English language flag EN
Polish language flag PL
◀
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.

Beta version#

BETA TEST VERSION OF THIS ITEM
This online calculator is currently under heavy development. It may or it may NOT work correctly.
You CAN try to use it. You CAN even get the proper results.
However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.
Feel free to send any ideas and comments !

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=lim⁡n→∞ 2n2−2+2+⋯+2⏟n\pi = \dfrac{\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=lim⁡n→∞(1+1n)n=∑n=0∞1n!=11+11+11⋅2+11⋅2⋅3+⋯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=lim⁡n→∞(−ln⁡n+∑k=1n1k)=∫1∞(−1x+1⌊x⌋) dx==∑n=1∞∑k=0∞(−1)k2n+k=∑n=1∞(1n−ln⁡(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:(δS−1)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 source229801∑n=0∞(4n)! (1103+26390  n)(n!)4 3964n\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=lim⁡n→∞n!  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 source∏n=1∞(4n24n2−1)=21⋅23⋅43⋅45⋅65⋅67⋅87⋅89⋯\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 source3 3 3 3 3 ⋯33333\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 sourceP 2{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 source∑n=1∞1n2n=∑n=1∞(−1)n+1n=11−12+13−14+⋯\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(∫01−log⁡(log⁡1x) 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+2cos⁡2π7=2+2+7+77+7 7+⋯3331+7+77+7 7+⋯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 source5 5 5 5 5 ⋯55555\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 source6 ζ(2)=6∑n=1∞1n2=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 source2 e2^{\,e}Show source2e2^e6.58088599101792097085
  • General usage in various math fields.
No dataNo data
Tribonacci constantShow sourceϕ3{\phi_{}}_3Show source1+19+3333+19−33333=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 source∑n=0∞enn!=lim⁡n→∞(1+nn)n−n(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 source∑n=0∞12nn!=∑n=0∞1(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+12n−1) ⁣= ⁣(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 source∑n=0∞2nn!=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 source∑n=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 source−1=ln⁡(−1)πei π=−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=0n−1e2k2πin=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=lim⁡n→∞ 2n2−2+2+⋯+2⏟n\pi = \dfrac{\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=lim⁡n→∞(1+1n)n=∑n=0∞1n!=11+11+11⋅2+11⋅2⋅3+⋯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)=2π∫01dx1−x4=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 source∫0∞1Γ(x) dx=e+∫0∞e−xπ2+ln⁡2x dx\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=0∞4(−1)n2n+1∑n=1∞(−1)n+1n=41−43+45−47+49−⋯11−12+13−14+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 sourcetanh⁡1\tanh 1Show source−itan⁡(i)=e−1ee+1e=e2−1e2+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 sourcelim⁡n→∞∫12n(−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 source∑n=0∞1n!!=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+25−25π=1π∫0π∣sin⁡(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 source∫0∞log⁡(s+1)es−1 ds= ⁣− ⁣∑n=1∞ennEi(−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 source−W(−1)-W(-1)Show sourcelim⁡n→∞f(x)=log⁡(log⁡(log⁡(log⁡(⋯log⁡(log⁡(x))))))  ⁣⏟log⁡s 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 source≈12π\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+∫0xcosh⁡t−1t dt\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=11−13+15−17+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 source∫01 ⁣xx dx=∑n=1∞(−1)n+1nn=111−122+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 source∫01 ⁣1xx dx=∑n=1∞1nn=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 source45−1929183+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 sourcelim⁡n→∞1−!nn!=lim⁡n→∞P(n)=∫01e−xdx=1−1e\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=4∫01dx(1−x2)(2−x2)\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 source∫01dt1+t3=∑n=0∞(−1)n3n+1=13(ln⁡2+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 sourcee2π∫0π3ttan⁡t dt=e ∫−1313 ln⁡⌊1+e2πit⌋dte^{^{\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 source∏n=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 source∑n=1∞1n3+n=∑n=1∞1n(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 sourceK−1{K_{-1}}Show sourcelog⁡2∑n=1∞1nlog⁡(1+1n(n+2))=lim⁡n→∞n1a1+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 sourceπ G=42π Γ(54)2=142π Γ(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 sourcee112−ζ′(−1)=e18−12∑n=0∞1n+1∑k=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 source−γ−π2−3ln⁡2=−γ+∑n=0∞(1n+1−1n+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 sourcee−1e{e}^{-\frac{1}{e}}0.69220062755534635386
  • Mathematical analysis.
No dataNo data
MRB constant, Marvin Ray BurnsShow sourceCMRBC_{{}_{MRB}}Show source∑n=1∞(−1)n(n1/n−1)=−11+22−33+⋯\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 sourcearctan⁡12\arctan \frac {1}{2}Show source∑n=0∞( ⁣−1 ⁣)n x2n+12n+1=12−13 ⁣⋅ ⁣23+15 ⁣⋅ ⁣25−17 ⁣⋅ ⁣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 source22⋅2+22⋅2+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 source∑n=1∞(−n)n−1n!= (1e)(1e)⋅⋅(1e)=e−Ω=e−e−e⋅⋅−e\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=lim⁡n→∞ 2n2−2+2+⋯+2⏟n\pi = \dfrac{\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(32−49 π+162 arctan⁡2)\frac{s^3}{12}(3\sqrt2 - 49 \, \pi + 162 \, \arctan\sqrt2)0.42215773311582662702
  • Geometry.
No dataNo data
Golden angleShow sourcebbShow source(4−2 Φ) π=(3−5) π(4-2\,\Phi)\,\pi = (3-\sqrt{5})\,\pi2.39996322972865332223
  • Geometry.
No dataNo data
Chi Function, hyperbolic cosine integralShow sourceChi()⁡{\operatorname{Chi()}}Show sourceγ+∫0xcosh⁡t−1t dt\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+π6−3)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 source4x4−28x3−7x2+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 source∑n=1∞1n(2nn)=1−12+14−15+17−18+⋯\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 source13⋅22/3(22/3+−23+3i2373+−23−3i2373){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 source6 ζ(2)=6∑n=1∞1n2=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 source∑n=1∞(−1)⌊n−12⌋2n+1=11+13−15−17+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 source23 ∑n=1∞1n(2nn)∑k=n2n−11k=6∫0π/3log⁡(12sin⁡t) 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 sourceK−1{K_{-1}}Show sourcelog⁡2∑n=1∞1nlog⁡(1+1n(n+2))=lim⁡n→∞n1a1+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 sourceπln⁡β{\pi \ln \beta}Show source334(1−∑n=0∞1(3n+2)2+∑n=1∞1(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=lim⁡n→∞(−ln⁡n+∑k=1n1k)=∫1∞(−1x+1⌊x⌋) dx==∑n=1∞∑k=0∞(−1)k2n+k=∑n=1∞(1n−ln⁡(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))log⁡2r\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 source∑m=1∞∑n=1∞12mn=∑n=1∞12n−1=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=lim⁡n→∞(∑p⩽n1p−ln⁡(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}1.902160583104191912
The Brun's constant for prime quadruplets (sum of inverse of prime quadruplets)Show sourceB4B_4Show sourceB4=∑(1p+1p+2+1p+6+1p+8)==(15+17+111+113)+(111+113+117+119)+(1101+1103+1107+1109)+⋯\begin{aligned}B_4 &= \sum\left(\frac{1}{p} + \frac{1}{p+2} + \frac{1}{p+6} + \frac{1}{p+8}\right) = \\ &= \left(\frac{1}{5} + \frac{1}{7} + \frac{1}{11} + \frac{1}{13}\right)\\ &+ \left(\frac{1}{11} + \frac{1}{13} + \frac{1}{17} + \frac{1}{19}\right)\\ &+ \left(\frac{1}{101} + \frac{1}{103} + \frac{1}{107} + \frac{1}{109}\right) + \cdots\end{aligned}0.8705883800No data8
The Stała Legendre's constantShow sourceBL′B'_LShow sourceBL′=lim⁡n→∞(ln⁡(n)−nπ(n))B'_L = \lim_{n \to \infty } \left( \ln(n) - {n \over \pi(n)} \right)1
  • Prime number theory,
  • currently unused (it has only historical significance),
  • originally approximated to 1.08366.
1808-
The Sierpiński's constantShow sourceKKShow sourcelim⁡n→∞[∑k=1nr2(k)k−πln⁡n]\lim_{n \to \infty}\left[\sum_{k=1}^n\frac{r_2(k)}{k} - \pi\ln n\right]2.58498175957925321706
  • Fractals.
1907No data
The twin primes constantShow sourceC2C_2Show source∏p⩾3p(p−2)(p−1)2\prod_{p\geqslant 3} \frac{p(p-2)}{(p-1)^2}0.66016181584686957392
  • Number theory,
  • twin primes.
19225020
The Ramanujan-Soldner constant, Soldner's constant, zero of the integral logarithmShow sourceμ\muShow sourceSolution of the equation:li(μ)=∫0μdtln⁡t=0\begin{aligned}&\text{Solution of the equation:}\\&\mathrm{li}(\mu) = \int_0^\mu \frac{dt}{\ln t} = 0\end{aligned}1.45136923488338105028
  • Specjal functions,
  • the zero of logarithmic integral function.
1792-180975500
De Bruijn-Newman's constantShow sourceΛ\LambdaShow sourceSolutions of below equation are realBπλ∫−∞∞exp⁡(−14λ(x−z)2)ξ(1/2+ix) dx=0if λ≥Λ.\begin{aligned}&\text{Solutions of below equation are real}\\&\frac{B\sqrt \pi}{\lambda} \int_{-\infty}^\infty \exp\left(\frac{-1}{4\lambda}(x-z)^{2}\right) \xi(1/2+ix) \, dx = 0\\&\text{if } \lambda \ge \Lambda.\end{aligned}Λ ∈ [0; 1/2)
  • Number theory,
  • prime numbers,
  • Riemann zeta function (special functions),
  • Riemann hypothesis.
1950-
Gauss-Kuzmin-Wirsing constantShow sourceλ2{\lambda}_{2}Show sourcelim⁡n→∞Fn(x)−ln⁡(1−x)(−λ)n=Ψ(x),\lim_{n \to \infty}\frac{F_n(x) - \ln(1 - x)}{(-\lambda)^n} = \Psi(x),0.30366300289873265859
  • Combinatoricts,
  • number theory.
1973468
Landau-Ramanujan constantShow sourceKKShow source12∏p≡3 ⁣ ⁣ ⁣ ⁣ ⁣mod   ⁣4 ⁣ ⁣(1−1p2)−12 ⁣ ⁣ ⁣ ⁣ ⁣ ⁣ ⁣ ⁣p: prime ⁣ ⁣=π4∏p≡1 ⁣ ⁣ ⁣ ⁣ ⁣mod   ⁣4 ⁣ ⁣(1−1p2)12 ⁣ ⁣ ⁣ ⁣p: prime\frac1{\sqrt2}\prod_{p\equiv3\!\!\!\!\!\mod \! 4}\!\! \underset{\!\!\!\!\!\!\!\! p: \text{ prime}}{\left(1-\frac1{p^2}\right)^{-\frac{1}{2}}}\!\!=\frac\pi4\prod_{p\equiv1\!\!\!\!\!\mod \!4}\!\! \underset{\!\!\!\! p: \text{ prime}}{\left(1-\frac1{p^2}\right)^\frac{1}{2}}0.76422365358922066299
  • Number theory,
  • prime numbers.
No data30010
Sum of the reciprocals of the averages of the twin prime pairs, JJGJJGShow sourceB1B_1Show source14+16+112+118+130+142+160+172+⋯\frac{1}{4}+\frac{1}{6}+\frac{1}{12}+\frac{1}{18}+\frac{1}{30}+\frac{1}{42}+\frac{1}{60}+\frac{1}{72}+\cdots0.9288358271
  • Number theory,
  • prime numbers.
2014No data
Smarandache constantShow sourceS1{S_1}Show source∑n=2∞1μ(n)!\sum_{n=2}^\infty \frac1{\mu(n)!}1.09317045919549089396
  • Number theory.
No dataNo data
Raabes formulaShow sourceζ′(0){\zeta'(0)}Show source∫aa+1log⁡Γ(t) dt=12log⁡2π+alog⁡a−a,a≥0\int\limits_a^{a+1}\log\Gamma(t)\,\mathrm dt = \tfrac12\log2\pi + a\log a - a,\quad a \ge 00.91893853320467274178
  • Number theory,
  • Riemann zeta function.
No dataNo data
Salem number, Lehmer's conjectureShow sourceσ10{\sigma_{_{10}}}Show sourcex10+x9−x7−x6−x5−x4−x3+x+1x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+11.17628081825991750654
  • Number theory.
1983 (?)No data
Artin's constantShow sourceCArtin{C}_{Artin}Show source∏n=1∞(1−1pn(pn−1))pn = prime\prod_{n=1}^{\infty} \left(1-\frac{1}{p_n(p_n-1)}\right)\quad p_n \scriptstyle \text{ = prime}0.37395581361920228805
  • Number theory,
  • prime numbers.
1999No data
Murata ConstantShow sourceCm{C_m}Show source∏n=1∞(1+1(pn−1)2)pn: prime\prod_{n = 1}^\infty \underset{p_{n}: \, {prime}}{ \Big(1 + \frac{1}{(p_n-1)^2}\Big)}2.82641999706759157554
  • Number theory,
  • prime numbers.
No dataNo data
Vardi constantShow sourceVc{V_c}Show source32∏n≥1(1+1(2en−1)2) ⁣1/2n+1\frac{\sqrt{3}}{\sqrt{2}}\prod_{n\ge1}\left(1+{1\over(2e_n-1)^2}\right)^{\!1/2^{n+1}}1.26408473530530111307
  • Number theory.
1991No data
Exponential factorial constantShow sourceSEf{S_{Ef}}Show source∑n=1∞1n(n−1)⋅⋅⋅21=1+121+1321+14321+154321+⋯\sum_{n=1}^{\infty} \frac{1}{n^{(n{-}1)^{\cdot^{\cdot^{\cdot^{2^1}}}}}} = 1 {+} \frac{1}{2^{1}} {+} \frac{1}{3^{2^{1}}} + \frac{1}{4^{3^{2^{1}}}} + \frac{1}{5^{4^{3^{2^{1}}}}} {+} \cdots1.611114925808376736111
  • Tetration (hyper-4),
  • number theory.
No dataNo data
Gelfond-Schneider constantShow sourceG GSG_{\,GS}Show source222^{\sqrt{2}}2.665144142690225188651934No data
Khinchin-Lévy constant (gamma)Show sourceγ\gammaShow sourceeπ2/(12ln⁡2)e^{\pi^2/(12\ln2)}3.27582291872181115978
  • Number theory.
1936No data
Viswanath constantShow sourceCVi{C}_{Vi}Show sourcelim⁡n→∞∣an∣1n\lim_{n \to \infty}|a_n|^\frac{1}{n}1.1319882487943
  • Fibonacci numbers,
