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Math constants table
Table constains 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 namesCommon symbolPossible definition or way of calculationApproximated valueExample usage or connotationsKnown at least sinceNumber of known digits after the point
(state on 2019)
The pi number, ludolfine, Archimedes numberShow sourceπ\piShow sourceπ=disk circumferencedisk diameter=limn2n22+2++2n\pi = \frac{\text{disk circumference}}{\text{disk diameter}} = \lim_{n\to \infty }\, 2^n \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_n3.14159265358979323846
  • Common in many branches of mathematics, natural and technical sciences,
  • Euclidean geometry.
2600 BC22459157718361
The e number, Euler's number, Neper's numberShow sourceeeShow sourcee=limn(1+1n)n=n=01n!=11+11+112+1123+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} + \cdots2.71828182845904523536
  • Common in many branches of mathematics, natural and technical sciences,
  • the base of natural logarithm.
1618100000000000
The Euler-Mascheroni constantShow sourceγ\gammaShow sourceγ=limn(lnn+k=1n1k)=1(1x+1x)dx==n=1k=0(1)k2n+k=n=1(1nln(1+1n))\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}0.577215664901532860601735477511832674
Golden ratio, golden mean, golden section, extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, golden numberShow sourceφ{\varphi}Show source1+52=1+1+1+1+\frac{1 + \sqrt{5}}{2} = \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}}1.61803398874989484820300-200 BC3000000000100
Silver ratioShow sourceδS\delta_SShow sourceSolution of the equation:(δS1)2=2\begin{aligned}&\text{Solution of the equation:}\\&(\delta_S - 1)^2 = 2\end{aligned}2.41421356237309504
  • Architecture (for aesthetic reasons).
Ancient timesNo data
Twice the pi numberShow sourceT\TauShow sourceT=2π\Tau = 2 \pi6.28318530717958648
  • Doubled value of the pi number,
  • sometimes used to simplify the expression (instead of 2π2\pi),
  • considered by some to be more intuitive than the number pi.
2600 BC22459157718361
Inverse of π numberShow source1π\frac{1}{\pi}Show source229801n=0(4n)!(1103+26390  n)(n!)43964n\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 dataNo data
Cube root of 2, Delian constantShow source23\sqrt[3]{2}Show source23\sqrt[3]{2}1.25992104989487316476
  • General usage in various math fields,
  • geometry.
No dataNo data
Square root of 2πShow source2π\sqrt{2 \pi}Show source2π=limnn!  ennnn\sqrt{2 \pi} = \lim_{n \to \infty} \frac {n! \; e^n}{n^n \sqrt{n}}2.50662827463100050241
  • General usage in various math fields.
1692, 1770No data
Square root of Tau × eShow sourceτe\sqrt{\tau e}Show source2πe\sqrt{2 \pi e}4.13273135412249293846
  • General usage in various math fields.
No dataNo data
Favard constant K1, Wallis productShow sourceπ2{\frac{\pi}{2}}Show sourcen=1(4n24n21)=2123434565678789\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} \cdots1.57079632679489661923
  • General usage in various math fields.
1655No data
Theodorus constantShow source3\sqrt{3}Show source3333333333\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 BCNo data
Universal parabolic constantShow sourceP2{P}_{\,2}Show sourceln(1+2)+2  =  arcsinh(1)+2\ln(1 + \sqrt2) + \sqrt2 \; = \; \operatorname{arcsinh}(1)+\sqrt{2}2.29558714939263807403
  • General usage in various math fields.
No dataNo data
Natural logarithm of 2Show sourceln(2)ln(2)Show sourcen=11n2n=n=1(1)n+1n=1112+1314+\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.
1550-1617No data
Reciprocal of the Euler-Mascheroni constantShow source1γ\frac {1}{\gamma}Show source(01log(log1x)dx)1=n=1(1)n(1+γ)n\left(\int_{0}^{1} -\log \left(\log \frac{1}{x}\right)\, dx\right)^{-1} = \sum_{n=1}^\infty (-1)^n (-1+\gamma)^n1.73245471460063347358
  • General usage in various math fields,
  • number theory.
