Mathematical constant

A mathematical constant is a number, which has a special meaning for calculations. For example, the constant π (pronounced "pie") means the ratio of a circle's circumference to its diameter. This value is always the same for any circle. A mathematical constant is often a real, non-integral number of interest.

In contrast to physical constants, mathematical constants do not come from physical measurements.

 Key mathematical constants 
The following table contains some important mathematical constants:
 Name Symbol Value MeaningPi, Archimedes' constant or Ludoph's number  π  ≈3.141592653589793 A transcendental number that is the ratio of the length of a circle's circumference to its diameter.  It is also the area of the unit circle.e, Napier's constant or Euler's (pronounced "oilers") number  e  ≈2.718281828459045  A transcendental number that is the base of natural logarithms.Phi, golden ratio φ  ≈1.618033988749894  It is the value of a larger value divided by a smaller value if this is equal to the value of the sum of the values divided by the larger value.Square root of 2, Pythagoras' constant  √2  ≈1.414213562373095  An irrational number that is the length of the diagonal of a square with sides of length 1. This number can not be written as a fraction.

 Constants and series 
The following table contains a list of constants and series in mathematics, with the following columns:
Value: Numerical value of the constant.
LaTeX: Formula or series in TeX format.
Formula: For use in programs such as Mathematica or Wolfram Alpha.
OEIS: Link to On-Line Encyclopedia of Integer Sequences (OEIS), where the constants are available with more details.
Continued fraction: In the simple form [to integer; frac1, frac2, frac3, ...] (in brackets if periodic)
Type:
 R - Rational number
 I - Irrational number
 T - Transcendental number
 C - Complex number
Note that the list can be ordered correspondingly by clicking on the header title at the top of the table.
 Value  Name  Symbol  LaTeX  Formula  Type  OEIS  Continued fraction 3.24697960371746706105000976800847962  Silver, Tutte–Beraha constant2+2 cos(2Pi/7)IA116425[3;4,20,2,3,1,6,10,5,2,2,1,2,2,1,18,1,1,3,2,...] 1.09864196439415648573466891734359621  Paris constantIA105415[1;10,7,3,1,3,1,5,1,4,2,7,1,2,3,22,1,2,5,2,1,...] 2.74723827493230433305746518613420282  Ramanujan nested radical R5 (2+sqrt(5)+sqrt(15-6 sqrt(5)))/2I[2;1,2,1,21,1,7,2,1,1,2,1,2,1,17,4,4,1,1,4,2,...] 2.23606797749978969640917366873127624  Square root of 5, Gauss sumSum[k=0 to 4]{e^(2k^2 pi i/5)}IA002163[2;4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,...]  = [2;(4),...] 3.62560990822190831193068515586767200  Gamma(1/4)4(1/4)!TA068466[3;1,1,1,2,25,4,9,1,1,8,4,1,6,1,1,19,1,1,4,1,...] 0.18785964246206712024851793405427323   MRB constant, Marvin Ray BurnsSum[n=1 to ∞]{(-1)^n (n^(1/n)-1)} TA037077[0;5,3,10,1,1,4,1,1,1,1,9,1,1,12,2,17,2,2,1,...] 