  • number theory.
No data8
Favard constantShow source34ζ(2)\tfrac34\zeta(2)Show sourceπ28=∑n=0∞1(2n−1)2=112+132+152+172+⋯\frac{\pi^2}{8} = \sum_{n = 0}^\infty \frac{1}{(2n-1)^2} = \frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\cdots1.23370055013616982735
  • Number theory,
  • Riemann zeta function.
1902, 1965No data
Lochs constantShow source£Lo{\text{£}_{_{Lo}}}Show source6ln⁡2ln⁡10π2\frac {6 \ln 2 \ln 10}{ \pi^2}0.97027011439203392574
  • Number theory.
1964No data
Carefree constantShow sourceC2{C}_2Show source∏n=1∞(1−1pn(pn+1))pn: prime\underset{ p_n: \, {prime}}{\prod_{n = 1}^\infty \left(1 - \frac{1}{p_n(p_n+1)}\right)}0.70444220099916559273
  • Number theory,
  • prime numbers.
No dataNo data
The value of Riemman Zeta function in point 2Show sourceζ( 2){\zeta}(\,2)Show sourceπ26=∑n=1∞1n2=112+122+132+142+⋯\frac{\pi^2}{6} = \sum_{n=1}^\infty\frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots1.64493406684822643647
  • Number theory,
  • prime numbers,
  • Riemann zeta function.
1826-1866No data
The Apéry's constantShow sourceA,ζ(3)A, \zeta(3)Show source∑n=1∞1n3=113+123+133+143+153+⋯\sum_{n=1}^\infty\frac{1}{n^3} = \frac{1}{1^3}+\frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \frac{1}{5^3} + \cdots1.20205690315959428539
  • Number theory,
  • special functions,
  • quantum electrodynamic,
  • random minimum spanning tree (computer science).
1979500000000000
The value of Riemman Zeta function in point 4Show sourceζ(4)\zeta(4)Show sourceπ490=∑n=1∞1n4=114+124+134+144+154+...\frac{\pi^4}{90} = \sum_{n=1}^\infty\frac{{1}}{n^4} = \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4} + \frac{1}{5^4} + ...1.08232323371113819151
  • Number theory,
  • prime numbers,
  • Riemann zeta function.
No dataNo data
The value of Riemman Zeta function in point 5Show sourceζ(5)\zeta(5)Show source1294π5−7235∑n=1∞1n5(e2πn−1)−235∑n=1∞1n5(e2πn+1)\frac{1}{294}\pi^5 -\frac{72}{35} \sum_{n=1}^\infty \frac{1}{n^5 (e^{2\pi n} -1)}-\frac{2}{35} \sum_{n=1}^\infty \frac{1}{n^5 (e^{2\pi n} +1)}1.0369277551433699263
  • Number theory,
  • prime numbers,
  • Riemann zeta function.
No data100000000000
The value of Riemman Zeta function in point 6Show sourceζ(6)\zeta(6)Show sourceπ6945 ⁣= ⁣∏n=1∞ ⁣11−pn−6pn: prime=11 ⁣− ⁣2−6 ⁣⋅ ⁣11 ⁣− ⁣3−6 ⁣⋅ ⁣11 ⁣− ⁣5−6⋯\frac{\pi^6}{945} \! = \! \prod_{n=1}^\infty \! \underset{p_n: \text{ prime}}{ \frac{1}{{1-p_n}^{-6}}} = \frac{1}{1 \! -\! 2^{-6}} \! \cdot \! \frac{1}{1 \! - \! 3^{-6}} \! \cdot \! \frac{1}{1 \! - \! 5^{-6}} \cdots1.01734306198444913971
  • Number theory,
  • prime numbers,
  • Riemann zeta function.
No dataNo data
Kasner numberShow sourceR{R}Show source1+2+3+4+⋯\sqrt{1 + \sqrt{2 + \sqrt{3 + \sqrt{4 + \cdots}}}}1.75793275661800453270
  • Number theory.
1878, 1955No data
Feller-Tornier constantShow sourceCFT{\mathcal{C}_{_{FT}}}Show source12∏n=1∞(1−2pn2)+12pn: prime=3π2∏n=1∞(1−1pn2−1)+12\underset{p_n: \, {prime}}{\frac{1}{2}\prod_{n = 1}^\infty \left(1-\frac{2}{p_n^2}\right){+}\frac{1}{2}} =\frac{3}{\pi^2}\prod_{n = 1}^\infty \left(1-\frac{1}{p_n^2-1}\right){+}\frac{1}{2}0.66131704946962233528
  • Number theory,
  • prime numbers.
1932No data
Niven's constantShow sourceC{C}Show source1+∑n=2∞(1−1ζ(n))1+\sum_{n = 2}^\infty \left(1-\frac{1}{\zeta(n)} \right)1.70521114010536776428
  • Number theory,
  • Riemann zeta function.
1969No data
Pell constantShow sourcePPell{\mathcal{P}_{_{Pell}}}Show source1−∏n=0∞(1−122n+1)1- \prod_{n = 0}^\infty \left(1-\frac{1}{2^{2n+1}}\right)0.58057755820489240229
  • Number theory.
No dataNo data
Hall-Montgomery ConstantShow sourceδ0{{\delta}_{_{0}}}Show source1+π26+2  Li2(−e  )Li2 = Dilogarithm integral1 + \frac{\pi^2}{6} +2 \; \mathrm{Li}_2 \left(-\sqrt{e}\;\right) \quad \mathrm{Li}_2 \, \scriptstyle \text{= Dilogarithm integral}0.17150049314153606586
  • Number theory.
No dataNo data
Value of Gamma function of 3/4Show sourceΓ(34)\Gamma(\tfrac34)Show source(−1+34)!=(−14)!\left(-1+\frac{3}{4}\right)! = \left(-\frac{1}{4}\right)!1.22541670246517764512
  • Number theory.
No dataNo data
Heath-Brown-Moroz constantShow sourceCHBM{C_{_{HBM}}}Show source∏n=1∞(1−1pn)7(1+7pn+1pn2)pn: prime\underset{p_n: \, {prime}}{\prod_{n = 1}^\infty \left(1-\frac{1}{p_n}\right)^7\left(1+\frac{7p_n+1}{p_n^2}\right)}0.00131764115485317810
  • Number theory,
  • prime numbers.
No dataNo data
Primorial constant, sum of the product of inverse of primesShow sourceP#{P_\#}Show source∑n=1∞1pn#=12+16+130+1210+...=∑k=1∞∏n=1k1pnpn: prime\underset{ p_n: \, {prime}}{\sum_{n = 1}^\infty \frac{1}{p_n\#} = \frac{1}{2} + \frac{1}{6} + \frac{1}{30} + \frac{1}{210} + ... = \sum_{k = 1}^\infty \prod_{n = 1}^k \frac {1}{p_n}}0.70523017179180096514
  • Number theory,
  • prime numbers.
No dataNo data
Minkowski-Siegel mass constantShow sourceF1F_1Show source∏n=1∞n!2πn(ne)n1+1n12\prod_{n=1}^{\infty} \frac{n!}{\sqrt{2\pi n}\left(\frac{n}{e}\right)^n \sqrt[12]{1+\tfrac1{n}}}1.04633506677050318098
  • Number theory.
1867-1885, 1935No data
Khinchin-Lévy constant (beta)Show sourceβ{\beta}Show sourceπ212 ln⁡2\frac {\pi^2}{12\,\ln 2}1.18656911041562545282
  • Number theory.
1935No data
Nielsen-Ramanujan constantShow sourceζ(2)2\frac{{\zeta}(2)}{2}Show sourceπ212=∑n=1∞(−1)n+1n2=112−122+132−142+152−⋯\frac{\pi^2}{12} = \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^2} = \frac{1}{1^2} {-} \frac{1}{2^2} {+} \frac{1}{3^2} {-} \frac{1}{4^2} {+} \frac{1}{5^2} {-} \cdots0.82246703342411321823
  • Number theory,
  • Riemann zeta function.
1909No data
Stephens constantShow sourceCSC_SShow source∏n=1∞(1−pp3−1)\prod_{n = 1}^\infty \left(1 - \frac{p}{p^3-1}\right)0.57595996889294543964
  • Number theory.
No dataNo data
Alladi-Grinstead constantShow sourceAAG{\mathcal{A}_{AG}}Show sourcee−1+∑k=2∞∑n=1∞1nkn+1=e−1−∑k=2∞1kln⁡(1−1k)e^{-1+\sum \limits_{k=2}^\infty \sum \limits_{n=1}^\infty \frac{1}{n k^{n+1}}} = e^{-1-\sum \limits_{k=2}^\infty \frac{1}{k} \ln \left( 1-\frac{1}{k}\right)}0.80939402054063913071
  • Number theory.
1977No data
Reciprocal of the Euler-Mascheroni constantShow source1γ\frac {1}{\gamma}Show source(∫01−log⁡(log⁡1x) 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
Euler totient constantShow sourceETETShow source∏p(1+1p(p−1))p= primes=ζ(2)ζ(3)ζ(6)=315ζ(3)2π4\underset {p \text{= primes}} {\prod_{p} \Big(1 + \frac{1}{p(p-1)}\Big)} = \frac {\zeta(2)\zeta(3)}{\zeta(6)}=\frac {315 \zeta(3)}{2\pi^4}1.94359643682075920505
  • Number theory,
  • prime numbers,
  • Riemann zeta function.
1750No data
Area of Ford circleShow sourceACFA_{CF}Show source∑q≥1∑(p,q)=11≤p<qπ(12q2)2=π4ζ(3)ζ(4)=452ζ(3)π3ζ()= Riemann Zeta Function\sum_{q\ge 1} \sum_{ (p, q)=1 \atop 1 \le p < q }\pi \left( \frac{1}{2 q^2} \right)^2 \underset {\zeta() \text{= Riemann Zeta Function}} {= \frac{\pi}{4} \frac{\zeta(3)}{\zeta(4)} = \frac{45}{2} \frac{\zeta(3)}{\pi^3}}0.87228404106562797617
  • Number theory,
  • Riemann zeta function.
No dataNo data
Triangular root of 2Show sourceR2{R_2}Show source17−12= 4+4+4+4+4+4+⋯−1\frac{\sqrt{17}-1}{2} = \,\scriptstyle \sqrt{4+\sqrt{4+\sqrt{4+\sqrt{4+\sqrt{4+\sqrt{4+\cdots}}}}}} \,\, -11.56155281280883027491
  • Number theory.
No dataNo data
Lévy constant (2)Show source2 ln γ2\,ln\,\gammaShow sourceπ26ln(2)\frac{\pi^2}{6ln(2)}2.37313822083125090564
  • Number theory.
1936No data
Liouville numberShow source£Li\text{£}_{Li}Show source∑n=1∞110n!=1101!+1102!+1103!+1104!+⋯\sum_{n=1}^\infty \frac{1}{10^{n!}} = \frac {1}{10^{1!}} + \frac{1}{10^{2!}} + \frac{1}{10^{3!}} + \frac{1}{10^{4!}} + \cdots0.110001000000000000000001
  • Number theory.
No dataNo data
Backhouse's constantShow sourceB{B}Show sourcelim⁡k→∞∣qk+1qk∣where:    Q(x)=1P(x)= ⁣∑k=1∞qkxk\lim_{k \to \infty}\left | \frac{q_{k+1}}{q_k} \right \vert \quad \scriptstyle \text {where:} \displaystyle \;\; Q(x)=\frac{1}{P(x)}= \! \sum_{k=1}^\infty q_k x^k1.45607494858268967139
  • Number theory,
  • prime numbers.
1995No data
Copeland-Erdős constantShow sourceCCE{\mathcal{C}_{CE}}Show source∑n=1∞pn10n+∑k=1n⌊log⁡10pk⌋\sum _{n=1}^\infty \frac{p_n} {10^{n + \sum \limits_{k=1}^n \lfloor \log_{10}{p_k} \rfloor }}0.23571113171923293137
  • Number theory.
No dataNo data
Mills' constantShow sourceθ{\theta}Show source⌊θ3n⌋\lfloor \theta^{3^{n}} \rfloor1.30637788386308069046
  • Number theory.
19476850
Glaisher-Kinkelin constantShow sourceA{A}Show sourcee112−ζ′(−1)=e18−12∑n=0∞1n+1∑k=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 source−γ−π2−3ln⁡2=−γ+∑n=0∞(1n+1−1n+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
Strongly Carefree constantShow sourceK2K_{2}Show source∏n=1∞(1−3pn−2pn3)pn: prime=6π2∏n=1∞(1−1pn(pn+1))pn: prime\prod_{n=1}^\infty \underset{p_n: \text{ prime}} {\left( 1-\frac{3 p_n-2}{{p_n}^{3}}\right)} = \frac {6}{\pi ^2}\prod_{n=1}^\infty \underset{p_n: \text{ prime}} {\left( 1-\frac{1}{{p_n(p_n+1)}}\right)}0.28674742843447873410
  • Number theory,
  • prime numbers.
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
Ramanujan-Forsyth seriesShow source4π\frac {4}{\pi}Show source∑n=0∞((2n−3)!!(2n)!!)2=1 ⁣+ ⁣(12)2 ⁣+ ⁣(12⋅4)2 ⁣+ ⁣(1⋅32⋅4⋅6)2+⋯\displaystyle \sum \limits_{n=0}^\infty \textstyle \left(\frac{(2n-3)!!}{(2n)!!}\right)^2 = {1 \! + \! \left(\frac {1}{2} \right)^2 \! + \! \left(\frac {1}{2 \cdot 4} \right)^2 \! + \! \left(\frac {1 \cdot 3}{2 \cdot 4 \cdot 6} \right)^2 + \cdots}1.27323954473516268615
  • Number theory.
No dataNo data
Porter's constantShow sourceC{C}Show source6ln⁡2π2(3ln⁡2+4 γ−24π2 ζ′(2)−2)−12\frac{6\ln 2}{\pi ^2} \left(3 \ln 2 + 4 \,\gamma -\frac{24}{\pi ^2} \,\zeta '(2)-2 \right)-\frac{1}{2}1.46707807943397547289
  • Number theory.
1974No data
Golomb-Dickman constantShow sourceλ{\lambda}Show source∫0∞f(x)x2 dxPara x>2=∫01eLi⁡(n)dnLi: Logarithmic integral\int \limits_0^\infty \underset{\text{Para } x>2}{\frac{f(x)}{x^2} \, dx} = \int \limits_0^1 e^{\operatorname{Li}(n)} dn \quad \scriptstyle \text{Li: Logarithmic integral}0.62432998854355087099
  • Combinatoricts,
  • number theory,
  • discrete structures.
1930, 1964No data
Gelfond's constantShow sourceeπ{e}^{\pi}Show source(−1)−i=i−2i=∑n=0∞πnn!=π11+π22!+π33!+⋯(-1)^{-i} = i^{-2i} = \sum_{n=0}^\infty \frac{\pi^{n}}{n!} = \frac{\pi^{1}}{1} + \frac{\pi^{2}}{2!} + \frac{\pi^{3}}{3!} + \cdots23.1406926327792690057
  • Number theory.
No dataNo data
Hafner-Sarnak-McCurley constant (1)Show sourceσ{\sigma}Show source∏k=1∞{1−[1−∏j=1n(1−pk−j)]2pk: prime}\prod_{k=1}^{\infty}\left\{1-[1-\prod_{j=1}^n \underset{p_k: \text{ prime}}{(1-p_k^{-j})]^2}\right\}0.35323637185499598454
  • Number theory,
  • prime numbers.
1993No data
Hafner-Sarnak-McCurley constant (2)Show source1ζ(2)\frac{1}{\zeta(2)}Show source6π2=∏n=0∞ ⁣( ⁣1−1pn2 ⁣)pn: prime ⁣= ⁣(1 ⁣− ⁣122) ⁣(1 ⁣− ⁣132) ⁣(1 ⁣− ⁣152)⋯\frac{6}{\pi^2} = \prod_{n = 0}^\infty \underset{p_n: \text{ prime}}{\! \left(\! 1- \frac{1}{{p_n}^2} \! \right)} \! = \! \textstyle \left(1 \! - \! \frac{1}{2^2}\right) \! \left(1 \! - \! \frac{1}{3^2}\right) \! \left(1 \! - \! \frac{1}{5^2}\right)\cdots0.60792710185402662866
  • Number theory,
  • prime numbers,
  • Riemann zeta function.
No dataNo data
Champernowne constantShow sourceC10C_{10}Show source∑n=1∞  ∑k=10n−110n−1k10kn−9∑j=0n−110j(n−j−1)\sum_{n=1}^\infty \; \sum_{k=10^{n-1}}^{10^n-1}\frac{k}{10^{kn-9\sum_{j=0}^{n-1}10^j(n-j-1)}}0.12345678910111213141
  • Number theory.
1933No data
Wright constantShow sourceω{\omega}Show source⌊222⋅⋅2ω ⁣⌋= primes:⌊2ω⌋=3,⌊22ω⌋=13,⌊222ω⌋=16381,…\left \lfloor 2^{2^{2^{\cdot^{\cdot^{2^{\omega}}}}}} \!\right \rfloor \scriptstyle \text{= primes:} \displaystyle\left\lfloor 2^\omega\right\rfloor \scriptstyle \text{=3,} \displaystyle\left\lfloor 2^{2^\omega} \right\rfloor \scriptstyle \text{=13,} \displaystyle \left\lfloor 2^{2^{2^\omega}} \right\rfloor \scriptstyle =16381, \ldots1.9287800
  • Tetration (hyper-4),
  • number theory,
  • prime numbers.
No dataNo data
Ramanujan nested radicalShow sourceR5R_{5}Show source5+5+5−5+5+5+5−⋯=2+5+15−652\scriptstyle \sqrt{5+\sqrt{5+\sqrt{5-\sqrt{5+ \sqrt{5+\sqrt{5+\sqrt{5-\cdots}}}}}}}\;= \textstyle\frac{2+\sqrt{5}+\sqrt{15-6\sqrt{5}}}{2}2.74723827493230433305
  • Number theory.
No dataNo data
Beta(3)Show sourceβ(3){\beta} (3)Show sourceπ332=∑n=1∞−1n+1(−1+2n)3=113−133+153−173+⋯\frac{\pi^3}{32} = \sum_{n=1}^\infty\frac{-1^{n+1}}{(-1+2n)^3} = \frac{1}{1^3} {-} \frac{1}{3^3} {+} \frac{1}{5^3} {-} \frac{1}{7^3} {+} \cdots0.96894614625936938048
  • Number theory.
No dataNo data

