Table constrains over 200 mathematical constants with common informations such as approximated value, date of discovery or last known precision (number of significant digits). This includes basic constants (e.g. pi number), but also less common constants such as Khinchin's constant are presented.

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General usage in various math fields#

Typical names	Common symbol	Possible definition or way of calculation	Approximated value	Example usage or connotations	Known at least since	Number of known digits after the point (state on 2019)
The pi number, ludolfine, Archimedes number	Show source $\pi$	Show source $\pi = \dfrac{\text{disk circumference}}{\text{disk diameter}} = \lim_{n\to \infty }\, 2^n \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_n$	3.14159265358979323846	Common in many branches of mathematics, natural and technical sciences, Euclidean geometry.	2600 BC	22459157718361
The e number, Euler's number, Neper's number	Show source $e$	Show source $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = \sum\limits_{n = 0}^{\infty} \frac{1}{n!} = \frac{1}{1} + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots$	2.71828182845904523536	Common in many branches of mathematics, natural and technical sciences, the base of natural logarithm.	1618	100000000000
The Euler-Mascheroni constant	Show source $\gamma$	Show source $\begin{aligned}\gamma &= \lim_{n\to\infty}\left(-\ln n + \sum_{k=1}^n \frac1{k}\right) = \int_1^\infty\left(-\frac1x+\frac1{\lfloor x\rfloor}\right)\,dx = \\&= \sum_{n=1}^\infty \sum_{k=0}^\infty \frac{(-1)^k}{2^n+k} = \sum_{n=1}^\infty \left(\frac{1}{n} -\ln \left(1+\frac{1}{n}\right)\right)\end{aligned}$	0.57721566490153286060	Integrals of exponential functions (mathematical analysis), Laplace transform of natural logarithm.	1735	477511832674
Golden ratio, golden mean, golden section, extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, golden number	Show source ${\varphi}$	Show source $\frac{1 + \sqrt{5}}{2} = \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}}$	1.61803398874989484820	Architecture and art (for aesthetic reasons), music theory, technical analysis of financial markets, golden-section search (algorithm).	300-200 BC	3000000000100
Silver ratio	Show source $\delta_S$	Show source $\begin{aligned}&\text{Solution of the equation:}\\&(\delta_S - 1)^2 = 2\end{aligned}$	2.41421356237309504	Architecture (for aesthetic reasons).	Ancient times	No data
Twice the pi number	Show source $\Tau$	Show source $\Tau = 2 \pi$	6.28318530717958648	Doubled value of the pi number, sometimes used to simplify the expression (instead of $2\pi$ ), considered by some to be more intuitive than the number pi.	2600 BC	22459157718361
Inverse of π number	Show source $\frac{1}{\pi}$	Show source $\frac{2\sqrt{2}}{9801} \sum^\infty_{n=0} \frac{(4n)!\,(1103+26390 \; n)}{(n!)^4 \, 396^{4n}}$	0.31830988618379067153	General usage in various math fields.	No data	No data
Cube root of 2, Delian constant	Show source $\sqrt[3]{2}$	Show source $\sqrt[3]{2}$	1.25992104989487316476	General usage in various math fields, geometry.	No data	No data