0.11494204485329620070104015746959874  Kepler–Bouwkamp constantprod[n=3 to ∞]{cos(pi/n)}TA085365[0;8,1,2,2,1,272,2,1,41,6,1,3,1,1,26,4,1,1,...] 1.78107241799019798523650410310717954  Exp(gamma)  G-Barnes functionProd[n=1 to ∞]{e^(1/n)}/{1 + 1/n}TA073004[1;1,3,1,1,3,5,4,1,1,2,2,1,7,9,1,16,1,1,1,2,...] 1.28242712910062263687534256886979172  Glaisher–Kinkelin constante^(1/2-zeta´{-1}) TA074962[1;3,1,1,5,1,1,1,3,12,4,1,271,1,1,2,7,1,35,...] 7.38905609893065022723042746057500781  Schwarzschild conic constant  Sum[n=0 to ∞]{2^n/n!}TA072334[7;2,1,1,3,18,5,1,1,6,30,8,1,1,9,42,11,1,...]= [7,2,(1,1,n,4*n+6,n+2)], n = 3, 6, 9, etc. 1.01494160640965362502120255427452028  Gieseking constant  
.TA143298[1;66,1,12,1,2,1,4,2,1,3,3,1,4,1,56,2,2,11,...] 2.62205755429211981046483958989111941  Lemniscata constant 4 sqrt(2/pi) (1/4!)^2TA062539[2;1,1,1,1,1,4,1,2,5,1,1,1,14,9,2,6,2,9,4,1,...] 0.83462684167407318628142973279904680  G, Gauss constant (4 sqrt(2)(1/4!)^2)/pi^(3/2)TA014549[0;1,5,21,3,4,14,1,1,1,1,1,3,1,15,1,3,7,1,...] 1.01734306198444913971451792979092052   Zeta(6)    Prod[n=1 to ∞] {1/(1-ithprime(n)^-6)} TA013664[1;57,1,1,1,15,1,6,3,61,1,5,3,1,6,1,3,3,6,1,...] 0,60792710185402662866327677925836583   
   Constante de Hafner-Sarnak-McCurleyProd{n=1 to ∞} (1-1/ithprime(n)^2) TA059956[0;1,1,1,1,4,2,4,7,1,4,2,3,4,10,1,2,1,1,1,...]  1.11072073453959156175397024751517342  The ratio of a square and circumscribed or inscribed circlessum[n=1 to ∞]{(-1)^(floor((n-1)/2))/(2n-1)}TA093954[1;9,31,1,1,17,2,3,3,2,3,1,1,2,2,1,4,9,1,3,...] 2.80777024202851936522150118655777293  Fransén–Robinson constantN[int[0 to ∞] {1/Gamma(x)}]TA058655[2;1,4,4,1,18,5,1,3,4,1,5,3,6,1,1,1,5,1,1,1...] 1.64872127070012814684865078781416357   Square root of esum[n=0 to ∞]{1/(2^n n!)}TA019774[1;1,1,1,5,1,1,9,1,1,13,1,1,17,1,1,21,1,1,...]  = [1;1,(1,1,4p+1)],  p∈ℕ i  
   i, imaginary unitsqrt(-1)C 262537412640768743.999999999999250073  Hermite-Ramanujan constant e^(π sqrt(163))TA060295 [262537412640768743;1,1333462407511,1,8,1,1,5,...] 4.81047738096535165547303566670383313  John constante^(π/2)TA042972 [4;1,4,3,1,1,1,1,1,1,1,1,7,1,20,1,3,6,10,3,...] 4.53236014182719380962768294571666681  Constante de Van der Pauw π/ln(2)TA163973[4;1,1,7,4,2,3,3,1,4,1,1,4,7,2,3,3,12,2,1,...] 0.76159415595576488811945828260479359  Hyperbolic tangent (1)(e-1/e)/(e+1/e)TA073744[0;1,3,5,7,9,11,13,15,17,19,21,23,25,27,...]  = [0;(2p+1)],  p∈ℕ 0.69777465796400798200679059255175260  Continued Fraction constant (sum {n=0 to inf} n/(n!n!)) /(sum {n=0 to inf} 1/(n!n!))A052119[0;1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,...]  = [0;(p+1)],  p∈ℕ 0.36787944117144232159552377016146086  Inverse Napier constantsum[n=2 to ∞]{(-1)^n/n!}TA068985[0;2,1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,...]  = [0;2,1,(1,2p,1)],  p∈ℕ 2.71828182845904523536028747135266250  Napier constantSum[n=0 to ∞]{1/n!}TA001113[2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,...]  = [2;(1,2p,1)],  p∈ℕ 0.49801566811835604271369111746219809     - 0.15494982830181068512495513048388 i Factorial of iGamma(1+i)CA212877  A212878[0;6,2,4,1,8,1,46,2,2,3,5,1,10,7,5,1,7,2,...]  - [0;2,125,2,18,1,2,1,1,19,1,1,1,2,3,34,...] i 0.43828293672703211162697516355126482   + 0.36059247187138548595294052690600 i   Infinite   Tetration of ii^i^i^...CA077589  A077590[0;2,3,1,1,4,2,2,1,10,2,1,3,1,8,2,1,2,1, ...]  + [0;2,1,3,2,2,3,1,5,5,1,2,1,10,10,6,1,1...] i 0.56755516330695782538461314419245334   Module of  Infinite  Tetration of  i Mod(i^i^i^...)A212479[0;1,1,3,4,1,58,12,1,51,1,4,12,1,1,2,2,3,...] 0.26149721284764278375542683860869585  Meissel-Mertens constant ..... p: primesA077761[0;3,1,4,1,2,5,2,1,1,1,1,13,4,2,4,2,1,33,...] 1.9287800...  Wright constant = primos:   =3,   =13,  =16381,  A086238[1; 1, 13, 24, 2, 1, 1, 3, 1, 1, 3] 0.37395581361920228805472805434641641   Artin constant ...... pn: primoTA005596[0;2,1,2,14,1,1,2,3,5,1,3,1,5,1,1,2,3,5,46,...] 4.66920160910299067185320382046620161  Feigenbaum constant δTA006890[4;1,2,43,2,163,2,3,1,1,2,5,1,2,3,80,2,5,...] 2.50290787509589282228390287321821578  Feigenbaum constant α TA006891[2;1,1,85,2,8,1,10,16,3,8,9,2,1,40,1,2,3,...] 5.97798681217834912266905331933922774  Hexagonal Madelung Constant 2 Pi Log[3]Sqrt[3] TA086055[5;1,44,2,2,1,15,1,1,12,1,65,11,1,3,1,1,...] 0.96894614625936938048363484584691860  Beta(3) Sum[n=1 to ∞]{(-1)^(n+1)/(-1+2n)^3}TA153071[0;1,31,4,1,18,21,1,1,2,1,2,1,3,6,3,28,1,...] 1.902160583104 Brun constant  2 = Σ inverse twin primesA065421[1; 1, 9, 4, 1, 1, 8, 3, 4, 4, 2, 2] 0.870588379975 Brun constant  4 = Σ inverse of twin primeA213007[0; 1, 6, 1, 2, 1, 2, 956, 3, 1, 1] 22.4591577183610454734271522045437350  pi^epi^e A059850[22;2,5,1,1,1,1,1,3,2,1,1,3,9,15,25,1,1,5,...] 3.14159265358979323846264338327950288   Pi, Archimedes constantSum[n=0 to ∞]{(-1)^n 4/(2n+1)}  TA000796[3;7,15,1,292,1,1,1,2,1,3,1,14,...] 0.06598803584531253707679018759684642    ... Lower limit of TetrationTA073230 [0;15,6,2,13,1,3,6,2,1,1,5,1,1,1,9,4,1,1,1,...] 0.20787957635076190854695561983497877  i^i  e^(-pi/2)TA049006 [0;4,1,4,3,1,1,1,1,1,1,1,1,7,1,20,1,3,6,10,...] 0.28016949902386913303643649123067200  Bernstein constantTA073001[0;3,1,1,3,9,6,3,1,3,13,1,16,3,3,4,…] 0.28878809508660242127889972192923078  Flajolet and Richmondprod[n=1 to ∞]{1-1/2^n}A048651 0.31830988618379067153776752674502872  Inverse of Pi, RamanujanTA049541 [0;3,7,15,292,1,1,1,2,1,3,1,14,2,1,1,...] 0.47494937998792065033250463632798297  Weierstraß constant(E^(Pi/8) Sqrt[Pi])/(4 2^(3/4) (1/4)!^2)TA094692[0;2,9,2,11,1,6,1,4,6,3,19,9,217,1,2,...] 0.56714329040978387299996866221035555  Omega constantsum[n=1 to ∞]{(-n)^(n-1)/n!}TA030178[0;1,1,3,4,2,10,4,1,1,1,1,2,7,306,1,5,1,...] 0.57721566490153286060651209008240243  Euler's numbersum[n=1 to ∞]|sum[k=0 to ∞]{((-1)^k)/(2^n+k)}?A001620[0;1,1,2,1,2,1,4,3,13,5,1,1,8,1,2,...]   