Architecture and art#

Typical namesCommon symbolPossible definition or way of calculationApproximated valueExample usage or connotationsKnown at least sinceNumber of known digits after the point
(state on 2019)
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:(δS−1)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
Bronze ratioShow sourceσ Rr{\sigma}_{\,Rr}Show source3+132=1+3+3+3+3+⋯\frac {3+\sqrt{13}}{2} = 1+\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\cdots}}}}3.30277563773199464655
  • Architecture and art.
No dataNo data
Plastic number, plastic constant, the plastic ratio, the minimal Pisot number, the platin numberShow sourceρ{\rho}Show source1+ ⁣1+ ⁣1+⋯333=12+69183+12−69183\sqrt[3]{1 + \! \sqrt[3]{1 + \! \sqrt[3]{1 + \cdots}}} = \textstyle \sqrt[3]{\frac{1}{2}+\frac{\sqrt{69}}{18}}+\sqrt[3]{\frac{1}{2}-\frac{\sqrt{69}}{18}}1.32471795724474602596
  • Architecture and art.
1928No data
Musical interval between each half toneShow source212\sqrt[12]{2}Show source ⁣2x12 ⁣ ⁣ ⁣0 ⁣ ⁣ ⁣ ⁣1 ⁣ ⁣2 ⁣ ⁣3 ⁣ ⁣45 ⁣ ⁣6 ⁣ ⁣7 ⁣ ⁣8 ⁣ ⁣9 ⁣ ⁣10 ⁣ ⁣11 ⁣ ⁣12 ⁣Key ⁣C1 ⁣ ⁣C# ⁣ ⁣D ⁣D# ⁣ ⁣EF ⁣F# ⁣ ⁣G ⁣G# ⁣ ⁣A ⁣A# ⁣ ⁣B ⁣C2\begin{array}{l|ccccccccccccr} \! 2^\frac{x}{12} \! & \!\!\scriptstyle{0} & \!\!\!\!\scriptstyle{1} & \!\!\scriptstyle{2} & \!\!\scriptstyle{3} & \!\!\scriptstyle{4} & \scriptstyle{5} & \!\!\scriptstyle{6} & \!\!\scriptstyle{7} & \!\!\scriptstyle{8} & \!\!\scriptstyle{9} & \!\! \scriptstyle{10} & \!\! \scriptstyle{11} & \!\! \scriptstyle{12} \\ \hline \! \scriptstyle{\textrm{Key}} \! & \!\scriptstyle{\mathrm{C_1}} & \!\!\scriptstyle{\mathrm{C^\#}} & \!\!\scriptstyle{\mathrm{D}} & \!\scriptstyle{\mathrm{D^\#}} & \!\!\scriptstyle{\mathrm{E}} & \scriptstyle{\mathrm{F}} & \!\scriptstyle{\mathrm{F^\#}} & \!\!\scriptstyle{\mathrm{G}} & \!\scriptstyle{\mathrm{G^\#}} & \!\!\scriptstyle{\mathrm{A}} & \!\scriptstyle{\mathrm{A^\#}} & \!\!\scriptstyle{\mathrm{B}} & \!\scriptstyle{\mathrm{C_2}} \end{array} 1.05946309435929526456
  • Architecture and art.
No dataNo data

