 2007. 978-0-691-11880-2. 316.
8. .
9. Kim Plofker (2009), Mathematics in India, Princeton University Press,, pp. 54–56.
10. Book: Plutarch. Quaestiones convivales VIII.ii. 718ef. And therefore Plato himself dislikes Eudoxus, Archytas, and Menaechmus for endeavoring to bring down the doubling the cube to mechanical operations. 2019-05-24. 2009-11-19. https://web.archive.org/web/20091119061142/http://ebooks.adelaide.edu.au/p/plutarch/symposiacs/chapter8.html#section80. dead.
11. Book: Koshy . Thomas . Fibonacci and Lucas Numbers with Applications . 2017 . John Wiley & Sons . 9781118742174 . 2 . 14 August 2018 . en.
12. Book: Mathematics: The New Golden Age. Keith J. Devlin. Columbia University Press. 1999. 978-0-231-11638-1. 66.
13. Book: Two-dimensional self-avoiding walks. Mireille Bousquet-Mélou. CNRS, LaBRI, Bordeaux, France. Mireille Bousquet-Mélou.
14. Book: The connective constant of the honeycomb lattice √ (2 + √ 2). Hugo Duminil-Copin. Stanislav Smirnov. Université de Geneve. 2011. amp.
15. 1301.6293. math.MG. Richard J. Mathar. Circumscribed Regular Polygons. 2013.
16. Book: Mathematics and the Imagination. E.Kasner y J.Newman.. Conaculta. 2007. 978-968-5374-20-0. 77.
17. Web site: [<!-- http://www.gap-system.org/~history/PrintHT/e.html -->http://www-history.mcs.st-and.ac.uk/HistTopics/e.html The number ''e'']. O'Connor. J J. Robertson. E F. MacTutor History of Mathematics.
18. Book: Handbook of Continued Fractions for Special Functions. Annie Cuyt. Annie Cuyt . Vigdis Brevik Petersen. Brigitte Verdonk. Haakon Waadeland. William B. Jones. Springer. 2008. 978-1-4020-6948-2. 182.
19. Book: A History of Mathematics. Cajori. Florian. 1991. AMS Bookstore. 0-8218-2102-4. 5th. 152. Florian Cajori.
20. Web site: The number e. O'Connor. J. J.. September 2001. The MacTutor History of Mathematics archive. 2009-02-02. E. F.. Robertson.
21. Book: L-Functions and Arithmetic. J. Coates. Martin J. Taylor. Cambridge University Press. 1991. 978-0-521-38619-7. 333.
22. 0806.4410. math.CA. Robert Baillie. Summing The Curious Series of Kempner and Irwin. 2013.
23. Book: Consideratio quarumdam serierum, quae singularibus proprietatibus sunt praeditae. Leonhard Euler. 1749. 108.
24. Book: Handbook of Continued Fractions for Special Functions. Annie Cuyt. Vigdis Brevik Petersen. Brigitte Verdonk. Haakon Waadelantl. William B. Jones.. Springer. 2008. 978-1-4020-6948-2. 188.
25. Book: Orbital Mechanics for Engineering Students. Orbital Mechanics for Engineering Students. Howard Curtis. Elsevier. 2014. 978-0-08-097747-8. 159.
26. Book: Théorie et tables d'une nouvelle fonction transcendante. Johann Georg Soldner. J. Lindauer, München. 1809. 42. fr.
27. Book: Adnotationes ad calculum integralem Euleri. Lorenzo Mascheroni. Petrus Galeatius, Ticini. 1792. 17. la.
28. Book: An Atlas of Functions: With Equator, the Atlas Function Calculator. Keith B. Oldham. Jan C. Myland. Jerome Spanier. Springer. 2009. 978-0-387-48806-6. 15.
29. Book: Nielsen, Mikkel Slot.. Undergraduate convexity : problems and solutions. July 2016. 9789813146211. 162. World Scientific . 951172848.
30. Book: Errata and Addenda to Mathematical Constants. Steven Finch. Harvard.edu. 2014. 2013-12-17. https://web.archive.org/web/20160316175639/http://www.people.fas.harvard.edu/~sfinch/csolve/erradd.pdf. 2016-03-16. dead.
31. Book: Mathematical Traveler: Exploring the Grand History of Numbers. Calvin C. Clawson. Perseus. 2003. 978-0-7382-0835-0. 187.
32. Amoretti . F. . Sur la fraction continue [0,1,2,3,4,...] ]. Nouvelles annales de mathématiques . 1855 . 1 . 14 . 40–44 .