No dataNo data
Silver root, Tutte-Beraha constantShow sourceς\varsigmaShow source2+2cos2π7=2+2+7+77+77+3331+7+77+77+3332+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 dataNo data
Fourth root of fiveShow source54\sqrt[4]{5}Show source5555555555\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 dataNo data
π squaredShow sourceπ2{\pi} ^2Show source6ζ(2)=6n=11n2=612+622+632+642+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}+ \cdots9.86960440108935861883
  • General usage in various math fields,
  • geometry,
  • Riemann zeta function.
No dataNo data
Froda constantShow source2e2^{\,e}Show source2e2^e6.58088599101792097085
  • General usage in various math fields.
No dataNo data
Tribonacci constantShow sourceϕ3{\phi_{}}_3Show source1+19+3333+1933333=1+(12+12+12+...333)1\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 dataNo data
π to π-ith powerShow sourceππ\pi ^\piShow sourceππ\pi ^\pi36.4621596072079117709
  • General usage in various math fields.
No dataNo data
Exponential reiterated constantShow sourceeee^eShow sourcen=0enn!=limn(1+nn)nn(1+n)1+n\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 dataNo data
Square root of the number eShow sourcee\sqrt {e}Show sourcen=012nn!=n=01(2n)!!=11+12+18+148+\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}+\cdots1.64872127070012814684
  • General usage in various math fields.
No dataNo data
Square root of 2, Pythagoras constantShow source2\sqrt{2}Show source ⁣n=1 ⁣(1 ⁣+ ⁣(1)n+12n1) ⁣= ⁣(1 ⁣+ ⁣11) ⁣(1 ⁣ ⁣13) ⁣(1 ⁣+ ⁣15)\! \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) \cdots1.41421356237309504880
  • General usage in various math fields.
No data10000000000000
Conic constant, Schwarzschild constantShow sourcee2e^2Show sourcen=02nn!=1+2+222!+233!+244!+255!+\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!}+\cdots7.38905609893065022723
  • General usage in various math fields.
No dataNo data
Inverse of number eShow source1e\frac{1}{e}Show sourcen=0(1)nn!=10!11!+12!13!+14!15!+\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!} +\cdots0.36787944117144232159
  • General usage in various math fields.
1618No data
Imaginary numberShow sourcei{i}Show source1=ln(1)πeiπ=1\sqrt{-1} = \frac{\ln(-1)}{\pi} \qquad\qquad \mathrm{e}^{i\,\pi} = -1i
  • General usage in various math fields,
  • complex analysis.
1501-1576-
Square root of 5, Gauss sumShow source5\sqrt{5}Show source(n=5)k=0n1e2k2πin=1+e2πi5+e8πi5+e18πi5+e32πi5\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 dataNo data

Mathematical analysis

Typical namesCommon symbolPossible definition or way of calculationApproximated valueExample usage or connotationsKnown at least sinceNumber of known digits after the point
(state on 2019)
The pi number, ludolfine, Archimedes numberShow sourceπ\piShow sourceπ=disk circumferencedisk diameter=limn2n22+2++2n\pi = \frac{\text{disk circumference}}{\text{disk diameter}} = \lim_{n\to \infty }\, 2^n \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_n3.14159265358979323846
  • Common in many branches of mathematics, natural and technical sciences,
  • Euclidean geometry.
2600 BC22459157718361
The e number, Euler's number, Neper's numberShow sourceeeShow sourcee=limn(1+1n)n=n=01n!=11+11+112+1123+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} + \cdots2.71828182845904523536
  • Common in many branches of mathematics, natural and technical sciences,
  • the base of natural logarithm.
1618100000000000
The Gauss's constantShow sourceGGShow sourceG=1agm(1,2)=2π01dx1x4=42(14!)2π3/2G = \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}}0.8346268416740731862830.05.1799No data
Fransen-Robinson constantShow sourceFFShow source01Γ(x)dx=e+0exπ2+ln2xdx\int_{0}^\infty \frac{1}{\Gamma(x)}\, dx = e + \int_0^\infty \frac{e^{-x}}{\pi^2 + \ln^2 x}\, dx2.80777024202851936522
  • Mathematical analysis,
  • approximation of functions.