Chaos 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 first Feigenbaum's constantShow sourceδ\deltaShow sourceδ=lim⁡n→∞an−1−an−2an−an−1\delta = \lim_{n \to \infty} \frac{a_{n-1} - a_{n-2}}{a_n - a_{n-1}}4.66920160910299067185
  • Chaos theory,
  • condition of bifurcation convergence,
  • fractals,
  • attractors theory (mathematics),
  • oscillations in quartz resonators,
  • turbulence (physics),
  • oscillating reactions (chemistry).
1975No data
The second Feigenbaum's constantShow sourceα\alphaShow sourceα=lim⁡n→∞dndn+1\alpha = \lim_{n \to \infty} \frac{d_n}{d_{n+1}}2.50290787509589282228
  • Chaos theory,
  • dynamical systems (mathematics).
1979No data
Dottie numberShow sourceddShow sourcelim⁡x→∞cos⁡[x](c)=lim⁡x→∞cos⁡(cos⁡(cos⁡(⋯(cos⁡(c)))))⏟x\lim_{x\to \infty} \cos^{[x]}(c) = \lim_{x\to \infty} \underbrace{\cos(\cos(\cos(\cdots(\cos(c)))))}_x0.73908513321516064165No dataNo data

Fractals#

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 first Feigenbaum's constantShow sourceδ\deltaShow sourceδ=lim⁡n→∞an−1−an−2an−an−1\delta = \lim_{n \to \infty} \frac{a_{n-1} - a_{n-2}}{a_n - a_{n-1}}4.66920160910299067185
  • Chaos theory,
  • condition of bifurcation convergence,
  • fractals,
  • attractors theory (mathematics),
  • oscillations in quartz resonators,
  • turbulence (physics),
  • oscillating reactions (chemistry).
1975No data
The second Feigenbaum's constantShow sourceα\alphaShow sourceα=lim⁡n→∞dndn+1\alpha = \lim_{n \to \infty} \frac{d_n}{d_{n+1}}2.50290787509589282228
  • Chaos theory,
  • dynamical systems (mathematics).
1979No data
The Sierpiński's constantShow sourceKKShow sourcelim⁡n→∞[∑k=1nr2(k)k−πln⁡n]\lim_{n \to \infty}\left[\sum_{k=1}^n\frac{r_2(k)}{k} - \pi\ln n\right]2.58498175957925321706
  • Fractals.
1907No data
Fractal dimension of the Apollonian packing of circlesShow sourceε\varepsilonShow source−-1.305686729
  • Fractals,
  • geometry.
1994, 1998No data
Fractal dimension of the Cantor setShow sourcedf(k)d_f(k)Show sourcelim⁡ε→0log⁡N(ε)log⁡(1/ε)=log⁡2log⁡3\lim_{\varepsilon \to 0} \frac {\log N(\varepsilon)}{\log (1/\varepsilon)} = \frac{\log 2}{\log 3}0.63092975357145743709
  • Fractals.
No dataNo data
Area of the Mandelbrot fractalShow sourceγ\gammaShow source−-1.5065918849 ± 0.0000000028
  • Fractals.
1912No data
Fractal dimension of the Koch snowflakeShow sourceCk{C_k}Show sourcelog⁡4log⁡3\frac{\log 4}{\log 3}1.26185950714291487419
  • Fractals.
No dataNo data
Fractal dimension of the boundary of the dragon curveShow sourceCd{C_d}Show sourcelog⁡(1+73−6873+73+68733)log⁡(2)\frac{\log\left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{87}}}{3}\right)} {\log(2)}1.52362708620249210627
  • Fractals.
No dataNo data
Hausdorff dimensionShow sourcelog23{log_2 3}Show sourcelog⁡3log⁡2=∑n=0∞122n+1(2n+1)∑n=0∞132n+1(2n+1)=12+124+1160+⋯13+181+11215+⋯\frac {\log 3}{\log 2} = \frac{\sum_{n=0}^\infty \frac{1}{2^{2n+1}(2n+1)}}{\sum_{n=0}^\infty \frac{1}{3^{2n+1}(2n+1)}} = \frac{\frac{1}{2}+\frac{1}{24}+\frac{1}{160}+\cdots}{\frac{1}{3}+\frac{1}{81}+\frac{1}{1215}+\cdots}1.58496250072115618145
  • Fractals.
No dataNo data

Combinatoricts#

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 Catalan's constantShow sourceG,CG, CShow sourceC=∫01 ⁣ ⁣∫01 ⁣ ⁣11+x2y2 dx dy== ⁣∑n=0∞ ⁣(−1)n(2n+1)2 ⁣= ⁣112−132+152−172+⋯\begin{aligned}C &= \int_0^1 \!\! \int_0^1 \!\! \frac{1}{1{+}x^2 y^2}\, dx \,dy = \\ &= \! \sum_{n = 0}^\infty \! \frac{(-1)^n}{(2n{+}1)^2} \! = \! \frac{1}{1^2}{-}\frac{1}{3^2}+\frac{1}{5^2} - \frac{1}{7^2}+\cdots\end{aligned}0.91596559417721901505
  • Combinatorics,
  • low-dimensional topology.
1864300000000000
Gauss-Kuzmin-Wirsing constantShow sourceλ2{\lambda}_{2}Show sourcelim⁡n→∞Fn(x)−ln⁡(1−x)(−λ)n=Ψ(x),\lim_{n \to \infty}\frac{F_n(x) - \ln(1 - x)}{(-\lambda)^n} = \Psi(x),0.30366300289873265859
  • Combinatoricts,
  • number theory.
1973468
Liebs square ice constantShow sourceW2D{W}_{2D}Show sourcelim⁡n→∞(f(n))n−2=(43)32=833\lim_{n\to\infty}(f(n))^{n^{-2}}=\left(\frac{4}{3}\right)^\frac{3}{2}=\frac{8}{3\sqrt3}1.53960071783900203869
  • Combinatoricts,
  • discrete structures.
1967No data
Madelung constant 2Show sourceH2(2){H}_{2}(2)Show sourceπln⁡(3)3\pi \ln(3) \sqrt 35.97798681217834912266
  • Combinatoricts,
  • discrete structures.
No dataNo data
Paris ConstantShow sourceCPaC_{Pa}Show source∏n=2∞2φφ+φn  ,  φ=1+52\prod_{n=2}^\infty \frac{2 \varphi}{\varphi+ \varphi_n} \; ,\; \varphi {=} \frac{1 + \sqrt{5}}{2}1.09864196439415648573
  • Combinatoricts.
1992No data
Komornik-Loreti constantShow sourceq{q}Show source1= ⁣∑n=1∞tkqkRaiz real de∏n=0∞ ⁣( ⁣1−1q2n ⁣) ⁣+q−2q−1=01 = \!\sum_{n=1}^\infty \frac{t_k}{q^k} \qquad \scriptstyle \text{Raiz real de} \displaystyle\prod_{n=0}^\infty \!\left (\! 1 {-} \frac{1}{q^{2^n}} \!\right ) \! {+} \frac{q{-}2}{q{-}1}=01.787231650182965933011998No data
Disk CoveringShow sourceC5C_5Show source1∑n=0∞1(3n+22)=332π{\frac{1}{{\sum\limits_{n=0}^\infty \frac{1}{\binom{3n+2}{2}}}}}=\frac{3\sqrt3}{2\pi}0.82699334313268807426
  • Combinatoricts,
  • discrete structures.
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
Golomb-Dickman constantShow sourceλ{\lambda}Show source∫0∞f(x)x2 dxPara x>2=∫01eLi⁡(n)dnLi: Logarithmic integral\int \limits_0^\infty \underset{\text{Para } x>2}{\frac{f(x)}{x^2} \, dx} = \int \limits_0^1 e^{\operatorname{Li}(n)} dn \quad \scriptstyle \text{Li: Logarithmic integral}0.62432998854355087099
  • Combinatoricts,
  • number theory,
  • discrete structures.
1930, 1964No data