33. Book: Modular Forms: A Classical and Computational Introduction. L. J. Lloyd James Peter Kilford. Imperial College Press. 2008. 978-1-84816-213-6. 107.
34. Book: Number Theory: Volume II: Analytic and Modern Tools. Henri Cohen. Springer. 2000. 978-0-387-49893-5. 127.
35. Book: Series Associated With the Zeta and Related Functions. H. M. Srivastava. Choi Junesang. Kluwer Academic Publishers. 2001. 978-0-7923-7054-3. 30.
36. Book: Mémoire sur la transformation des séries, et sur quelques intégrales définies, Comptes rendus hebdomadaires des séances de l'Académie des sciences 59. E. Catalan. Kluwer Academic éditeurs. 1864. 618.
37. Book: Single Variable Calculus: Concepts and Contexts. James Stewart. Brooks/Cole. 2010. 978-0-495-55972-6. 314.
38. Book: Gamma: Exploring Euler's Constant. Julian Havil. Princeton University Press. 2003. 9780691141336. 64.
39. Book: Errata and Addenda to Mathematical Constants. Steven Finch. Harvard.edu. 2014. 59. 2013-12-17. https://web.archive.org/web/20160316175639/http://www.people.fas.harvard.edu/~sfinch/csolve/erradd.pdf. 2016-03-16. dead.
40. Book: Osborne, George Abbott. An Elementary Treatise on the Differential and Integral Calculus. Leach, Shewell, and Sanborn. 1891. 250.
41. Book: Series representations for some mathematical constants. Yann Bugeaud. 2004. 978-0-521-82329-6. 72. Cambridge University Press .
42. Book: The Penguin Dictionary of Curious and Interesting Numbers. David Wells. David G. Wells. Penguin Books Ltd.. 1997. 9780141929408. 4.
43. Book: Tijdeman, Robert. Mathematical Developments Arising from Hilbert Problems. American Mathematical Society. 1976. 0-8218-1428-1. Felix E. Browder. Felix Browder. Proceedings of Symposia in Pure Mathematics. XXVIII.1. 241–268. On the Gel'fond–Baker method and its applications. 0341.10026. Robert Tijdeman.
44. Book: Precalculus: With Unit Circle Trigonometry. David Cohen. Thomson Learning Inc.. 2006. 978-0-534-40230-3. 328.
45. Book: Quadrature Theory: The Theory of Numerical Integration on a Compact Interval. Helmut Brass. Knut Petras. AMS. 2010. 978-0-8218-5361-0. 274.
46. Book: Ángulo áureo.
47. Book: CRC Concise Encyclopedia of Mathematics, Second Edition. Eric W. Weisstein. CRC Press. 2002. 9781420035223. 1356.
48. Book: Prime Numbers: A Computational Perspective. Richard E. Crandall. Carl B. Pomerance. Springer. 2005. 978-0387-25282-7. 80.
49. Book: Nielsen – Ramanujan (costanti di). Mauro Fiorentini.
50. Book: Volumes of Hyperbolic 3-Manifolds. Steven Finch. Harvard University. https://web.archive.org/web/20150919160427/http://www.people.fas.harvard.edu/~sfinch/csolve/hyp.pdf. 2015-09-19. dead.
51. Book: Approximation Theory and Approximation Practice. Lloyd N. Trefethen. SIAM. 2013. 978-1-611972-39-9. 211.
52. Agronomof . M. . 1914 . Sur une suite récurrente. . Mathesis . 4 . 125–126.
53. Book: Thomas Koshy . Elementary Number Theory with Applications . Elsevier . 2007 . 978-0-12-372-487-8 . 119.
54. Book: Professor Stewart's Cabinet of Mathematical Curiosities. Ian Stewart. Birkhäuser Verlag. 1996. 978-1-84765-128-0.
55. Book: CRC Concise Encyclopedia of Mathematics, Second Edition. Eric W. Weisstein. CRC Press. 2003. 978-1-58488-347-0. 1688.
56. Web site: Engineering Statistics Handbook: Confidence Limits for the Mean . 2008-02-04 . National Institute of Standards and Technology . Although the choice of confidence coefficient is somewhat arbitrary, in practice 90%, 95%, and 99% intervals are often used, with 95% being the most commonly used. . https://web.archive.org/web/20080205120031/http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm . 5 February 2008 . dead . dmy-all.