19781025
Van der Pauw constantShow sourceα{\alpha}Show sourceπln(2)=n=04(1)n2n+1n=1(1)n+1n=4143+4547+491112+1314+15\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}4.53236014182719380962No dataNo data
Hyperbolic tangent of 1Show sourcetanh1\tanh 1Show sourceitan(i)=e1ee+1e=e21e2+1-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 dataNo data
Chebyshev constantShow sourceλCh\lambda_\text{Ch}Show sourceΓ(14)24π3/2=4(14!)2π3/2\frac{\Gamma(\tfrac14)^2}{4 \pi^{3/2}} = \frac{4 (\tfrac14 !)^2}{\pi^{3/2}}0.59017029950804811302
  • Mathematical analysis,
  • approximation of functions.
No dataNo data
MKB constantShow sourceMIM_IShow sourcelimn12n(1)x xx dx=12neiπx x1/x dx\lim_{n\rightarrow \infty} \int_{1}^{2n} (-1)^x ~ \sqrt[x]{x} ~ dx = \int_{1}^{2n} e^{i \pi x} ~ x^{1/x} ~ dx0.07077603931152880353
- 0.684000389437932129 i
  • Mathematical analysis.
2009No data
Double factorial constantShow sourceCn!!{C_{_{n!!}}}Show sourcen=01n!!=e[12+γ(12,12)]\sum_{n=0}^{\infty} \frac{1}{n!!} = \sqrt{e} \left[\frac {1}{\sqrt{2}}+\gamma(\tfrac12 ,\tfrac12)\right]3.05940740534257614453
  • Mathematical analysis.
No dataNo data
Lebesgue constant L2Show sourceL2{L2}Show source15+2525π=1π0πsin(5t2)sin(t2)dt\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 t1.64218843522212113687
  • Mathematical analysis,
  • approximation of functions.
1910No data
Goh-Schmutz constantShow sourceCGSC_{GS}Show source0log(s+1)es1 ds= ⁣ ⁣n=1ennEi(n)\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 dataNo data
Fixed points super-logarithm tetrationShow sourceW(1)-W(-1)Show sourcelimnf(x)=log(log(log(log(log(log(x)))))) ⁣logs n times\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 dataNo data
Bernsteins constantShow sourceβ{\beta}Show source12π\approx \frac {1}{2\sqrt {\pi}}0.28016949902386913303
  • Mathematical analysis,
  • approximation of functions.
1913No data
Chi Function, hyperbolic cosine integralShow sourceChi(){\operatorname{Chi()}}Show sourceγ+0xcosht1tdt\gamma + \int_0^x\frac{\cosh t-1}{t}\,dt0.52382257138986440645
  • Mathematical analysis,
  • geometry.
No dataNo data
Laplace limitShow sourceλ{\lambda}Show sourcexex2+1x2+1+1=1\frac{x e^{\sqrt{x^2+1}}} {\sqrt{x^2+1}+1} = 10.66274341934918158097
  • Mathematical analysis,
  • approximation of functions.
1782No data
Beta(1)Show sourceβ(1){\beta}(1)Show sourceπ4=n=0(1)n2n+1=1113+1517+19\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} - \cdots0.78539816339744830961
  • Mathematical analysis.
1805-1859No data
Sophomores dream I1Show sourceI1{I}_{1}Show source01 ⁣xxdx=n=1(1)n+1nn=111122+133\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.
1697No data
Sophomores dream I2Show sourceI2{I}_{2}Show source01 ⁣1xxdx=n=11nn=111+122+133+144+\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}+ \cdots1.29128599706266354040
  • Mathematical analysis.
1697No data
Wallis ConstantShow sourceWWShow source451929183+45+1929183\sqrt[3]{\frac{45-\sqrt{1929}}{18}}+\sqrt[3]{\frac{45+\sqrt{1929}}{18}}2.09455148154232659148
  • Mathematical analysis.
1616-1703No data
Time constantShow sourceτ{\tau}Show sourcelimn1!nn!=limnP(n)=01exdx=11e\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 dataNo data
Lemniscate constantShow source2ϖ2\varpiShow source[Γ(14)]22π=401dx(1x2)(2x2)\frac{[\Gamma(\tfrac14)]^2}{\sqrt{2 \pi}} = 4\int^1_0 \frac{dx}{\sqrt{(1-x^2)(2-x^2)}}5.24411510858423962092
  • Mathematical analysis.