Square root of 2π	Show source $\sqrt{2 \pi}$	Show source $\sqrt{2 \pi} = \lim_{n \to \infty} \frac {n! \; e^n}{n^n \sqrt{n}}$	2.50662827463100050241	General usage in various math fields.	1692, 1770	No data
Square root of Tau × e	Show source $\sqrt{\tau e}$	Show source $\sqrt{2 \pi e}$	4.13273135412249293846	General usage in various math fields.	No data	No data
Favard constant K1, Wallis product	Show source ${\frac{\pi}{2}}$	Show source $\prod_{n=1}^{\infty} \left(\frac{4n^2}{4n^2 - 1}\right) = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdots$	1.57079632679489661923	General usage in various math fields.	1655	No data
Theodorus constant	Show source $\sqrt{3}$	Show source $\sqrt[3]{3 \,\sqrt[3]{3 \, \sqrt[3]{3 \,\sqrt[3]{3 \,\sqrt[3]{3 \,\cdots}}}}}$	1.73205080756887729352	General usage in various math fields.	465-398 BC	No data
Universal parabolic constant	Show source ${P}_{\,2}$	Show source $\ln(1 + \sqrt2) + \sqrt2 \; = \; \operatorname{arcsinh}(1)+\sqrt{2}$	2.29558714939263807403	General usage in various math fields.	No data	No data
Natural logarithm of 2	Show source $ln(2)$	Show source $\sum_{n=1}^\infty \frac{1}{n 2^n} = \sum_{n=1}^\infty \frac{({-}1)^{n+1}}{n} = \frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+{\cdots}$	0.69314718055994530941	General usage in various math fields.	1550-1617	No data
Reciprocal of the Euler-Mascheroni constant	Show source $\frac {1}{\gamma}$	Show source $\left(\int_{0}^{1} -\log \left(\log \frac{1}{x}\right)\, dx\right)^{-1} = \sum_{n=1}^\infty (-1)^n (-1+\gamma)^n$	1.73245471460063347358	General usage in various math fields, number theory.	No data	No data
Silver root, Tutte-Beraha constant	Show source $\varsigma$	Show source $2+2 \cos \frac {2\pi} 7 = \textstyle 2+\frac{2+\sqrt[3]{7 + 7 \sqrt[3]{7 + 7 \sqrt[3]{\, 7 + \cdots}}}}{1+\sqrt[3]{7 + 7 \sqrt[3]{7 + 7 \sqrt[3]{\, 7 + \cdots}}}}$	3.24697960371746706105	General usage in various math fields.	No data	No data
Fourth root of five	Show source $\sqrt[4]{5}$	Show source $\sqrt[5]{5 \,\sqrt[5]{5 \, \sqrt[5]{5 \,\sqrt[5]{5 \,\sqrt[5]{5 \,\cdots}}}}}$	1.49534878122122054191	General usage in various math fields.	No data	No data
π squared	Show source ${\pi} ^2$	Show source $6\, \zeta(2) = 6 \sum_{n=1}^\infty \frac{1}{n^2} = \frac{6}{1^2} + \frac{6}{2^2} + \frac{6}{3^2} + \frac{6}{4^2}+ \cdots$	9.86960440108935861883	General usage in various math fields, geometry, Riemann zeta function.	No data	No data
Froda constant	Show source $2^{\,e}$	Show source $2^e$	6.58088599101792097085	General usage in various math fields.	No data	No data
Tribonacci constant	Show source ${\phi_{}}_3$	Show source $\textstyle \frac{1+\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}}{3} = \scriptstyle \, 1+ \left(\sqrt[3]{\tfrac12 + \sqrt[3]{\tfrac12 + \sqrt[3]{\tfrac12 + ...}}}\right)^{-1}$	1.83928675521416113255	General usage in various math fields.	No data	No data
π to π-ith power	Show source $\pi ^\pi$	Show source $\pi ^\pi$	36.4621596072079117709	General usage in various math fields.	No data	No data
Exponential reiterated constant	Show source $e^e$	Show source $\sum_{n=0}^\infty \frac{e^n}{n!} = \lim_{n \to \infty} \left(\frac {1+n}{n} \right)^{n^{-n}(1+n)^{1+n}}$	15.1542622414792641897	General usage in various math fields.	No data	No data