57. 10.1080/03610920802255856. Swift. MB. Comparison of Confidence Intervals for a Poisson Mean – Further Considerations. Communications in Statistics – Theory and Methods. 2009 . 38. 5. 748–759. 120748700 . In modern applied practice, almost all confidence intervals are stated at the 95% level..
58. Book: Errata and Addenda to Mathematical Constants. Steven Finch. Harvard.edu. 2014. 53. 2013-12-17. https://web.archive.org/web/20160316175639/http://www.people.fas.harvard.edu/~sfinch/csolve/erradd.pdf. 2016-03-16. dead.
59. Book: CRC Concise Encyclopedia of Mathematics. Eric W. Weisstein. Crc Press. 2002. 9781420035223. 1212.
60. Horst Alzer. 2002. Journal of Computational and Applied Mathematics, Volume 139, Issue 2. Journal of Computational and Applied Mathematics. 139. 2. 215–230. 10.1016/S0377-0427(01)00426-5. free.
61. Book: SOME ASYMPTOTIC FORMULAS IN THE THEORY OF NUMBERS. ECKFORD COHEN. University of Tennessee. 1962. 220.
62. Book: Computation, Physics and Beyond. Michael J. Dinneen. Bakhadyr Khoussainov. Prof. Andre Nies. Springer. 2012. 978-3-642-27653-8. 110.
63. Book: Distribution Theory of Algebraic Numbers. Pei-Chu Hu, Chung-Chun. Hong Kong University. 2008. 978-3-11-020536-7. 246.
64. Book: Gamma: Exploring Euler's Constant. Julian Havil. Princeton University Press. 2003. 9780691141336. 161.
65. Book: Continued Fractions. Aleksandr I͡Akovlevich Khinchin. Courier Dover Publications. 1997. 978-0-486-69630-0. 66.
66. 1002.4171. Marek Wolf. Two arguments that the nontrivial zeros of the Riemann zeta function are irrational. Computational Methods in Science and Technology. 2018. 24. 4. 215–220. 10.12921/cmst.2018.0000049. 115174293.
67. Book: Distribution Modulo One and Diophantine Approximation. Yann Bugeaud. Cambridge University Press. 2012. 978-0-521-11169-0. 87.
68. Book: Stealth Ciphers. Laith Saadi. Trafford Publishing. 2004. 978-1-4120-2409-9. 160.
69. Book: Handbook of continued fractions for special functions. Annie Cuyt. Viadis Brevik Petersen. Brigitte Verdonk. William B. Jones. Springer Science. 2008. 978-1-4020-6948-2. 190.
70. Book: Discrete Geometry. Andras Bezdek. Marcel Dekkcr, Inc.. 2003. 978-0-8247-0968-6. 150.
71. Lowe. I. J.. 1959-04-01. Free Induction Decays of Rotating Solids. Physical Review Letters. en. 2. 7. 285–287. 10.1103/PhysRevLett.2.285. 1959PhRvL...2..285L . 0031-9007.
72. Book: My Numbers, My Friends: Popular Lectures on Number Theory. Paulo Ribenboim. Springer. 2000. 978-0-387-98911-2. 66.
73. Book: Constructive, Experimental, and Nonlinear Analysis. Michel A. Théra. CMS-AMS. 2002. 978-0-8218-2167-1. 77.
74. Book: Continued Fraction Transformation. Steven Finch. Harvard University. 2007. 7. 2015-02-28. https://web.archive.org/web/20160419114446/http://www.people.fas.harvard.edu/~sfinch/csolve/kz.pdf. 2016-04-19. dead.
75. Book: Lieb's Square Ice Theorem. Robin Whitty.
76. Book: Averages of exponents in factoring integers. Ivan Niven.
77. Book: Class Number Theory. Steven Finch. Harvard University. 2005. 8. 2014-04-15. https://web.archive.org/web/20160419150530/http://www.people.fas.harvard.edu/~sfinch/csolve/clss.pdf. 2016-04-19. dead.
78. Book: Tools for visualizing real numbers.. Francisco J. Aragón Artacho. David H. Baileyy. Jonathan M. Borweinz. Peter B. Borwein. 2012. 33. 2014-01-20. 2017-02-20. https://web.archive.org/web/20170220175429/https://carma.newcastle.edu.au/jon/tools1.pdf. dead.