1718250000000000
Baker constantShow sourceβ3\beta_3Show source01dt1+t3=n=0(1)n3n+1=13(ln2+π3)\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 dataNo data
Kempner-Reihe Kempner Serie(0)Show sourceK0{K_0}Show source1+12+13++19+111++119+121+1{+}\frac12{+}\frac13{+}\cdots{+}\frac19{+}\frac1{11}{+}\cdots{+}\frac1{19}{+}\frac1{21}{+}\cdots23.1034479094205416160
  • Mathematical analysis.
No dataNo data
Kneser-Mahler polynomial constantShow sourceβ\betaShow sourcee2π0π3ttant dt=e1313ln1+e2πitdte^{^{\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.
1963No data
Infinite product constantShow sourcePr1Pr_1Show sourcen=2(1+1n)1n\prod_{n = 2}^\infty \Big(1 + \frac{1}{n}\Big)^\frac{1}{n}1.75874362795118482469
  • Mathematical analysis.
1977No data
Spiral of TheodorusShow source\partialShow sourcen=11n3+n=n=11n(n+1)\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 BCNo data
Nested radical S5Show sourceS5S_{5}Show source21+12=5+5+5+5+5+\displaystyle \frac{\sqrt{21}+1}{2} = \scriptstyle \, \sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+\cdots}}}}}2.79128784747792000329
  • Mathematical analysis.
No dataNo data
Ioachimescu constantShow source2+ζ(12)2+\zeta(\tfrac12)Show source2(1+2)n=1(1)n+1n=γ+n=1(1)2n  γn2nn!{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 dataNo data
Khinchin harmonic meanShow sourceK1{K_{-1}}Show sourcelog2n=11nlog(1+1n(n+2))=limnn1a1+1a2++1an\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 dataNo data
Lemniscate constantShow sourceϖ{\varpi}Show sourceπG=42πΓ(54)2=142πΓ(14)2=42π(14!)2\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)^22.62205755429211981046
  • Functional iteration,
  • mathematical analysis.
1798No data
Glaisher-Kinkelin constantShow sourceA{A}Show sourcee112ζ(1)=e1812n=01n+1k=0n(1)k(nk)(k+1)2ln(k+1)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 dataNo data
The value of Digamma function in point 1/4Show sourceψ(14){\psi} (\tfrac14)Show sourceγπ23ln2=γ+n=0(1n+11n+14)-\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 dataNo data
The value of Gamma function in point 1/4Show sourceΓ(14)\Gamma(\tfrac14)Show source4(14)!=(34)!4 \left(\frac{1}{4}\right)! = \left(-\frac{3}{4}\right)!3.62560990822190831193
  • Number theory,
  • mathematical analysis.
1729100000000000
Magic angleShow sourceθm{\theta_m}Show sourcearctan(2)=arccos(13)54.7356\arctan \left(\sqrt{2}\right) = \arccos \left(\sqrt{\tfrac13}\right) \approx \textstyle {54.7356} ^{ \circ }0.955316618124509278163
  • Geometry,
  • mathematical analysis.
No dataNo data
Minimum value of function ƒ(x) = xxShow source(1e)1e{\left(\frac{1}{e}\right)}^\frac{1}{e}Show sourcee1e{e}^{-\frac{1}{e}}0.69220062755534635386
  • Mathematical analysis.
No dataNo data
MRB constant, Marvin Ray BurnsShow sourceCMRBC_{{}_{MRB}}Show sourcen=1(1)n(n1/n1)=11+2233+\sum_{n=1}^{\infty} (-1)^n (n^{1/n}-1) = - \sqrt[1]{1} + \sqrt[2]{2} - \sqrt[3]{3} + \cdots0.18785964246206712024
  • Mathematical analysis.
19996
Machin-Gregory serieShow sourcearctan12\arctan \frac {1}{2}Show sourcen=0( ⁣1 ⁣)nx2n+12n+1=1213 ⁣ ⁣23+15 ⁣ ⁣2517 ⁣ ⁣27+For x=1/2\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 dataNo data
Buffon constantShow source2π\frac{2}{\pi}Show source222+222+2+22\frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdots0.63661977236758134307
  • Mathematical analysis.