Square root of the number e	Show source $\sqrt {e}$	Show source $\sum_{n = 0}^\infty \frac{1}{2^n n!} = \sum_{n = 0}^\infty \frac{1}{(2n)!!} = \frac{1}{1}+\frac{1}{2}+\frac{1}{8}+\frac{1}{48}+\cdots$	1.64872127070012814684	General usage in various math fields.	No data	No data
Square root of 2, Pythagoras constant	Show source $\sqrt{2}$	Show source $\! \prod_{n=1}^\infty \! \left( 1 \! + \! \frac{(-1)^{n+1}}{2n-1} \right) \! = \! \left(1 \! + \! \frac{1}{1}\right) \! \left(1 \! - \! \frac{1}{3} \right) \! \left(1 \! + \! \frac{1}{5} \right) \cdots$	1.41421356237309504880	General usage in various math fields.	No data	10000000000000
Conic constant, Schwarzschild constant	Show source $e^2$	Show source $\sum_{n = 0}^\infty \frac{2^n}{n!} = 1+2+\frac{2^2}{2!}+\frac{2^3}{3!}+\frac{2^4}{4!}+\frac{2^5}{5!}+\cdots$	7.38905609893065022723	General usage in various math fields.	No data	No data
Inverse of number e	Show source $\frac{1}{e}$	Show source $\sum_{n = 0}^\infty \frac{(-1)^n}{n!} = \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!} +\cdots$	0.36787944117144232159	General usage in various math fields.	1618	No data
Imaginary number	Show source ${i}$	Show source $\sqrt{-1} = \frac{\ln(-1)}{\pi} \qquad\qquad \mathrm{e}^{i\,\pi} = -1$	i	General usage in various math fields, complex analysis.	1501-1576	-
Square root of 5, Gauss sum	Show source $\sqrt{5}$	Show source $\scriptstyle (n = 5) \displaystyle \sum_{k=0}^{n-1} e^{\frac{2 k^2 \pi i}{n}} = 1 + e^\frac{2 \pi i} {5} + e^\frac{8 \pi i} {5} + e^\frac{18 \pi i} {5} + e^\frac{32 \pi i} {5}$	2.23606797749978969640	General usage in various math fields.	No data	No data

Mathematical analysis#

Typical names	Common symbol	Possible definition or way of calculation	Approximated value	Example usage or connotations	Known at least since	Number of known digits after the point (state on 2019)
The pi number, ludolfine, Archimedes number	Show source $\pi$	Show source $\pi = \dfrac{\text{disk circumference}}{\text{disk diameter}} = \lim_{n\to \infty }\, 2^n \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_n$	3.14159265358979323846	Common in many branches of mathematics, natural and technical sciences, Euclidean geometry.	2600 BC	22459157718361
The e number, Euler's number, Neper's number	Show source $e$	Show source $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = \sum\limits_{n = 0}^{\infty} \frac{1}{n!} = \frac{1}{1} + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots$	2.71828182845904523536	Common in many branches of mathematics, natural and technical sciences, the base of natural logarithm.	1618	100000000000
The Gauss's constant	Show source $G$	Show source $G = \frac{1}{\operatorname{agm}\left(1, \sqrt{2}\right)} = \frac{2}{\pi}\int_0^1\frac{dx}{\sqrt{1 - x^4}} = \frac{4 \sqrt{2} \,(\tfrac14 !)^2}{\pi ^{3/2}}$	0.83462684167407318628	Arithmetic–geometric mean	30.05.1799	No data
Fransen-Robinson constant	Show source $F$	Show source $\int_{0}^\infty \frac{1}{\Gamma(x)}\, dx = e + \int_0^\infty \frac{e^{-x}}{\pi^2 + \ln^2 x}\, dx$	2.80777024202851936522	Mathematical analysis, approximation of functions.	1978	1025