Prime numbers#

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 twin primes constantShow sourceC2C_2Show source∏p⩾3p(p−2)(p−1)2\prod_{p\geqslant 3} \frac{p(p-2)}{(p-1)^2}0.66016181584686957392
  • Number theory,
  • twin primes.
19225020
Landau-Ramanujan constantShow sourceKKShow source12∏p≡3 ⁣ ⁣ ⁣ ⁣ ⁣mod   ⁣4 ⁣ ⁣(1−1p2)−12 ⁣ ⁣ ⁣ ⁣ ⁣ ⁣ ⁣ ⁣p: prime ⁣ ⁣=π4∏p≡1 ⁣ ⁣ ⁣ ⁣ ⁣mod   ⁣4 ⁣ ⁣(1−1p2)12 ⁣ ⁣ ⁣ ⁣p: prime\frac1{\sqrt2}\prod_{p\equiv3\!\!\!\!\!\mod \! 4}\!\! \underset{\!\!\!\!\!\!\!\! p: \text{ prime}}{\left(1-\frac1{p^2}\right)^{-\frac{1}{2}}}\!\!=\frac\pi4\prod_{p\equiv1\!\!\!\!\!\mod \!4}\!\! \underset{\!\!\!\! p: \text{ prime}}{\left(1-\frac1{p^2}\right)^\frac{1}{2}}0.76422365358922066299
  • Number theory,
  • prime numbers.
No data30010
π to e-th powerShow sourceπe\pi^{e}Show sourceπe\pi^e22.45915771836104547342
  • Prime numbers.
No dataNo data
Sum of the reciprocals of the averages of the twin prime pairs, JJGJJGShow sourceB1B_1Show source14+16+112+118+130+142+160+172+⋯\frac{1}{4}+\frac{1}{6}+\frac{1}{12}+\frac{1}{18}+\frac{1}{30}+\frac{1}{42}+\frac{1}{60}+\frac{1}{72}+\cdots0.9288358271
  • Number theory,
  • prime numbers.
2014No data
Artin's constantShow sourceCArtin{C}_{Artin}Show source∏n=1∞(1−1pn(pn−1))pn = prime\prod_{n=1}^{\infty} \left(1-\frac{1}{p_n(p_n-1)}\right)\quad p_n \scriptstyle \text{ = prime}0.37395581361920228805
  • Number theory,
  • prime numbers.
1999No data
Murata ConstantShow sourceCm{C_m}Show source∏n=1∞(1+1(pn−1)2)pn: prime\prod_{n = 1}^\infty \underset{p_{n}: \, {prime}}{ \Big(1 + \frac{1}{(p_n-1)^2}\Big)}2.82641999706759157554
  • Number theory,
  • prime numbers.
No dataNo data
Carefree constantShow sourceC2{C}_2Show source∏n=1∞(1−1pn(pn+1))pn: prime\underset{ p_n: \, {prime}}{\prod_{n = 1}^\infty \left(1 - \frac{1}{p_n(p_n+1)}\right)}0.70444220099916559273
  • Number theory,
  • prime numbers.
No dataNo data
The value of Riemman Zeta function in point 2Show sourceζ( 2){\zeta}(\,2)Show sourceπ26=∑n=1∞1n2=112+122+132+142+⋯\frac{\pi^2}{6} = \sum_{n=1}^\infty\frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots1.64493406684822643647
  • Number theory,
  • prime numbers,
  • Riemann zeta function.
1826-1866No data
The Apéry's constantShow sourceA,ζ(3)A, \zeta(3)Show source∑n=1∞1n3=113+123+133+143+153+⋯\sum_{n=1}^\infty\frac{1}{n^3} = \frac{1}{1^3}+\frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \frac{1}{5^3} + \cdots1.20205690315959428539
  • Number theory,
  • special functions,
  • quantum electrodynamic,
  • random minimum spanning tree (computer science).
1979500000000000
The value of Riemman Zeta function in point 4Show sourceζ(4)\zeta(4)Show sourceπ490=∑n=1∞1n4=114+124+134+144+154+...\frac{\pi^4}{90} = \sum_{n=1}^\infty\frac{{1}}{n^4} = \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4} + \frac{1}{5^4} + ...1.08232323371113819151
  • Number theory,
  • prime numbers,
  • Riemann zeta function.
No dataNo data
The value of Riemman Zeta function in point 5Show sourceζ(5)\zeta(5)Show source1294π5−7235∑n=1∞1n5(e2πn−1)−235∑n=1∞1n5(e2πn+1)\frac{1}{294}\pi^5 -\frac{72}{35} \sum_{n=1}^\infty \frac{1}{n^5 (e^{2\pi n} -1)}-\frac{2}{35} \sum_{n=1}^\infty \frac{1}{n^5 (e^{2\pi n} +1)}1.0369277551433699263
  • Number theory,
  • prime numbers,
  • Riemann zeta function.
No data100000000000
The value of Riemman Zeta function in point 6Show sourceζ(6)\zeta(6)Show sourceπ6945 ⁣= ⁣∏n=1∞ ⁣11−pn−6pn: prime=11 ⁣− ⁣2−6 ⁣⋅ ⁣11 ⁣− ⁣3−6 ⁣⋅ ⁣11 ⁣− ⁣5−6⋯\frac{\pi^6}{945} \! = \! \prod_{n=1}^\infty \! \underset{p_n: \text{ prime}}{ \frac{1}{{1-p_n}^{-6}}} = \frac{1}{1 \! -\! 2^{-6}} \! \cdot \! \frac{1}{1 \! - \! 3^{-6}} \! \cdot \! \frac{1}{1 \! - \! 5^{-6}} \cdots1.01734306198444913971
  • Number theory,
  • prime numbers,
  • Riemann zeta function.
No dataNo data
Silverman constantShow sourceSm{\mathcal{S}_{_{m}}}Show source∑n=1∞1ϕ(n)σ1(n)=∏n=1∞(1+∑k=1∞1pn2k−pnk−1)pn: prime\sum_{n = 1}^\infty \frac {1}{\phi (n)\sigma_1(n)} = \underset{ p_n: \, {prime}}{ \prod_{n = 1}^\infty \left( 1 + \sum_{k = 1}^\infty \frac {1}{p_n^{2k} - p_n^{k-1}}\right)}1.78657645936592246345
  • Prime numbers.
No dataNo data
Feller-Tornier constantShow sourceCFT{\mathcal{C}_{_{FT}}}Show source12∏n=1∞(1−2pn2)+12pn: prime=3π2∏n=1∞(1−1pn2−1)+12\underset{p_n: \, {prime}}{\frac{1}{2}\prod_{n = 1}^\infty \left(1-\frac{2}{p_n^2}\right){+}\frac{1}{2}} =\frac{3}{\pi^2}\prod_{n = 1}^\infty \left(1-\frac{1}{p_n^2-1}\right){+}\frac{1}{2}0.66131704946962233528
  • Number theory,
  • prime numbers.
1932No data
Heath-Brown-Moroz constantShow sourceCHBM{C_{_{HBM}}}Show source∏n=1∞(1−1pn)7(1+7pn+1pn2)pn: prime\underset{p_n: \, {prime}}{\prod_{n = 1}^\infty \left(1-\frac{1}{p_n}\right)^7\left(1+\frac{7p_n+1}{p_n^2}\right)}0.00131764115485317810
  • Number theory,
  • prime numbers.
No dataNo data
Primorial constant, sum of the product of inverse of primesShow sourceP#{P_\#}Show source∑n=1∞1pn#=12+16+130+1210+...=∑k=1∞∏n=1k1pnpn: prime\underset{ p_n: \, {prime}}{\sum_{n = 1}^\infty \frac{1}{p_n\#} = \frac{1}{2} + \frac{1}{6} + \frac{1}{30} + \frac{1}{210} + ... = \sum_{k = 1}^\infty \prod_{n = 1}^k \frac {1}{p_n}}0.70523017179180096514
  • Number theory,
  • prime numbers.
No dataNo data
Sarnak constantShow sourceCsa{C_{sa} }Show source∏p>2(1−p+2p3)\prod_{p>2} \Big(1 - \frac{p+2}{p^3}\Big)0.72364840229820000940
  • Prime numbers.
No dataNo data
Foias constant (α)Show sourceFαF_\alphaShow sourcexn+1=(1+1xn)n for n=1,2,3,…x_{n+1} = \left( 1 + \frac{1}{x_n} \right)^n\text{ for }n=1,2,3,\ldots1.18745235112650105459
  • Prime numbers.
2000No data
Foias constant (β)Show sourceFβF_\betaShow sourcexx+1=(x+1)xx^{x+1} = (x+1)^x2.29316628741186103150
  • Prime numbers.
2000No data
Taniguchi constantShow sourceCTC_TShow source∏n=1∞(1−3pn3+2pn4+1pn5−1pn6)\prod_{n = 1}^\infty \left(1 - \frac{3}{{p_n}^3}+\frac{2}{{p_n}^4}+\frac{1}{{p_n}^5}-\frac{1}{{p_n}^6}\right)0.67823449191739197803
  • Prime numbers.
No dataNo data
Euler totient constantShow sourceETETShow source∏p(1+1p(p−1))p= primes=ζ(2)ζ(3)ζ(6)=315ζ(3)2π4\underset {p \text{= primes}} {\prod_{p} \Big(1 + \frac{1}{p(p-1)}\Big)} = \frac {\zeta(2)\zeta(3)}{\zeta(6)}=\frac {315 \zeta(3)}{2\pi^4}1.94359643682075920505
  • Number theory,
  • prime numbers,
  • Riemann zeta function.
1750No data
Backhouse's constantShow sourceB{B}Show sourcelim⁡k→∞∣qk+1qk∣where:    Q(x)=1P(x)= ⁣∑k=1∞qkxk\lim_{k \to \infty}\left | \frac{q_{k+1}}{q_k} \right \vert \quad \scriptstyle \text {where:} \displaystyle \;\; Q(x)=\frac{1}{P(x)}= \! \sum_{k=1}^\infty q_k x^k1.45607494858268967139
  • Number theory,
  • prime numbers.
1995No data
Glaisher-Kinkelin constantShow sourceA{A}Show sourcee112−ζ′(−1)=e18−12∑n=0∞1n+1∑k=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
Strongly Carefree constantShow sourceK2K_{2}Show source∏n=1∞(1−3pn−2pn3)pn: prime=6π2∏n=1∞(1−1pn(pn+1))pn: prime\prod_{n=1}^\infty \underset{p_n: \text{ prime}} {\left( 1-\frac{3 p_n-2}{{p_n}^{3}}\right)} = \frac {6}{\pi ^2}\prod_{n=1}^\infty \underset{p_n: \text{ prime}} {\left( 1-\frac{1}{{p_n(p_n+1)}}\right)}0.28674742843447873410
  • Number theory,
  • prime numbers.
No dataNo data
Hafner-Sarnak-McCurley constant (1)Show sourceσ{\sigma}Show source∏k=1∞{1−[1−∏j=1n(1−pk−j)]2pk: prime}\prod_{k=1}^{\infty}\left\{1-[1-\prod_{j=1}^n \underset{p_k: \text{ prime}}{(1-p_k^{-j})]^2}\right\}0.35323637185499598454
  • Number theory,
  • prime numbers.
1993No data
Hafner-Sarnak-McCurley constant (2)Show source1ζ(2)\frac{1}{\zeta(2)}Show source6π2=∏n=0∞ ⁣( ⁣1−1pn2 ⁣)pn: prime ⁣= ⁣(1 ⁣− ⁣122) ⁣(1 ⁣− ⁣132) ⁣(1 ⁣− ⁣152)⋯\frac{6}{\pi^2} = \prod_{n = 0}^\infty \underset{p_n: \text{ prime}}{\! \left(\! 1- \frac{1}{{p_n}^2} \! \right)} \! = \! \textstyle \left(1 \! - \! \frac{1}{2^2}\right) \! \left(1 \! - \! \frac{1}{3^2}\right) \! \left(1 \! - \! \frac{1}{5^2}\right)\cdots0.60792710185402662866
  • Number theory,
  • prime numbers,
  • Riemann zeta function.
No dataNo data
Wright constantShow sourceω{\omega}Show source⌊222⋅⋅2ω ⁣⌋= primes:⌊2ω⌋=3,⌊22ω⌋=13,⌊222ω⌋=16381,…\left \lfloor 2^{2^{2^{\cdot^{\cdot^{2^{\omega}}}}}} \!\right \rfloor \scriptstyle \text{= primes:} \displaystyle\left\lfloor 2^\omega\right\rfloor \scriptstyle \text{=3,} \displaystyle\left\lfloor 2^{2^\omega} \right\rfloor \scriptstyle \text{=13,} \displaystyle \left\lfloor 2^{2^{2^\omega}} \right\rfloor \scriptstyle =16381, \ldots1.9287800
  • Tetration (hyper-4),
  • number theory,
  • prime numbers.
No dataNo data

Statistics#

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 Gauss's constantShow sourceGGShow sourceG=1agm⁡(1,2)=2π∫01dx1−x4=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
Median of the Gumbel distributionShow sourcell2{ll_2}Show source−ln⁡(ln⁡(2))-\ln(\ln(2))0.36651292058166432701
  • Statistics.
No dataNo data
Khinchin harmonic meanShow sourceK−1{K_{-1}}Show sourcelog⁡2∑n=1∞1nlog⁡(1+1n(n+2))=lim⁡n→∞n1a1+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

Approximation of functions#

Typical namesCommon symbolPossible definition or way of calculationApproximated valueExample usage or connotationsKnown at least sinceNumber of known digits after the point
(state on 2019)
Fransen-Robinson constantShow sourceFFShow source∫0∞1Γ(x) dx=e+∫0∞e−xπ2+ln⁡2x dx\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
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
Lebesgue constant L2Show sourceL2{L2}Show source15+25−25π=1π∫0π∣sin⁡(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
Bernsteins constantShow sourceβ{\beta}Show source≈12π\approx \frac {1}{2\sqrt {\pi}}0.28016949902386913303
  • Mathematical analysis,
  • approximation of functions.
1913No 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
Lebesgue constantShow sourceC1{C_1}Show sourcelim⁡n→∞ ⁣ ⁣( ⁣Ln−4π2ln⁡(2n+1) ⁣ ⁣) ⁣=4π2 ⁣(∑k=1∞ ⁣2ln⁡k4k2−1−Γ′(12)Γ(12) ⁣ ⁣)\lim_{n\to\infty}\!\! \left(\!{L_n{-}\frac{4}{\pi^2}\ln(2n{+}1)}\!\!\right)\!{=} \frac{4}{\pi^2}\!\left({\sum_{k=1}^\infty \!\frac{2\ln k}{4k^2{-}1}} {-}\frac{\Gamma'(\tfrac12)}{\Gamma(\tfrac12)}\!\!\right)0.98943127383114695174
  • Approximation of functions.
No dataNo data
Lebesgue constant (interpolation)Show sourceL1{L_1}Show source∏i=0j≠inx−xixj−xi=1π∫0π⌊sin⁡3t2⌋sin⁡t2 dt=13+23π\prod_{\begin{matrix}i=0\\ j\neq i\end{matrix}}^{n} \frac{x-x_i}{x_j-x_i} = \frac {1}{\pi} \int_0^{\pi} \frac {\lfloor \sin{\frac{3 t}{2}}\rfloor}{\sin{\frac{t}{2}}}\, dt = \frac {1}{3} + \frac {2 \sqrt{3}}{\pi}1.43599112417691743235
  • Approximation of functions.
1902No data
Gibbs constantShow sourceSi(π){Si(\pi)}Show source∫0πsin⁡tt dt=∑n=1∞(−1)n−1π2n−1(2n−1)(2n−1)!\int_0^{\pi} \frac {\sin t}{t}\, dt = \sum \limits_{n=1}^\infty (-1)^{n-1} \frac{\pi^{2n-1}}{(2n-1)(2n-1)!}1.85193705198246617036
  • Approximation of functions.
No dataNo data

Topology#

Typical namesCommon symbolPossible definition or way of calculationApproximated valueExample usage or connotationsKnown at least sinceNumber of known digits after the point
(state on 2019)
Hermite constant sphere packing 3D Kepler conjectureShow sourceμK{\mu_{_{K}}}Show sourceπ32\frac{\pi}{3\sqrt{2}}0.74048048969306104116
  • Geometry,
  • topology.
1611No data