0.60459978807807261686469275254738524  Dirichlet serieSum[1/(n Binomial[2 n, n]), {n, 1, ∞}] TA073010 [0;1,1,1,1,8,10,2,2,3,3,1,9,2,5,4,1,27,27,...] 0.63661977236758134307553505349005745  2/Pi, François VièteTA060294[0;1,1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1,4,...] 0.66016181584686957392781211001455577  Twin prime constantprod[p=3 to ∞]{p(p-2)/(p-1)^2A005597[0;1,1,1,16,2,2,2,2,1,18,2,2,11,1,1,2,4,1,...] 0.66274341934918158097474209710925290  Laplace Limit constantA033259[0;1,1,1,27,1,1,1,8,2,154,2,4,1,5,...] 0.69314718055994530941723212145817657  Logarithm de 2 Sum[n=1 to ∞]{(-1)^(n+1)/n}TA002162  [0;1,2,3,1,6,3,1,1,2,1,1,1,1,3,10,...]   0.78343051071213440705926438652697546   Sophomore's Dream 1 J.BernoulliSum[ -(-1)^n /n^n]TA083648[0;1,3,1,1,1,1,1,1,2,4,7,2,1,2,1,1,1,...] 0.78539816339744830961566084581987572   Dirichlet beta(1)Sum[n=0 to ∞]{(-1)^n/(2n+1)}TA003881[0; 1,3,1,1,1,15,2,72,1,9,1,17,1,2,1,5,...] 0.82246703342411321823620758332301259   Traveling Salesman Nielsen-Ramanujan  Sum[n=1 to ∞]{((-1)^(k+1))/n^2}TA072691[0;1,4,1,1,1,2,1,1,1,1,3,2,2,4,1,1,1,...] 0.91596559417721901505460351493238411  Catalan constantSum[n=0 to ∞]{(-1)^n/(2n+1)^2} IA006752 [0;1,10,1,8,1,88,4,1,1,7,22,1,2,...] 1.05946309435929526456182529494634170  Ratio of the distance between semi-tones  2^(1/12)IA010774[1;16,1,4,2,7,1,1,2,2,7,4,1,2,1,60,1,3,1,2,...] 1,.08232323371113819151600369654116790   Zeta(04) Sum[n=1 to ∞]{1/n^4}TA013662[1;12,6,1,3,1,4,183,1,1,2,1,3,1,1,5,4,2,7,...]1.1319882487943 ...  Viswanaths  constant A078416 [1;7,1,1,2,1,3,2,1,2,1,8,1,5,1,1,1,9,1,...] 1.20205690315959428539973816151144999  Apéry constant Sum[n=1 to ∞]{1/n^3}IA010774[1;4,1,18,1,1,1,4,1,9,9,2,1,1,1,2,...] 1.22541670246517764512909830336289053 Gamma(3/4) (-1+3/4)!TA068465 [1;4,2,3,2,2,1,1,1,2,1,4,7,1,171,3,2,3,1,1,...] 1.23370055013616982735431137498451889  Favard constantsum[n=1 to ∞]{1/((2n-1)^2)}TA111003 [1;4,3,1,1,2,2,5,1,1,1,1,2,1,2,1,10,4,3,1,1,...] 1.25992104989487316476721060727822835  Cube root of 2, constante Delian2^(1/3)IA002580[1;3,1,5,1,1,4,1,1,8,1,14,1,10,...] 1.29128599706266354040728259059560054  Sophomore's Dream 2 J.BernoulliSum[1/(n^n]), {n, 1, ∞}] A073009 [1;3,2,3,4,3,1,2,1,1,6,7,2,5,3,1,2,1,8,1,...] 1.32471795724474602596090885447809734  Plastic numberIA060006[1;3,12,1,1,3,2,3,2,4,2,141,80,2,5,1,2,8,...] 1.41421356237309504880168872420969808  Square root of 2, Pythagoras constantprod[n=1 to ∞]{1+(-1)^(n+1)/(2n-1)}IA002193[1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...] = [1;(2),...] 1.44466786100976613365833910859643022  Steiner number  ... Upper Limit of TetrationA073229 [1;2,4,55,27,1,1,16,9,3,2,8,3,2,1,1,4,1,9,...] 1.53960071783900203869106341467188655  Lieb's Square Ice constant  (4/3)^(3/2)IA118273 [1;1,1,5,1,4,2,1,6,1,6,1,2,4,1,5,1,1,2,...] 1.57079632679489661923132169163975144  Wallis productTA019669 [1;1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1...] 