Van der Pauw constant	Show source ${\alpha}$	Show source $\frac{\pi}{\ln(2)}=\frac{\sum\limits_{n=0}^\infty \frac{4(-1)^n}{2n+1}}{\sum\limits_{n=1}^\infty \frac{(-1)^{n+1}}{n}}=\frac{\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\cdots}{\frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots}$	4.53236014182719380962	Van der Pauw method, Hall coefficient, Hall effect.	No data	No data
Hyperbolic tangent of 1	Show source $\tanh 1$	Show source $-i \tan (i) = \frac{e-\frac{1}{e}}{e+\frac{1}{e}} = \frac{e^2-1}{e^2+1}$	0.76159415595576488811	Mathematical analysis, complex analysis.	No data	No data
Chebyshev constant	Show source $\lambda_\text{Ch}$	Show source $\frac{\Gamma(\tfrac14)^2}{4 \pi^{3/2}} = \frac{4 (\tfrac14 !)^2}{\pi^{3/2}}$	0.59017029950804811302	Mathematical analysis, approximation of functions.	No data	No data
MKB constant	Show source $M_I$	Show source $\lim_{n\rightarrow \infty} \int_{1}^{2n} (-1)^x ~ \sqrt[x]{x} ~ dx = \int_{1}^{2n} e^{i \pi x} ~ x^{1/x} ~ dx$	0.07077603931152880353 - 0.684000389437932129 i	Mathematical analysis.	2009	No data
Double factorial constant	Show source ${C_{_{n!!}}}$	Show source $\sum_{n=0}^{\infty} \frac{1}{n!!} = \sqrt{e} \left[\frac {1}{\sqrt{2}}+\gamma(\tfrac12 ,\tfrac12)\right]$	3.05940740534257614453	Mathematical analysis.	No data	No data
Lebesgue constant L2	Show source ${L2}$	Show source $\frac{1}{5} + \frac{\sqrt{25-2\sqrt{5}}}{\pi} = \frac{1}{\pi} \int_0^\pi \frac {\left\|\sin(\frac{5t}{2})\right\|} {\sin(\frac{t}{2})} \,d t$	1.64218843522212113687	Mathematical analysis, approximation of functions.	1910	No data
Goh-Schmutz constant	Show source $C_{GS}$	Show source $\int^\infty_0\frac{\log(s+1)}{e^s-1} \ ds = \! - \! \sum_{n=1}^\infty \frac {e^n}{n} Ei(-n)$	1.11786415118994497314	Algebra, mathematical analysis.	No data	No data
Fixed points super-logarithm tetration	Show source $-W(-1)$	Show source $\lim_{n\rightarrow \infty} f(x) = \underbrace{\log(\log(\log(\log(\cdots\log(\log(x)))))) \,\! }\atop {\log_s \text{ }n\text{ times}}$	0.31813150520476413531 ± 1.33723570143068940 i	Algebra, mathematical analysis, tetration (hyper-4).	No data	No data
Bernsteins constant	Show source ${\beta}$	Show source $\approx \frac {1}{2\sqrt {\pi}}$	0.28016949902386913303	Mathematical analysis, approximation of functions.	1913	No data
Chi Function, hyperbolic cosine integral	Show source ${\operatorname{Chi()}}$	Show source $\gamma + \int_0^x\frac{\cosh t-1}{t}\,dt$	0.52382257138986440645	Mathematical analysis, geometry.	No data	No data
Laplace limit	Show source ${\lambda}$	Show source $\frac{x e^{\sqrt{x^2+1}}} {\sqrt{x^2+1}+1} = 1$	0.66274341934918158097	Mathematical analysis, approximation of functions.	1782	No data
Beta(1)	Show source ${\beta}(1)$	Show source $\frac{\pi}{4} = \sum_{n = 0}^\infty \frac{(-1)^n}{2n+1} = \frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots$	0.78539816339744830961	Mathematical analysis.	1805-1859	No data
Sophomores dream I1	Show source ${I}_{1}$	Show source $\int_0^1 \! x^{x}\,dx = \sum_{n = 1}^\infty \frac{(-1)^{n+1}}{n^n} = \frac{1}{1^1} - \frac{1}{2^2} + \frac{1}{3^3} - {\cdots}$	0.78343051071213440705	Mathematical analysis.	1697	No data
Sophomores dream I2	Show source ${I}_{2}$	Show source $\int_0^1 \! \frac{1}{x^x}\, dx = \sum_{n = 1}^\infty \frac{1}{n^n} = \frac{1}{1^1} + \frac{1}{2^2} + \frac{1}{3^3} + \frac{1}{4^4}+ \cdots$	1.29128599706266354040	Mathematical analysis.	1697	No data