Complex 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)
Infinite tetration of iShow source∞i{}^\infty iShow sourcelim⁡n→∞ni=lim⁡n→∞ii⋅⋅i⏟n\lim_{n \to \infty} {}^n i = \lim_{n \to \infty} \underbrace{i^{i^{\cdot^{\cdot^i}}}}_n0.43828293672703211162
+ 0.360592471871385485 i
  • Complex analysis,
  • tetration (hyper-4).
No dataNo data
i to i-th powerShow sourceiii^iShow sourcee−π2e^{-\frac{\pi}{2}}0.20787957635076190854
  • Complex analysis.
1746No data
Hyperbolic tangent of 1Show sourcetanh⁡1\tanh 1Show source−itan⁡(i)=e−1ee+1e=e2−1e2+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
Generalized continued fraction of iShow sourceFCG(i)F_{CG(i)}Show sourcei+ii+ii+ii+ii+ii+ii+i/⋯ =17−18+i(12+217−1)\textstyle i+\cfrac i{i+\cfrac i{i+\cfrac i{i+\cfrac i{i+\cfrac i{i+\cfrac i{i+i\left/\cdots\right.}}}}}} = \sqrt{\frac{\sqrt{17}-1}{8}} + i \left(\tfrac12 + \sqrt{\frac{2}{\sqrt{17}-1}}\right)0.62481053384382658687
+ 1.300242590220120419 i
  • Complex analysis.
No dataNo data
Module of infinite tetration of iShow source∣∞i∣|{}^\infty {i} |Show sourcelim⁡n→∞∣ni∣=∣lim⁡n→∞ii⋅⋅i⏟n∣\lim_{n \to \infty} \left | {}^n i \right | =\left | \lim_{n \to \infty} \underbrace{i^{i^{\cdot^{\cdot^{i}}}}}_n \right |0.56755516330695782538
  • Complex analysis,
  • tetration (hyper-4).
No dataNo data
Factorial(i)Show sourcei!i!Show sourceΓ(1+i)=i Γ(i)=∫0∞tietdt\Gamma (1+i) = i \, \Gamma (i) = \int\limits_0^\infty \frac{t^i}{e^t} \mathrm{d} t0.49801566811835604271
+ 0.15494982830181068512 i
  • Complex analysis.
No dataNo data
Square root of iShow sourcei\sqrt{i}Show source−14=1+i2=eiπ4=cos⁡(π4)+isin⁡(π4)\sqrt[4]{-1} = \frac{1+i}{\sqrt{2}} = e^ \frac{i\pi}{4} = \cos\left (\frac{\pi}{4} \right ) + i\sin\left ( \frac{\pi}{4} \right )0.70710678118654752440 + 0.707106781186547524 i
  • Complex analysis.
No dataNo data
John constantShow sourceγ\gammaShow sourceii=i−i=(ii)−1=(((i)i)i)i=eπ2=∑n=0∞πnn!\sqrt[i]{i} = i^{-i} = (i^i)^{-1} = (((i)^i)^i)^i = e^{\frac{\pi}{2}} = \sqrt{\sum_{n=0}^\infty \frac{\pi^{n}}{n!}}4.81047738096535165547
  • Complex analysis.
No dataNo data
Cube root of 1Show source13\sqrt[3]{1}Show source{  1−12+32i−12−32i.\begin{cases} \ \ 1 \\ -\frac{1}{2}+\frac{\sqrt{3}}{2}i \\ -\frac{1}{2}-\frac{\sqrt{3}}{2}i. \end{cases}-0.5 ± 0.86602540378443864676 i
  • Complex 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
Masser-Gramain constantShow sourceC{C}Show sourceγβ(1) ⁣+ ⁣β′(1) ⁣=π ⁣(− ⁣ln⁡Γ(14)+34π+12ln⁡2+12γ)\gamma {\beta}(1) \! + \! {\beta}'(1) \! = \pi \! \left(-\!\ln \Gamma(\tfrac14)+\tfrac34 \pi+\tfrac12 \ln 2+\tfrac12 \gamma \right)0.64624543989481330426
  • Complex analysis.
No dataNo data
Exp.gamma, Barnes G-functionShow sourceeγe^{\gamma}Show source∏n=1∞e1n1+1n=∏n=0∞(∏k=0n(k+1)(−1)k+1(nk))1n+1=\prod_{n=1}^\infty \frac{e^{\frac{1}{n}}}{1+\tfrac1n} = \prod_{n=0}^\infty \left(\prod_{k=0}^n (k+1)^{(-1)^{k+1}{n \choose k}}\right)^{\frac{1}{n+1}} =1.78107241799019798523
  • Complex analysis.
No dataNo data
Bloch-Landau constantShow sourceL{L}Show source=Γ(13)  Γ(56)Γ(16)=(−23)!  (−1+56)!(−1+16)!= \frac {\Gamma(\tfrac13)\;\Gamma(\tfrac{5}{6})} {\Gamma(\tfrac{1}{6})} = \frac {(-\tfrac23)!\;(-1+\tfrac56)!} {(-1+\tfrac16)!}0.54325896534297670695
  • Complex analysis.
1929No data
Imaginary numberShow sourcei{i}Show source−1=ln⁡(−1)πei π=−1\sqrt{-1} = \frac{\ln(-1)}{\pi} \qquad\qquad \mathrm{e}^{i\,\pi} = -1i
  • General usage in various math fields,
  • complex analysis.
1501-1576-

Tetration (hyper-4)#

Typical namesCommon symbolPossible definition or way of calculationApproximated valueExample usage or connotationsKnown at least sinceNumber of known digits after the point
(state on 2019)
Infinite tetration of iShow source∞i{}^\infty iShow sourcelim⁡n→∞ni=lim⁡n→∞ii⋅⋅i⏟n\lim_{n \to \infty} {}^n i = \lim_{n \to \infty} \underbrace{i^{i^{\cdot^{\cdot^i}}}}_n0.43828293672703211162
+ 0.360592471871385485 i
  • Complex analysis,
  • tetration (hyper-4).
No dataNo data
Exponential factorial constantShow sourceSEf{S_{Ef}}Show source∑n=1∞1n(n−1)⋅⋅⋅21=1+121+1321+14321+154321+⋯\sum_{n=1}^{\infty} \frac{1}{n^{(n{-}1)^{\cdot^{\cdot^{\cdot^{2^1}}}}}} = 1 {+} \frac{1}{2^{1}} {+} \frac{1}{3^{2^{1}}} + \frac{1}{4^{3^{2^{1}}}} + \frac{1}{5^{4^{3^{2^{1}}}}} {+} \cdots1.611114925808376736111
  • Tetration (hyper-4),
  • number theory.
No dataNo data
Fixed points super-logarithm tetrationShow source−W(−1)-W(-1)Show sourcelim⁡n→∞f(x)=log⁡(log⁡(log⁡(log⁡(⋯log⁡(log⁡(x))))))  ⁣⏟log⁡s 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
Module of infinite tetration of iShow source∣∞i∣|{}^\infty {i} |Show sourcelim⁡n→∞∣ni∣=∣lim⁡n→∞ii⋅⋅i⏟n∣\lim_{n \to \infty} \left | {}^n i \right | =\left | \lim_{n \to \infty} \underbrace{i^{i^{\cdot^{\cdot^{i}}}}}_n \right |0.56755516330695782538
  • Complex analysis,
  • tetration (hyper-4).
No dataNo data
Lower limit of TetrationShow sourcee−e{e}^{-e}Show source(1e)e\left(\frac {1}{e}\right)^e0.06598803584531253707
  • Tetration (hyper-4).
No dataNo data
Upper iterated exponentialShow sourceH2n+1{H}_{2n+1}Show sourcelim⁡n→∞H2n+1=(12)(13)(14)⋅⋅(12n+1)=2−3−4⋅⋅−2n−1\lim_{n \to \infty} {H}_{2n+1} = \textstyle \left(\frac{1}{2}\right) ^{\left(\frac{1}{3}\right) ^{\left(\frac{1}{4}\right) ^{\cdot^{\cdot^{\left(\frac{1}{2n+1}\right)}}}}} = {2}^{-3^{-4^{\cdot^{\cdot^{{-2n-1}}}}}}0.69034712611496431946
  • Tetration (hyper-4).
No dataNo data
Lower limit iterated exponentialShow sourceH2n{H}_{2n}Show sourcelim⁡n→∞H2n=(12)(13)(14)⋅⋅(12n)=2−3−4⋅⋅−2n\lim_{n \to \infty} {H}_{2n} = \textstyle \left(\frac{1}{2}\right) ^{\left(\frac{1}{3}\right) ^{\left(\frac{1}{4}\right) ^{\cdot^{\cdot^{\left(\frac{1}{2n}\right)}}}}} = {2}^{-3^{-4^{\cdot^{\cdot^{{-2n}}}}}}0.6583655992
  • Tetration (hyper-4).
No dataNo data
Wright constantShow sourceω{\omega}Show source⌊222⋅⋅2ω ⁣⌋= primes:⌊2ω⌋=3,⌊22ω⌋=13,⌊222ω⌋=16381,…\left \lfloor 2^{2^{2^{\cdot^{\cdot^{2^{\omega}}}}}} \!\right \rfloor \scriptstyle \text{= primes:} \displaystyle\left\lfloor 2^\omega\right\rfloor \scriptstyle \text{=3,} \displaystyle\left\lfloor 2^{2^\omega} \right\rfloor \scriptstyle \text{=13,} \displaystyle \left\lfloor 2^{2^{2^\omega}} \right\rfloor \scriptstyle =16381, \ldots1.9287800
  • Tetration (hyper-4),
  • number theory,
  • prime numbers.
No dataNo data

Algebra#

Typical namesCommon symbolPossible definition or way of calculationApproximated valueExample usage or connotationsKnown at least sinceNumber of known digits after the point
(state on 2019)
Flajolet-Richmond constantShow sourceQ{Q}Show source∏n=1∞(1−12n)=(1−121)(1−122)(1−123)⋯\prod_{n=1}^\infty \left(1 - \frac{1}{2^n}\right) = \left(1-\frac{1}{2^1}\right) \left(1-\frac{1}{2^2} \right)\left(1-\frac{1}{2^3} \right) \cdots0.288788095086602421271992No data
Goh-Schmutz constantShow sourceCGSC_{GS}Show source∫0∞log⁡(s+1)es−1 ds= ⁣− ⁣∑n=1∞ennEi(−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 source−W(−1)-W(-1)Show sourcelim⁡n→∞f(x)=log⁡(log⁡(log⁡(log⁡(⋯log⁡(log⁡(x))))))  ⁣⏟log⁡s 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
Hermite-Ramanujan constantShow sourceR{R}Show sourceeπ163e^{\pi\sqrt{163}}262537412640768743
+ 0.999999999999250073
  • Algebra.
1859No data