1540-1603No data
Omega constant, Lambert W functionShow sourceΩ{\Omega}Show sourcen=1(n)n1n!=(1e)(1e)(1e)=eΩ=eeee\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 dataNo data

Geometry

Typical namesCommon symbolPossible definition or way of calculationApproximated valueExample usage or connotationsKnown at least sinceNumber of known digits after the point
(state on 2019)
The pi number, ludolfine, Archimedes numberShow sourceπ\piShow sourceπ=disk circumferencedisk diameter=limn2n22+2++2n\pi = \frac{\text{disk circumference}}{\text{disk diameter}} = \lim_{n\to \infty }\, 2^n \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_n3.14159265358979323846
  • Common in many branches of mathematics, natural and technical sciences,
  • Euclidean geometry.
2600 BC22459157718361
Twice the pi numberShow sourceT\TauShow sourceT=2π\Tau = 2 \pi6.28318530717958648
  • Doubled value of the pi number,
  • sometimes used to simplify the expression (instead of 2π2\pi),
  • considered by some to be more intuitive than the number pi.
2600 BC22459157718361
Hermite constant sphere packing 3D Kepler conjectureShow sourceμK{\mu_{_{K}}}Show sourceπ32\frac{\pi}{3\sqrt{2}}0.74048048969306104116
  • Geometry,
  • topology.
1611No data
Fractal dimension of the Apollonian packing of circlesShow sourceε\varepsilonShow source-1.305686729
  • Fractals,
  • geometry.
1994, 1998No data
Cube root of 2, Delian constantShow source23\sqrt[3]{2}Show source23\sqrt[3]{2}1.25992104989487316476
  • General usage in various math fields,
  • geometry.
No dataNo data
Volume of Reuleaux tetrahedronShow sourceVR{V_{_{R}}}Show sources312(3249π+162arctan2)\frac{s^3}{12}(3\sqrt2 - 49 \, \pi + 162 \, \arctan\sqrt2)0.42215773311582662702
  • Geometry.
No dataNo data
Golden angleShow sourcebbShow source(42Φ)π=(35)π(4-2\,\Phi)\,\pi = (3-\sqrt{5})\,\pi2.39996322972865332223
  • Geometry.
No dataNo data
Chi Function, hyperbolic cosine integralShow sourceChi(){\operatorname{Chi()}}Show sourceγ+0xcosht1tdt\gamma + \int_0^x\frac{\cosh t-1}{t}\,dt0.52382257138986440645
  • Mathematical analysis,
  • geometry.
No dataNo data
Area bounded by the eccentric rotation of Reuleaux triangleShow sourceTR{T}_RShow sourcea2(23+π63)a^2 \cdot \left( 2\sqrt{3} + {\frac{\pi}{6}} - 3 \right)0.98770039073605346013
  • Geometry.
No dataNo data
Area of the regular hexagon with side equal to 1Show sourceA6{A}_6Show source332\frac{3 \sqrt{3}}{2}2.59807621135331594029
  • Geometry.
No dataNo data
DeVicci's tesseract constantShow sourcef(3,4){f_{(3,4)}}Show source4x428x37x2+16x+16=04x^4{-}28x^3{-}7x^2{+}16x{+}16=01.00743475688427937609
  • Geometry.
No dataNo data
Relationship among the area of an equilateral triangle and the inscribed circleShow sourceπ33\frac{\pi}{3 \sqrt 3}Show sourcen=11n(2nn)=112+1415+1718+\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} + \cdots0.60459978807807261686
  • Geometry.
No dataNo data
Hermite constantShow sourceγ2\gamma_{_{2}}Show source23=1cos(π6)\frac{2}{\sqrt{3}} = \frac{1}{\cos \, (\frac{\pi}{6})}1.15470053837925152901
  • Geometry,
  • combinatoricts,
  • discrete structures.
No dataNo data
Calabi triangle constantShow sourceCCR{C_{_{CR}}}Show source1322/3(22/3+23+3i2373+233i2373){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.