Wallis Constant	Show source $W$	Show source $\sqrt[3]{\frac{45-\sqrt{1929}}{18}}+\sqrt[3]{\frac{45+\sqrt{1929}}{18}}$	2.09455148154232659148	Mathematical analysis.	1616-1703	No data
Time constant	Show source ${\tau}$	Show source $\lim_{n \to \infty} 1-\frac {!n}{n!}=\lim_{n \to \infty} P(n)= \int_{0}^{1}e^{-x}dx = 1{-}\frac{1}{e}$	0.63212055882855767840	Mathematical analysis.	No data	No data
Lemniscate constant	Show source $2\varpi$	Show source $\frac{[\Gamma(\tfrac14)]^2}{\sqrt{2 \pi}} = 4\int^1_0 \frac{dx}{\sqrt{(1-x^2)(2-x^2)}}$	5.24411510858423962092	Mathematical analysis.	1718	250000000000
Baker constant	Show source $\beta_3$	Show source $\int^1_0 \frac{{\mathrm{d} t}}{1 + t^3}=\sum_{n = 0}^\infty \frac{(-1)^n}{3n+1}= \frac{1}{3}\left(\ln 2+\frac{\pi}{\sqrt{3}}\right)$	0.83564884826472105333	Mathematical analysis.	No data	No data
Kempner-Reihe Kempner Serie(0)	Show source ${K_0}$	Show source $1{+}\frac12{+}\frac13{+}\cdots{+}\frac19{+}\frac1{11}{+}\cdots{+}\frac1{19}{+}\frac1{21}{+}\cdots$	23.1034479094205416160	Mathematical analysis.	No data	No data
Kneser-Mahler polynomial constant	Show source $\beta$	Show source $e^{^{\textstyle{\frac{2}{\pi}} \displaystyle{\int_0^{\frac{\pi}{3}}} \textstyle{t \tan t\ dt}}} = e^{^{\displaystyle{\,\int_{\frac{-1}{3}}^{\frac{1}{3}}} \textstyle{\,\ln \lfloor 1+e^{2 \pi i t}} \rfloor dt}}$	1.38135644451849779337	Mathematical analysis.	1963	No data
Infinite product constant	Show source $Pr_1$	Show source $\prod_{n = 2}^\infty \Big(1 + \frac{1}{n}\Big)^\frac{1}{n}$	1.75874362795118482469	Mathematical analysis.	1977	No data
Spiral of Theodorus	Show source $\partial$	Show source $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^3} + \sqrt{n}} = \sum_{n=1}^{\infty} \frac{1}{\sqrt{n} (n+1)}$	1.86002507922119030718	Mathematical analysis.	460-399 BC	No data
Nested radical S5	Show source $S_{5}$	Show source $\displaystyle \frac{\sqrt{21}+1}{2} = \scriptstyle \, \sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+\cdots}}}}}$	2.79128784747792000329	Mathematical analysis.	No data	No data
Ioachimescu constant	Show source $2+\zeta(\tfrac12)$	Show source ${2{-}(1{+}\sqrt{2})\sum_{n=1}^\infty \frac{(-1)^{n+1}}{\sqrt{n}}} = \gamma + \sum_{n=1}^\infty \frac{(-1)^{2n} \; \gamma_n}{2^n n!}$	0.53964549119041318711	Mathematical analysis, complex analysis, Riemann zeta function.	No data	No data
Khinchin harmonic mean	Show source ${K_{-1}}$	Show source $\frac {\log 2} {\sum \limits_{n=1}^\infty \frac {1}{n} \log\bigl(1{+}\frac{1}{n(n+2)}\bigr)} = \lim_{n \to \infty} \frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}}$	1.74540566240734686349	Mathematical analysis, statistics, geometry.	No data	No data
Lemniscate constant	Show source ${\varpi}$	Show source $\pi \, {G} = 4 \sqrt{\tfrac2\pi}\,\Gamma{\left(\tfrac54 \right)^2} = \tfrac14 \sqrt{\tfrac{2}{\pi}}\,\Gamma {\left(\tfrac14 \right)^2} = 4 \sqrt{\tfrac2\pi}\left(\tfrac14 !\right)^2$	2.62205755429211981046	Functional iteration, mathematical analysis.	1798	No data