Discrete structures#

Typical namesCommon symbolPossible definition or way of calculationApproximated valueExample usage or connotationsKnown at least sinceNumber of known digits after the point
(state on 2019)
Liebs square ice constantShow sourceW2D{W}_{2D}Show sourcelim⁡n→∞(f(n))n−2=(43)32=833\lim_{n\to\infty}(f(n))^{n^{-2}}=\left(\frac{4}{3}\right)^\frac{3}{2}=\frac{8}{3\sqrt3}1.53960071783900203869
  • Combinatoricts,
  • discrete structures.
1967No data
Madelung constant 2Show sourceH2(2){H}_{2}(2)Show sourceπln⁡(3)3\pi \ln(3) \sqrt 35.97798681217834912266
  • Combinatoricts,
  • discrete structures.
No dataNo data
Disk CoveringShow sourceC5C_5Show source1∑n=0∞1(3n+22)=332π{\frac{1}{{\sum\limits_{n=0}^\infty \frac{1}{\binom{3n+2}{2}}}}}=\frac{3\sqrt3}{2\pi}0.82699334313268807426
  • Combinatoricts,
  • discrete structures.
No dataNo data
Connective constantShow sourcemu{mu}Show source2+2  =lim⁡n→∞cn1/n\sqrt{2 + \sqrt{2}} \; = \lim_{n \rightarrow \infty} c_n^{1/n}1.84775906502257351225
  • Discrete structures.
No dataNo data
Baxter's four-coloring constantShow sourceC2\mathcal{C}^2Show source∏n=1∞(3n−1)2(3n−2)(3n)=34π2 Γ(13)3\prod_{n = 1}^\infty \frac{(3n-1)^2}{(3n-2)(3n)} = \frac {3}{4\pi^2} \,\Gamma \left(\frac {1}{3}\right)^31.46099848620631835815
  • Discrete structures.
1970No 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
Dimer constant 2D, Domino tilingShow sourceCπ{\frac{C}{\pi}}Show source∫−ππcosh⁡−1(cos⁡(t)+32)4π dt\int\limits_{-\pi}^{\pi} \frac{\cosh^{-1}\left(\frac{\sqrt{\cos(t)+3}}{\sqrt2}\right)}{4\pi}\,dt0.29156090403081878013
  • Discrete structures.
No dataNo data
Rényi's Parking ConstantShow sourcem{m}Show source∫0∞exp( ⁣−2∫0x1−e−yydy) ⁣dx=e−2γ∫0∞e−2Γ(0,n)n2\int \limits_{0}^{\infty} exp \left(\! -2 \int \limits_{0}^{x} \frac {1-e^{-y}}{y} dy\right)\! dx = {e^{-2 \gamma}} \int \limits_{0}^{\infty} \frac{e^{-2 \Gamma(0,n)}}{n^2}0.74759792025341143517
  • Discrete structures.
No dataNo data
Pólya Random walk constantShow sourcep(3){p(3)}Show source1− ⁣ ⁣(3(2π)3∫−ππ∫−ππ∫−ππdx dy dz3− ⁣cos⁡x− ⁣cos⁡y− ⁣cos⁡z) ⁣−11- \!\!\left({3\over(2\pi)^3}\int\limits_{-\pi}^{\pi} \int\limits_{-\pi}^{\pi} \int\limits_{-\pi}^{\pi} {dx\,dy\,dz\over 3-\!\cos x-\!\cos y-\!\cos z}\right)^{\!-1}0.34053732955099914282
  • Discrete structures.
No dataNo data
Golomb-Dickman constantShow sourceλ{\lambda}Show source∫0∞f(x)x2 dxPara x>2=∫01eLi⁡(n)dnLi: Logarithmic integral\int \limits_0^\infty \underset{\text{Para } x>2}{\frac{f(x)}{x^2} \, dx} = \int \limits_0^1 e^{\operatorname{Li}(n)} dn \quad \scriptstyle \text{Li: Logarithmic integral}0.62432998854355087099
  • Combinatoricts,
  • number theory,
  • discrete structures.
1930, 1964No data

Riemann zeta function#

Typical namesCommon symbolPossible definition or way of calculationApproximated valueExample usage or connotationsKnown at least sinceNumber of known digits after the point
(state on 2019)
Raabes formulaShow sourceζ′(0){\zeta'(0)}Show source∫aa+1log⁡Γ(t) dt=12log⁡2π+alog⁡a−a,a≥0\int\limits_a^{a+1}\log\Gamma(t)\,\mathrm dt = \tfrac12\log2\pi + a\log a - a,\quad a \ge 00.91893853320467274178
  • Number theory,
  • Riemann zeta function.
No dataNo data
Favard constantShow source34ζ(2)\tfrac34\zeta(2)Show sourceπ28=∑n=0∞1(2n−1)2=112+132+152+172+⋯\frac{\pi^2}{8} = \sum_{n = 0}^\infty \frac{1}{(2n-1)^2} = \frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\cdots1.23370055013616982735
  • Number theory,
  • Riemann zeta function.
1902, 1965No data
The value of Riemman Zeta function in point 2Show sourceζ( 2){\zeta}(\,2)Show sourceπ26=∑n=1∞1n2=112+122+132+142+⋯\frac{\pi^2}{6} = \sum_{n=1}^\infty\frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots1.64493406684822643647
  • Number theory,
  • prime numbers,
  • Riemann zeta function.
1826-1866No data
The Apéry's constantShow sourceA,ζ(3)A, \zeta(3)Show source∑n=1∞1n3=113+123+133+143+153+⋯\sum_{n=1}^\infty\frac{1}{n^3} = \frac{1}{1^3}+\frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \frac{1}{5^3} + \cdots1.20205690315959428539
  • Number theory,
  • special functions,
  • quantum electrodynamic,
  • random minimum spanning tree (computer science).
1979500000000000
The value of Riemman Zeta function in point 4Show sourceζ(4)\zeta(4)Show sourceπ490=∑n=1∞1n4=114+124+134+144+154+...\frac{\pi^4}{90} = \sum_{n=1}^\infty\frac{{1}}{n^4} = \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4} + \frac{1}{5^4} + ...1.08232323371113819151
  • Number theory,
  • prime numbers,
  • Riemann zeta function.
No dataNo data
The value of Riemman Zeta function in point 5Show sourceζ(5)\zeta(5)Show source1294π5−7235∑n=1∞1n5(e2πn−1)−235∑n=1∞1n5(e2πn+1)\frac{1}{294}\pi^5 -\frac{72}{35} \sum_{n=1}^\infty \frac{1}{n^5 (e^{2\pi n} -1)}-\frac{2}{35} \sum_{n=1}^\infty \frac{1}{n^5 (e^{2\pi n} +1)}1.0369277551433699263
  • Number theory,
  • prime numbers,
  • Riemann zeta function.
No data100000000000
The value of Riemman Zeta function in point 6Show sourceζ(6)\zeta(6)Show sourceπ6945 ⁣= ⁣∏n=1∞ ⁣11−pn−6pn: prime=11 ⁣− ⁣2−6 ⁣⋅ ⁣11 ⁣− ⁣3−6 ⁣⋅ ⁣11 ⁣− ⁣5−6⋯\frac{\pi^6}{945} \! = \! \prod_{n=1}^\infty \! \underset{p_n: \text{ prime}}{ \frac{1}{{1-p_n}^{-6}}} = \frac{1}{1 \! -\! 2^{-6}} \! \cdot \! \frac{1}{1 \! - \! 3^{-6}} \! \cdot \! \frac{1}{1 \! - \! 5^{-6}} \cdots1.01734306198444913971
  • Number theory,
  • prime numbers,
  • Riemann zeta function.
No dataNo data
Niven's constantShow sourceC{C}Show source1+∑n=2∞(1−1ζ(n))1+\sum_{n = 2}^\infty \left(1-\frac{1}{\zeta(n)} \right)1.70521114010536776428
  • Number theory,
  • Riemann zeta function.
1969No data
Nielsen-Ramanujan constantShow sourceζ(2)2\frac{{\zeta}(2)}{2}Show sourceπ212=∑n=1∞(−1)n+1n2=112−122+132−142+152−⋯\frac{\pi^2}{12} = \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^2} = \frac{1}{1^2} {-} \frac{1}{2^2} {+} \frac{1}{3^2} {-} \frac{1}{4^2} {+} \frac{1}{5^2} {-} \cdots0.82246703342411321823
  • Number theory,
  • Riemann zeta function.
1909No data
Euler totient constantShow sourceETETShow source∏p(1+1p(p−1))p= primes=ζ(2)ζ(3)ζ(6)=315ζ(3)2π4\underset {p \text{= primes}} {\prod_{p} \Big(1 + \frac{1}{p(p-1)}\Big)} = \frac {\zeta(2)\zeta(3)}{\zeta(6)}=\frac {315 \zeta(3)}{2\pi^4}1.94359643682075920505
  • Number theory,
  • prime numbers,
  • Riemann zeta function.
1750No data
Area of Ford circleShow sourceACFA_{CF}Show source∑q≥1∑(p,q)=11≤p<qπ(12q2)2=π4ζ(3)ζ(4)=452ζ(3)π3ζ()= Riemann Zeta Function\sum_{q\ge 1} \sum_{ (p, q)=1 \atop 1 \le p < q }\pi \left( \frac{1}{2 q^2} \right)^2 \underset {\zeta() \text{= Riemann Zeta Function}} {= \frac{\pi}{4} \frac{\zeta(3)}{\zeta(4)} = \frac{45}{2} \frac{\zeta(3)}{\pi^3}}0.87228404106562797617
  • Number theory,
  • Riemann zeta function.
No dataNo data
π squaredShow sourceπ2{\pi} ^2Show source6 ζ(2)=6∑n=1∞1n2=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
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
Hafner-Sarnak-McCurley constant (2)Show source1ζ(2)\frac{1}{\zeta(2)}Show source6π2=∏n=0∞ ⁣( ⁣1−1pn2 ⁣)pn: prime ⁣= ⁣(1 ⁣− ⁣122) ⁣(1 ⁣− ⁣132) ⁣(1 ⁣− ⁣152)⋯\frac{6}{\pi^2} = \prod_{n = 0}^\infty \underset{p_n: \text{ prime}}{\! \left(\! 1- \frac{1}{{p_n}^2} \! \right)} \! = \! \textstyle \left(1 \! - \! \frac{1}{2^2}\right) \! \left(1 \! - \! \frac{1}{3^2}\right) \! \left(1 \! - \! \frac{1}{5^2}\right)\cdots0.60792710185402662866
  • Number theory,
  • prime numbers,
  • Riemann zeta function.
No dataNo data

Fibonacci numbers#

Typical namesCommon symbolPossible definition or way of calculationApproximated valueExample usage or connotationsKnown at least sinceNumber of known digits after the point
(state on 2019)
Fibonacci factorial constantShow sourceFFShow source∏n=1∞(1−(−1φ2)n)=∏n=1∞(1−(5−32)n)\prod_{n = 1}^\infty \left(1 - \left( -\frac{1}{{\varphi}^2}\right)^n \right)= \prod_{n = 1}^\infty \left(1 - \left( \frac{\sqrt{5}-3}{2}\right)^n \right)1.22674201072035324441
  • Fibonacci numbers.
No dataNo data
Gelfond-Schneider constantShow sourceG GSG_{\,GS}Show source222^{\sqrt{2}}2.665144142690225188651934No data
Viswanath constantShow sourceCVi{C}_{Vi}Show sourcelim⁡n→∞∣an∣1n\lim_{n \to \infty}|a_n|^\frac{1}{n}1.1319882487943
  • Fibonacci numbers,
  • number theory.
No data8
Tetranacci constantShow sourceT\mathcal{T}Show sourcex4−x3−x2−x−1=0x^4-x^3-x^2-x-1=01.92756197548292530426
  • Fibonacci numbers.
No dataNo data
Prévost constant, Reciprocal Fibonacci constantShow sourceΨ\PsiShow source∑n=1∞1Fn=11+11+12+13+15+18+113+⋯\sum_{n=1}^{\infty} \frac{1}{F_n} = \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{8} + \frac{1}{13} + \cdots3.35988566624317755317
  • Fibonacci numbers.
No dataNo data

