Math constants table
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) = xx 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}