Glaisher-Kinkelin constant	Show source ${A}$	Show source $e^{\frac{1}{12}-\zeta^\prime(-1)} = e^{\frac{1}{8}-\frac{1}{2}\sum\limits_{n=0}^\infty \frac{1}{n+1} \sum\limits_{k=0}^n \left(-1\right)^k \binom{n}{k} \left(k+1\right)^2 \ln(k+1)}$	1.28242712910062263687	Number theory, prime numbers, mathematical analysis.	No data	No data
The value of Digamma function in point 1/4	Show source ${\psi} (\tfrac14)$	Show source $-\gamma -\frac{\pi}{2} - 3\ln{2} = -\gamma+\sum_{n=0}^{\infty}\left(\frac{1}{n+1}-\frac{1}{n+\tfrac14}\right)$	-4.227453533376265408	Number theory, mathematical analysis.	No data	No data
The value of Gamma function in point 1/4	Show source $\Gamma(\tfrac14)$	Show source $4 \left(\frac{1}{4}\right)! = \left(-\frac{3}{4}\right)!$	3.62560990822190831193	Number theory, mathematical analysis.	1729	100000000000
Magic angle	Show source ${\theta_m}$	Show source $\arctan \left(\sqrt{2}\right) = \arccos \left(\sqrt{\tfrac13}\right) \approx \textstyle {54.7356} ^{ \circ }$	0.955316618124509278163	Geometry, mathematical analysis.	No data	No data
Minimum value of function ƒ(x) = x^x	Show source ${\left(\frac{1}{e}\right)}^\frac{1}{e}$	Show source ${e}^{-\frac{1}{e}}$	0.69220062755534635386	Mathematical analysis.	No data	No data
MRB constant, Marvin Ray Burns	Show source $C_{{}_{MRB}}$	Show source $\sum_{n=1}^{\infty} (-1)^n (n^{1/n}-1) = - \sqrt[1]{1} + \sqrt[2]{2} - \sqrt[3]{3} + \cdots$	0.18785964246206712024	Mathematical analysis.	1999	6
Machin-Gregory serie	Show source $\arctan \frac {1}{2}$	Show source $\underset{\text{For } x = 1/2 \qquad \qquad} {\sum_{n=0}^\infty \frac{(\!-1\!)^n \, x^{2n+1}}{2n+1} = \frac {1}{2} {-} \frac{1}{3 \! \cdot \! 2^3} {+} \frac{1}{5 \! \cdot \! 2^5} {-} \frac{1}{7 \! \cdot \! 2^7} {+} \cdots}$	0.46364760900080611621	Mathematical analysis.	No data	No data
Buffon constant	Show source $\frac{2}{\pi}$	Show source $\frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdots$	0.63661977236758134307	Mathematical analysis.	1540-1603	No data
Omega constant, Lambert W function	Show source ${\Omega}$	Show source $\sum_{n=1}^\infty \frac{(-n)^{n-1}}{n!} =\,\left(\frac{1}{e}\right) ^{\left(\frac{1}{e}\right) ^{\cdot^{\cdot^{\left(\frac{1}{e}\right)}}}} = e^{-\Omega} = e^{-e^{-e^{\cdot^{\cdot^{{-e}}}}}}$	0.56714329040978387299	Mathematical analysis.	No data	No data

Geometry#

Typical names	Common symbol	Possible definition or way of calculation	Approximated value	Example usage or connotations	Known at least since	Number of known digits after the point (state on 2019)
The pi number, ludolfine, Archimedes number	Show source $\pi$	Show source $\pi = \dfrac{\text{disk circumference}}{\text{disk diameter}} = \lim_{n\to \infty }\, 2^n \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_n$	3.14159265358979323846	Common in many branches of mathematics, natural and technical sciences, Euclidean geometry.	2600 BC	22459157718361
Twice the pi number	Show source $\Tau$	Show source $\Tau = 2 \pi$	6.28318530717958648	Doubled value of the pi number, sometimes used to simplify the expression (instead of $2\pi$ ), considered by some to be more intuitive than the number pi.	2600 BC	22459157718361
Hermite constant sphere packing 3D Kepler conjecture	Show source ${\mu_{_{K}}}$	Show source $\frac{\pi}{3\sqrt{2}}$	0.74048048969306104116	Geometry, topology.	1611	No data