1946No data
Robbins constantShow sourceΔ(3)\Delta(3)Show source4 ⁣+ ⁣172 ⁣63 ⁣7π105 ⁣+ ⁣ln(1 ⁣+ ⁣2)5 ⁣+ ⁣2ln(2 ⁣+ ⁣3)5\frac{4 \! + \! 17\sqrt2 \! -6 \sqrt3 \! -7\pi}{105} \! + \! \frac{\ln(1 \! + \! \sqrt2)}{5} \! + \! \frac{2\ln(2 \! + \! \sqrt3)}{5}0.66170718226717623515
  • Geometry.
1978No data
Golden spiralShow sourceccShow sourceφ2π=(1+52)2π\varphi ^ \frac{2}{\pi} = \left(\frac{1 + \sqrt{5}}{2}\right)^{\frac{2}{\pi}}1.35845627418298843520
  • Geometry.
No dataNo data
π squaredShow sourceπ2{\pi} ^2Show source6ζ(2)=6n=11n2=612+622+632+642+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}+ \cdots9.86960440108935861883
  • General usage in various math fields,
  • geometry,
  • Riemann zeta function.
No dataNo data
The ratio of a square and circle circumscribedShow sourceπ22\frac{\pi}{2\sqrt 2}Show sourcen=1(1)n122n+1=11+131517+19+111\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 dataNo data
Figure eight knot hyperbolic volumeShow sourceV8{V_{8}}Show source23n=11n(2nn)k=n2n11k=60π/3log(12sint)dt=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 dataNo data
Khinchin harmonic meanShow sourceK1{K_{-1}}Show sourcelog2n=11nlog(1+1n(n+2))=limnn1a1+1a2++1an\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 dataNo data
Gieseking-Konstante constantShow sourceπlnβ{\pi \ln \beta}Show source334(1n=01(3n+2)2+n=11(3n+1)2)\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.
1912No data
Magic angleShow sourceθm{\theta_m}Show sourcearctan(2)=arccos(13)54.7356\arctan \left(\sqrt{2}\right) = \arccos \left(\sqrt{\tfrac13}\right) \approx \textstyle {54.7356} ^{ \circ }0.955316618124509278163
  • Geometry,
  • mathematical analysis.
No dataNo data
Steiner number, Iterated exponential constantShow sourceee\sqrt[e]{e}Show sourcee1ee^{\frac{1}{e}}1.44466786100976613365
  • Geometry.
No dataNo data

Number theory

Typical namesCommon symbolPossible definition or way of calculationApproximated valueExample usage or connotationsKnown at least sinceNumber of known digits after the point
(state on 2019)
The Euler-Mascheroni constantShow sourceγ\gammaShow sourceγ=limn(lnn+k=1n1k)=1(1x+1x)dx==n=1k=0(1)k2n+k=n=1(1nln(1+1n))\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}0.577215664901532860601735477511832674
The Khinchin's constantShow sourceκ,K0\kappa, K_0Show sourceκ=r=1(1+1r(r+2))log2r\kappa = \prod_{r=1}^\infty {\left( 1+{1\over r(r+2)}\right)}^{\log_2 r}2.68545200106530644
  • Number theory.
19347350
The Erdős-Borwein's constantShow sourceEBE_BShow sourcem=1n=112mn=n=112n1=11 ⁣+ ⁣13 ⁣+ ⁣17 ⁣+ ⁣115 ⁣+ ⁣...\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).
1949No data
The Meissel-Mertens's constant, Mertens constant, Kronecker's constant, Hadamard–de la Vallée-Poussin constant, the prime reciprocal constantShow sourceM,M1M, M_1Show sourceM=limn(pn1pln(ln(n)))== ⁣γ ⁣+ ⁣ ⁣p ⁣( ⁣ln ⁣( ⁣1 ⁣ ⁣1p ⁣) ⁣ ⁣+ ⁣1p ⁣)\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}0.261497212847642783751866, 18738010
The Brun's constant for twin primes (sum of inverse of twin primes)Show sourceB2B_2Show sourceB2=(1p+1p+2)==(13+15)+(15+17)+(111+113)+(117+119)+(129+131)+\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}