1.60669515241529176378330152319092458  Erdős–Borwein constantsum[n=1 to ∞]{1/(2^n-1)}IA065442[1;1,1,1,1,5,2,1,2,29,4,1,2,2,2,2,6,1,7,1,...] 1.61803398874989484820458633436563812  Phi, Golden ratio(1+5^(1/2))/2IA001622[0;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...] = [0;(1),...] 1.64493406684822643647241516664602519  Zeta(2) Sum[n=1 to ∞]{1/n^2}TA013661[1;1,1,1,4,2,4,7,1,4,2,3,4,10 1,2,1,1,1,15,...] 1.66168794963359412129581892274995074  Somos' quadratic recurrence constantTA065481 [1;1,1,1,21,1,1,1,6,4,2,1,1,2,1,3,1,13,13,...] 1.73205080756887729352744634150587237  Theodorus constant3^(1/2)IA002194  [1;1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,...]  = [1;(1,2),...] 1.75793275661800453270881963821813852  Kasner numberA072449[1;1,3,7,1,1,1,2,3,1,4,1,1,2,1,2,20,1,2,2,...] 1.77245385090551602729816748334114518  Carlson-Levin constant sqrt (pi)TA002161 [1;1,3,2,1,1,6,1,28,13,1,1,2,18,1,1,1,83,1,...] 2.29558714939263807403429804918949038  P, Universal parabolic constantln(1+sqrt 2)+sqrt 2TA103710 [2;3,2,1,1,1,1,3,3,1,1,4,2,3,2,7,1,6,1,8,...] 2.30277563773199464655961063373524797  Bronze Number(3+sqrt 13)/2IA098316 [3;3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,...]  = [3;(3),...] 2.37313822083125090564344595189447424  Lévy constant2  Pi^(2)/(6*ln(2))TA174606 [2;2,1,2,8,57,9,32,1,1,2,1,2,1,2,1,2,1,3,2,...] 2.50662827463100050241576528481104525  square root of 2 pi sqrt (2*pi)TA019727 [2;1,1,37,4,1,1,1,1,9,1,1,2,8,6,1,2,2,1,3,...] 2.66514414269022518865029724987313985  Gelfond-Schneider constant 2^sqrt{2}TA007507 [2;1,1,1,72,3,4,1,3,2,1,1,1,14,1,2,1,1,3,1,...] 2.68545200106530644530971483548179569  Khintchin constantprod[n=1 to ∞]{(1+1/(n(n+2)))^((ln(n)/ln(2))}?A002210[2;1,2,5,1,1,2,1,1,3,10,2,1,3,2,24,1,3,2,...] 3.27582291872181115978768188245384386  Khinchin-Lévy constante^(\pi^2/(12 ln(2))A086702[3;3,1,1,1,2,29,1,130,1,12,3,8,2,4,1,3,55,...] 3.35988566624317755317201130291892717  Reciprocal Fibonacci constantA079586[3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,...] 4.13273135412249293846939188429985264  Root of 2 e pisqrt(2e pi)    TA019633 [4;7,1,1,6,1,5,1,1,1,8,3,1,2,2,15,2,1,1,2,4,...] 6.58088599101792097085154240388648649  Froda constant2^e[6;1,1,2,1,1,2,3,1,14,11,4,3,1,1,7,5,5,2,7,...] 9.86960440108935861883449099987615114   Pi Squared6 Sum[n=1 to ∞]{1/n^2}TA002388[9;1,6,1,2,47,1,8,1,1,2,2,1,1,8,3,1,10,5,...] 23.1406926327792690057290863679485474  Gelfond constantSum[n=0 to ∞]{(pi^n)/n!}TA039661[23;7,9,3,1,1,591,2,9,1,2,34,1,16,1,30,1,...]

 Related pages 
 Speed of light
 Differentiation
 Constant function
 Quadratic function
 Differential equation
 Multivariable calculus
 List of mathematical symbols
 Eigenvalues and eigenvectors

 References 

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 Online bibliography 
 On-Line Encyclopedia of Integer Sequences (OEIS)
 Simon Plouffe, Tables of Constants 
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