Fractal dimension of the Apollonian packing of circles	Show source $\varepsilon$	Show source $-$	1.305686729	Fractals, geometry.	1994, 1998	No data
Cube root of 2, Delian constant	Show source $\sqrt[3]{2}$	Show source $\sqrt[3]{2}$	1.25992104989487316476	General usage in various math fields, geometry.	No data	No data
Volume of Reuleaux tetrahedron	Show source ${V_{_{R}}}$	Show source $\frac{s^3}{12}(3\sqrt2 - 49 \, \pi + 162 \, \arctan\sqrt2)$	0.42215773311582662702	Geometry.	No data	No data
Golden angle	Show source $b$	Show source $(4-2\,\Phi)\,\pi = (3-\sqrt{5})\,\pi$	2.39996322972865332223	Geometry.	No data	No data
Chi Function, hyperbolic cosine integral	Show source ${\operatorname{Chi()}}$	Show source $\gamma + \int_0^x\frac{\cosh t-1}{t}\,dt$	0.52382257138986440645	Mathematical analysis, geometry.	No data	No data
Area bounded by the eccentric rotation of Reuleaux triangle	Show source ${T}_R$	Show source $a^2 \cdot \left( 2\sqrt{3} + {\frac{\pi}{6}} - 3 \right)$	0.98770039073605346013	Geometry.	No data	No data
Area of the regular hexagon with side equal to 1	Show source ${A}_6$	Show source $\frac{3 \sqrt{3}}{2}$	2.59807621135331594029	Geometry.	No data	No data
DeVicci's tesseract constant	Show source ${f_{(3,4)}}$	Show source $4x^4{-}28x^3{-}7x^2{+}16x{+}16=0$	1.00743475688427937609	Geometry.	No data	No data
Relationship among the area of an equilateral triangle and the inscribed circle	Show source $\frac{\pi}{3 \sqrt 3}$	Show source $\sum_{n = 1}^\infty \frac{1}{n{2n \choose n}} = 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{5} + \frac{1}{7} - \frac{1}{8} + \cdots$	0.60459978807807261686	Geometry.	No data	No data
Hermite constant	Show source $\gamma_{_{2}}$	Show source $\frac{2}{\sqrt{3}} = \frac{1}{\cos \, (\frac{\pi}{6})}$	1.15470053837925152901	Geometry, combinatoricts, discrete structures.	No data	No data
Calabi triangle constant	Show source ${C_{_{CR}}}$	Show source ${1 \over 3 \cdot 2^{2/3}} \bigg( 2^{2/3} + \sqrt[3]{-23 + 3i \sqrt{237}} + \sqrt[3]{-23 - 3i \sqrt{237}} \bigg)$	1.55138752454832039226	Geometry.	1946	No data
Robbins constant	Show source $\Delta(3)$	Show source $\frac{4 \! + \! 17\sqrt2 \! -6 \sqrt3 \! -7\pi}{105} \! + \! \frac{\ln(1 \! + \! \sqrt2)}{5} \! + \! \frac{2\ln(2 \! + \! \sqrt3)}{5}$	0.66170718226717623515	Geometry.	1978	No data
Golden spiral	Show source $c$	Show source $\varphi ^ \frac{2}{\pi} = \left(\frac{1 + \sqrt{5}}{2}\right)^{\frac{2}{\pi}}$	1.35845627418298843520	Geometry.	No data	No data
π squared	Show source ${\pi} ^2$	Show source $6\, \zeta(2) = 6 \sum_{n=1}^\infty \frac{1}{n^2} = \frac{6}{1^2} + \frac{6}{2^2} + \frac{6}{3^2} + \frac{6}{4^2}+ \cdots$	9.86960440108935861883	General usage in various math fields, geometry, Riemann zeta function.	No data	No data
The ratio of a square and circle circumscribed	Show source $\frac{\pi}{2\sqrt 2}$	Show source $\sum_{n = 1}^\infty \frac{({-}1)^{\lfloor \frac{n-1}{2}\rfloor}}{2n+1} = \frac{1}{1} + \frac{1}{3} - \frac{1}{5} - \frac{1}{7} + \frac{1}{9} + \frac{1}{11} - {\cdots}$	1.11072073453959156175	Geometry.	No data	No data
Figure eight knot hyperbolic volume	Show source ${V_{8}}$	Show source $2 \sqrt{3}\, \sum_{n=1}^\infty \frac{1}{n {2n \choose n}} \sum_{k=n}^{2n-1} \frac{1}{k} = 6 \int \limits_{0}^{\pi / 3} \log \left( \frac{1}{2 \sin t} \right) \, dt =$	2.02988321281930725004	Geometry.	No data	No data