79. Book: Papierfalten. 1998.
80. Book: Numerical Constants. Gérard P. Michon. Numericana. 2005.
81. Book: Chaos: An Introduction to Dynamical Systems. Kathleen T. Alligood. Springer. 1996. 978-0-387-94677-1.
82. Book: The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes. David Darling. Wiley & Sons inc.. 2004. 978-0-471-27047-8. 63.
83. Book: Mathematical Constants. registration. Schmutz.. Steven R. Finch. Cambridge University Press. 2003. 978-3-540-67695-9. 479.
84. Book: CRC Concise Encyclopedia of Mathematics, Second Edition. Eric W. Weisstein. CRC Press. 2003. 978-1-58488-347-0. 151.
85. Waldschmidt, M. "Nombres transcendants et fonctions sigma de Weierstrass." C. R. Math. Rep. Acad. Sci. Canada 1, 111-114, 1978/79.
86. Book: Orthogonal and diagonal dimension fluxes of hyperspherical function. Dusko Letic. Nenad Cakic. Branko Davidovic. Ivana Berkovic. Springer.
87. Book: Chaos in Electric Drive Systems: Analysis, Control and Application. K. T. Chau. Zheng Wang. John Wiley & Son. 201. 978-0-470-82633-1. 7.
88. Book: Mathematical Constants. registration. Steven R. Finch. Cambridge University Press. 2003. 978-3-540-67695-9. 238.
89. Book: Mathematics Frontiers. Facts On File, Incorporated. 1997. 978-0-8160-5427-5. 46. Infobase .
90. Book: Mathematical Constants. Steven R. Finch. 2003. 978-3-540-67695-9. 110. Cambridge University Press .
91. Book: CRC Concise Encyclopedia of Mathematics, Second Edition. Eric W. Weisstein. CRC Press. 2003. 978-1-58488-347-0. 151.
92. Book: RANDOM FIBONACCI SEQUENCES AND THE NUMBER 1.13198824.... DIVAKAR VISWANATH. MATHEMATICS OF COMPUTATION. 1999.
93. Book: k-Automatic Reals. Christoph Lanz. Technischen Universität Wien.
94. Book: Analytic Number Theory. J. B. Friedlander. A. Perelli. C. Viola. D.R. Heath-Brown. H.Iwaniec. J. Kaczorowski. Springer. 2002. 978-3-540-36363-7. 29.
95. Book: Unified algorithms for polylogarithm, L-series, and zeta variants. Richard E. Crandall. perfscipress.com. 2012. https://web.archive.org/web/20130430193005/http://www.perfscipress.com/papers/UniversalTOC25.pdf. 2013-04-30. dead.
96. 0912.3844. math.CA. RICHARD J. MATHAR. NUMERICAL EVALUATION OF THE OSCILLATORY INTEGRAL OVER exp(I pi x)x^1/x BETWEEN 1 AND INFINITY. 2010.
97. Book: Root constant. M.R.Burns. Marvin Ray Burns. 1999.
98. [oeis:wiki/MRB constant|MRB constant]
99. Book: Hardy, G. H. . An introduction to the theory of numbers . 2008 . Oxford University Press . E. M. Wright, D. R. Heath-Brown, Joseph H. Silverman . 978-0-19-921985-8 . 6th . Oxford . 214305907.
100. 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. math/0506319. 10.1007/s11139-007-9102-0. 119131640.
101. Book: Gazeta Matemetica Seria a revista de cultur Matemetica Anul XXV(CIV)Nr. 1, Constante de tip Euler generalízate. Andrei Vernescu. 2007. 14.
102. Book: Electrical Capacitance. Steven Finch. Harvard.edu. 2014. 1. 2015-10-12. https://web.archive.org/web/20160419150944/http://www.people.fas.harvard.edu/~sfinch/csolve/capa.pdf. 2016-04-19. dead.
103. Ransford . Thomas . 10.1007/BF03321780 . 2 . Computational Methods and Function Theory . 2791324 . 555–578 . Computation of logarithmic capacity . 10 . 2010.
104. Bugeaud . Yann . Queffélec . Martine . On Rational Approximation of the Binary Thue-Morse-Mahler Number . Journal of Integer Sequences . 2013 . 16 . 13.2.3 .
105. Wolf . Marek . 22 February 2010 . Remark on the irrationality of the Brun's constant . 1002.4174 . math.NT.
106. Web site: Hermite Constants .
107. Book: Process Algebra and Probabilistic Methods.. Holger Hermanns. Roberto Segala. Springer-Verlag. 2000. 978-3-540-67695-9. 270.
108. Web site: Favard Constants .