Functional iteration#

Typical namesCommon symbolPossible definition or way of calculationApproximated valueExample usage or connotationsKnown at least sinceNumber of known digits after the point
(state on 2019)
Kepler-Bouwkamp constantShow sourceho{ ho}Show source∏n=3∞cos⁡(πn)=cos⁡(π3)cos⁡(π4)cos⁡(π5)...\prod_{n=3}^\infty \cos\left(\frac{\pi}{n} \right) = \cos\left(\frac{\pi}{3} \right) \cos\left(\frac{\pi}{4} \right) \cos\left(\frac{\pi}{5}\right) ...0.11494204485329620070
  • Functional iteration.
No dataNo data
Prouhet-Thue-Morse constantShow sourceτ\tauShow source∑n=0∞tn2n+1\sum_{n=0}^{\infty} \frac{t_n}{2^{n+1}}0.41245403364010759778
  • Functional iteration.
No dataNo data
Plouffe's gamma constantShow sourceC{{C}}Show source1πarctan⁡12=1π∑n=0∞(−1)n(22n+1)(2n+1)\frac{1}{\pi} \arctan {\frac{1}{2}} = \frac{1}{\pi}\sum_{n=0}^\infty \frac {(-1)^n}{(2^{2n+1})(2n+1)}0.14758361765043327417
  • Functional iteration.
No dataNo data
Plouffe's A constantShow sourceA{A}Show source12π\frac{1}{2 \pi}0.15915494309189533576
  • Functional iteration.
No dataNo data
Conway constantShow sourceλ{\lambda}Show sourcex71−x69−2x68−x67+2x66+2x65+x64−x63−x62−x61−x60−x59+2x58+5x57+3x56−2x55−10x54−3x53−2x52+6x51+6x50+x49+9x48−3x47−7x46−8x45−8x44+10x43+6x42+8x41−5x40−12x39+7x38−7x37+7x36+x35−3x34+10x33+x32−6x31−2x30−10x29−3x28+2x27+9x26−3x25+14x24−8x23−7x21+9x20+3x19−4x18−10x17−7x16+12x15+7x14+2x13−12x12−4x11−2x10+5x9+x7−7x6+7x5−4x4+12x3−6x2+3x−6=0\begin{matrix}x^{71} & -x^{69} & -2x^{68} & -x^{67} & +2x^{66} & \\+2x^{65} & +x^{64} & -x^{63} & -x^{62} & -x^{61} & \\-x^{60} & -x^{59} & +2x^{58} & +5x^{57} & +3x^{56} & \\-2x^{55} & -10x^{54} & -3x^{53} & -2x^{52} & +6x^{51} & \\+6x^{50} & +x^{49} & +9x^{48} & -3x^{47} & -7x^{46} & \\-8x^{45} & -8x^{44} & +10x^{43} & +6x^{42} & +8x^{41} & \\-5x^{40} & -12x^{39} & +7x^{38} & -7x^{37} & +7x^{36} & \\+x^{35} & -3x^{34} & +10x^{33} & +x^{32} & -6x^{31} & \\-2x^{30} & -10x^{29} & -3x^{28} & +2x^{27} & +9x^{26} & \\-3x^{25} & +14x^{24} & -8x^{23} & -7x^{21} & +9x^{20} & \\+3x^{19} & -4x^{18} & -10x^{17} & -7x^{16} & +12x^{15} & \\+7x^{14} & +2x^{13} & -12x^{12} & -4x^{11} & -2x^{10} & \\+5x^{9} & +x^{7} & -7x^{6} & +7x^{5} & -4x^{4} & \\ & +12x^{3} & -6x^{2} & +3x & -6 & = 0\end{matrix}1.30357726903429639125
  • Functional iteration.
1987No data
Weierstrass constantShow sourceσ(12)\sigma(\tfrac12)Show sourceeπ8π4⋅23/4(14!)2\frac{e^{\frac{\pi}{8}}\sqrt{\pi}}{4 \cdot 2^{3/4} {(\frac {1}{4}!)^2}}0.47494937998792065033
  • Functional iteration.
1872No data
Gauss's Lemniscate constantShow sourceL/2L \text{/}\sqrt{2}Show source∫0∞dx1+x4=14π Γ(14)2=4(14!)2π\int\limits_0^\infty \frac{{\mathrm{d} x}}{\sqrt{1 + x^4}} = \frac {1}{4\sqrt{\pi}} \,\Gamma \left(\frac {1}{4}\right)^2 = \frac{4 \left(\frac {1}{4}!\right)^2} {\sqrt{\pi}}1.85407467730137191843
  • Functional iteration.
No dataNo data
Regular paperfolding sequenceShow sourcePf{P_f}Show source∑n=0∞82n22n+2−1=∑n=0∞122n1−122n+2\sum_{n=0}^{\infty} \frac {8^{2^n}}{2^{2^{n+2}}-1} = \sum_{n=0}^{\infty} \cfrac {\tfrac {1}{2^{2^n}}} {1-\tfrac{1}{2^{2^{n+2}}}}0.85073618820186726036
  • Functional iteration.
No dataNo data
Cubic recurrence constantShow sourceσ3{\sigma_3}Show source∏n=1∞n3−n=123⋯333=11/3  21/9  31/27⋯\prod_{n=1}^\infty n^{{3}^{-n}} = \sqrt[3] {1 \sqrt[3] {2 \sqrt[3]{3 \cdots}}} = 1^{1/3} \; 2^{1/9} \; 3^{1/27} \cdots1.15636268433226971685
  • Functional iteration.
No dataNo data
Cahens constantShow sourceξ2\xi _{2}Show source∑k=1∞(−1)ksk−1=11−12+16−142+11806 ±⋯\sum_{k=1}^{\infty} \frac{(-1)^{k}}{s_k-1} = \frac{1}{1} - \frac{1}{2} + \frac{1}{6} - \frac{1}{42} + \frac{1}{1806} {\,\pm \cdots}0.64341054628833802618
  • Functional iteration.
1891No data
Lemniscate constantShow sourceϖ{\varpi}Show sourceπ G=42π Γ(54)2=142π Γ(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
Somos' quadratic recurrence constantShow sourceσ{\sigma}Show source∏n=1∞n1/2n=123⋯=11/2  21/4  31/8⋯\prod_{n=1}^\infty n^{{1/2}^n} = \sqrt {1 \sqrt {2 \sqrt{3 \cdots}}} = 1^{1/2} \; 2^{1/4} \; 3^{1/8} \cdots1.66168794963359412129
  • Functional iteration.
No dataNo data
Euler-Gompertz constantShow sourceG{G}Show source ⁣∫0∞ ⁣ ⁣e−n1+n dn= ⁣ ⁣∫01 ⁣ ⁣11−ln⁡n dn=11+11+11+21+21+31+3/⋯\! \int \limits_0^\infty \!\! \frac{e^{-n}}{1{+}n} \, dn = \!\! \int \limits_0^1 \!\! \frac{1}{1{-}\ln n} \, dn = \textstyle {\tfrac 1 {1+\tfrac 1{1+\tfrac 1{1+\tfrac 2{1+\tfrac 2{1+\tfrac 3{1+3{/\cdots}} }}}}}}0.59634736232319407434
  • Functional iteration,
  • continued fractions.
No dataNo data

Analytic inequalities#

Typical namesCommon symbolPossible definition or way of calculationApproximated valueExample usage or connotationsKnown at least sinceNumber of known digits after the point
(state on 2019)
2nd du Bois-Reymond constantShow sourceC2{C_2}Show sourcee2−72=∫0∞∣ddt(sin⁡tt)n∣ dt−1\frac{e^2-7}{2} = \int_0^\infty \left|{\frac{d}{dt}\left(\frac{\sin t}{t}\right)^n}\right|\,dt-10.19452804946532511361
  • Analytic inequalities.
No dataNo data
Grothendieck constantShow sourceKR{K_{R}}Show sourceπ2log⁡(1+2)\frac {\pi}{2 \log(1+\sqrt{2})}1.78221397819136911177
  • Analytic inequalities.
No dataNo data
Carlson-Levin constantShow sourceΓ(12){\Gamma}(\tfrac12)Show sourceπ=(−12)!=∫−∞∞1ex2 dx=∫011−ln⁡x dx\sqrt{\pi} = \left(-\frac{1}{2}\right)! = \int_{-\infty }^{\infty } \frac {1}{e^{x^2}} \, dx = \int_{0 }^{1} \frac {1}{\sqrt{-\ln x}} \, dx1.77245385090551602729
  • Analytic inequalities.
No dataNo data

Continued fractions#

Typical namesCommon symbolPossible definition or way of calculationApproximated valueExample usage or connotationsKnown at least sinceNumber of known digits after the point
(state on 2019)
Lüroth constantShow sourceCLC_LShow source∑n=2∞ln⁡(nn−1)n\sum_{n = 2}^\infty \frac{\ln\left(\frac{n}{n-1}\right)}{n}0.78853056591150896106
  • Continued fractions.
No dataNo data
Trott constantShow sourceT1\mathrm{T}_1Show source11+10+18+14+11+10+1/⋯\tfrac 1{1+\tfrac 1{0+\tfrac 1{8+\tfrac 1{4+\tfrac 1{1+\tfrac 1{0+1{/\cdots}}}}}}}0.10841015122311136151
  • Continued fractions.
No dataNo data
Continued fraction constantShow sourceCCF{C}_{CF}Show sourceI1(2)I0(2)=∑n=0∞nn!n!∑n=0∞1n!n!=11+12+13+14+15+16+1/⋯\frac{I_1(2)}{I_0(2)} = \frac{ \sum \limits_{n = 0}^\infty \frac{n}{n!n!}} {{ \sum \limits_{n = 0}^{\infty} \frac{1}{n!n!}}} = \textstyle \tfrac 1{1+\tfrac 1{2+\tfrac 1{3+\tfrac 1{4+\tfrac 1{5+\tfrac 1{6+1{/\cdots}}}}}}}0.69777465796400798200
  • Continued fractions.
No dataNo data
Euler-Gompertz constantShow sourceG{G}Show source ⁣∫0∞ ⁣ ⁣e−n1+n dn= ⁣ ⁣∫01 ⁣ ⁣11−ln⁡n dn=11+11+11+21+21+31+3/⋯\! \int \limits_0^\infty \!\! \frac{e^{-n}}{1{+}n} \, dn = \!\! \int \limits_0^1 \!\! \frac{1}{1{-}\ln n} \, dn = \textstyle {\tfrac 1 {1+\tfrac 1{1+\tfrac 1{1+\tfrac 2{1+\tfrac 2{1+\tfrac 3{1+3{/\cdots}} }}}}}}0.59634736232319407434
  • Functional iteration,
  • continued fractions.
No dataNo data
Embree-Trefethen constantShow sourceB∗\Beta^*Show source−-0.70258
  • Continued fractions.
No dataNo data

Information 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)
Dottie numberShow sourceddShow sourcelim⁡x→∞cos⁡[x](c)=lim⁡x→∞cos⁡(cos⁡(cos⁡(⋯(cos⁡(c)))))⏟x\lim_{x\to \infty} \cos^{[x]}(c) = \lim_{x\to \infty} \underbrace{\cos(\cos(\cos(\cdots(\cos(c)))))}_x0.73908513321516064165No dataNo data
Chaitin's constantShow sourceΩ\OmegaShow source∑p∈P2−∣p∣\sum_{p \in P} 2^{-|p|}0.0078749969978123844
  • Information theory.
1975No data

Some facts#

  • It's rather easy to say what are physical constants such as the speed of light in a vacuum, but defining mathematical constants can pose a lot of difficulty.
  • In practice various fields of mathematics provide different definitions of what the constant is. So, the concept of a mathematical constant depends on the context.
  • Most often when we talk about mathematical constants, we mean specific numbers which for some reason are relevant in given context or simply useful. Therefore, this is not a strict definition, and the main criterion here is the usefulness of such and no other selection of constants. Examples of such constants are:
    • the number π\pi (read as: number pi) being the ratio of the circumference to the diameter:
      π=disk circumferencedisk diameter≈3.14159265358979324…\pi = \dfrac{\text{disk circumference}}{\text{disk diameter}} \approx 3.14159265358979324\ldots
    • and the number T=2π\Tau = 2 \pi (read as: number tau) which is twice the number π\pi:
      T=2π=2⋅disk circumferencedisk diameter≈6.28318530717958648…\Tau = 2 \pi = \dfrac{2 \cdot \text{disk circumference}}{\text{disk diameter}} \approx 6.28318530717958648\ldots
    Although second number is result of simple conversion from the first one, sometimes mathematicians use constant T\Tau to keep expressions simply. The introduction of the new constant has in this case the strictly practical aspect.
  • Some fields of mathematics try to define a constant in a more formal way. For example, an abstract algebra defines a constant as a zero-argument function. So, the mathemical constant is a function that does not accept any arguments and always returns the same number.
  • Often a given constant can be calculated in different ways e.g. there are many methods for calculating the e number:
    • as the limit of the sequence:
      e=lim⁡n→∞(1+1n)n≈2.71828182845904524…e = \lim_{n\to\infty}\left(1+\tfrac{1}{n}\right)^n \approx 2.71828182845904524\ldots
    • but also as an infinite sum of a serie:
      e=∑n=0∞1n!=10!+11!+12!+13!+14!+⋯≈2.71828182845904524…e = \sum_{n=0}^\infty \dfrac{1}{n!} = \dfrac{1}{0!} + \dfrac{1}{1!} + \dfrac{1}{2!} + \dfrac{1}{3!} + \dfrac{1}{4!} + \dots \approx 2.71828182845904524\ldots
    • etc.

Tags and links to this website#

Tags:
Tags to Polish version:

What tags this calculator has#

Permalink#

This is permalink. Permalink is the link containing your input data. Just copy it and share your work with friends:

Links to external sites (leaving Calculla?)#

Calculla 2.0 by Kemu StudioAll rights reserved
JavaScript failed !
So this is static version of this website.
This website works a lot better in JavaScript enabled browser.
Please enable JavaScript.