Khinchin harmonic mean	Show source ${K_{-1}}$	Show source $\frac {\log 2} {\sum \limits_{n=1}^\infty \frac {1}{n} \log\bigl(1{+}\frac{1}{n(n+2)}\bigr)} = \lim_{n \to \infty} \frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}}$	1.74540566240734686349	Mathematical analysis, statistics, geometry.	No data	No data
Gieseking-Konstante constant	Show source ${\pi \ln \beta}$	Show source $\frac{3\sqrt{3}}{4} \left(1- \sum_{n=0}^\infty \frac{1}{(3n+2)^2}+ \sum_{n=1}^\infty\frac{1}{(3n+1)^2} \right)$	1.01494160640965362502	Geometry.	1912	No data
Magic angle	Show source ${\theta_m}$	Show source $\arctan \left(\sqrt{2}\right) = \arccos \left(\sqrt{\tfrac13}\right) \approx \textstyle {54.7356} ^{ \circ }$	0.955316618124509278163	Geometry, mathematical analysis.	No data	No data
Steiner number, Iterated exponential constant	Show source $\sqrt[e]{e}$	Show source $e^{\frac{1}{e}}$	1.44466786100976613365	Geometry.	No data	No data

Number theory#

Typical names	Common symbol	Possible definition or way of calculation	Approximated value	Example usage or connotations	Known at least since	Number of known digits after the point (state on 2019)
The Euler-Mascheroni constant	Show source $\gamma$	Show source $\begin{aligned}\gamma &= \lim_{n\to\infty}\left(-\ln n + \sum_{k=1}^n \frac1{k}\right) = \int_1^\infty\left(-\frac1x+\frac1{\lfloor x\rfloor}\right)\,dx = \\&= \sum_{n=1}^\infty \sum_{k=0}^\infty \frac{(-1)^k}{2^n+k} = \sum_{n=1}^\infty \left(\frac{1}{n} -\ln \left(1+\frac{1}{n}\right)\right)\end{aligned}$	0.57721566490153286060	Integrals of exponential functions (mathematical analysis), Laplace transform of natural logarithm.	1735	477511832674
The Khinchin's constant	Show source $\kappa, K_0$	Show source $\kappa = \prod_{r=1}^\infty {\left( 1+{1\over r(r+2)}\right)}^{\log_2 r}$	2.68545200106530644	Number theory.	1934	7350
The Erdős-Borwein's constant	Show source $E_B$	Show source $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty}\frac{1}{2^{mn}} =\sum_{n=1}^{\infty}\frac{1}{2^n-1} = \frac{1}{1} \! + \! \frac{1}{3} \! + \! \frac{1}{7} \! + \! \frac{1}{15} \! + \! ...$	1.60669515241529176378	Number theory, heapsort algorithm (computer science).	1949	No data
The Meissel-Mertens's constant, Mertens constant, Kronecker's constant, Hadamard–de la Vallée-Poussin constant, the prime reciprocal constant	Show source $M, M_1$	Show source $\begin{aligned}M &=\lim_{n \rightarrow \infty} \left(\sum_{p \leqslant n} \frac{1}{p} - \ln(\ln(n)) \right) = \\&= {\! \gamma \! + \!\! \sum_{p} \!\left( \! \ln \! \left( \! 1 \! - \! \frac{1}{p} \! \right) \!\! + \! \frac{1}{p} \! \right)}\end{aligned}$	0.26149721284764278375	Number theory, Mersenne primes (kind of prime numbers).	1866, 1873	8010
The Brun's constant for twin primes (sum of inverse of twin primes)	Show source $B_2$	Show source $\begin{aligned}B_2 &= \sum\left(\frac{1}{p} + \frac{1}{p+2}\right) = \\ &= \left(\frac{1}{3} + \frac{1}{5}\right) + \left(\frac{1}{5} + \frac{1}{7}\right)\\ &+ \left(\frac{1}{11} + \frac{1}{13}\right) + \left(\frac{1}{17} + \frac{1}{19}\right)\\ &+ \left(\frac{1}{29} + \frac{1}{31}\right) + \dots\end{aligned}$