List of mathematical constants

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A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a symbol or by mathematicians' names to facilitate using it across multiple mathematical problems.[1] For example, the constant π may be defined as the ratio of the length of a circle's circumference to its diameter. The following list includes the decimal expansion and set containing each number, ordered by year of discovery.

Antiquity[edit]

Name Symbol Decimal Expansion Formula Year Set
One 1 1 None[nb 1] Prehistory N
Two 2 2 Prehistory N
One half 1/2 0.5 Prehistory Q
,

Archimedes

constant [2]

3.14159 26535 89793 23846 [Mw 1][OEIS 1] Ratio of a circle's circumference to its diameter. 1900 to 1600 BCE [3]
Square root of 2,

Pythagoras constant.[4]

1.41421 35623 73095 04880 [Mw 2][OEIS 2] Positive root of 1800 to 1600 BCE[5]
Square root of 3,

Theodorus constant[6]

1.73205 08075 68877 29352 [Mw 3][OEIS 3] Positive root of 465 to 398 BCE
Phi, Golden ratio [7] 1.61803 39887 49894 84820 [Mw 4][OEIS 4] Positive root of ~300 BCE
Zero 0 0 The additive identity: 300-100 century BCE[8] Z
Negative one -1 -1 300-200 BCE Z
Square root of 5[9] 2.23606 79774 99789 69640[OEIS 5] Positive root of
Cube root of 2
Delian Constant
1.25992 10498 94873 16476 [Mw 5][OEIS 6] Real root of 46 -120 CE

[10]

Gallery[edit]

Pi is half the wavelength of sine
The square root of 2 is the length of the hypotenuse of a right triangle with other side-lengths 1
The square root of 3 is the length of the distance between diametrically opposite corners of a 1 by 1 by 1 cube
Phi is the ratio between two quantities (orange and blue segments) that is the same as the ratio of their sum to the larger of the two quantities
The square root of 5 is the length of the hypotenuse of a right triangle with other side-lengths 1 and 2

Medieval and Early Modern[edit]

Name Symbol Decimal Expansion Formula Year Set
Imaginary number [11]
i [Mw 6]
One root of [nb 2] 1501
to
1576
Buffon constant[12]

0.63661 97723 67581 34307[Mw 7][Ow 1][OEIS 7]

Viète product 1540
to
1603
Hermite constant Sphere packing 3D Kepler conjecture [13] 0.74048 04896 93061 04116 [Mw 8][OEIS 8] [14] 1611
Wallis Constant 2.09455 14815 42326 59148 [Mw 9][OEIS 9] 1616
to
1703
e, Euler's number [15] 2.71828 18284 59045 23536 [Mw 10][OEIS 10] [nb 3] 1618[16]
Natural logarithm of 2 [17] 0.69314 71805 59945 30941 [Mw 11][OEIS 11] 1619,[18]1668[19]
Favard constant[citation needed] K1[citation needed]
Wallis product [20]
1.57079 63267 94896 61923 [Mw 12][OEIS 12] 1656[Mw 12]
Square root of 2 pi 2.50662 82746 31000 50241 [OEIS 13] [nb 4] 1667 to 1754[21][22][23]
Sophomore's dream1
J.Bernoulli [24]
0.78343 05107 12134 40705 [OEIS 14] 1697
Sophomore's dream2
J.Bernoulli [25]
1.29128 59970 62663 54040 [Mw 13] [OEIS 15] 1697
Lemniscate constant[26] 2.62205 75542 92119 81046 [Mw 14] [OEIS 16] 1718 to 1798
Euler–Mascheroni constant 0.57721 56649 01532 86060 [Mw 15] [OEIS 17]

1735 ?
[27] Infinitely many values, with principal value

0.20787 95763 50761 90854 [Mw 6] [OEIS 18]

1746[27]
Erdős–Borwein constant[28] 1.60669 51524 15291 76378 [Mw 16] [OEIS 19] 1749[29]
Euler Totient
constant
[30][31]
1.94359 64368 20759 20505 [Mw 17] [OEIS 20] 1750
Laplace limit [32] 0.66274 34193 49181 58097 [Mw 18] [OEIS 21] ~1782 ?
Gauss's constant [33] 0.83462 68416 74073 18628 [Mw 19][OEIS 22] AGM = Arithmetic–geometric mean 1799[34] ?
Pi Squared 9.86960 44010 89358 61883 [OEIS 23]
Fourth root of five [35] 1.49534 87812 21220 54191 [OEIS 24] Positive root of
John constant [36] ,

4.81047 73809 65351 65547 [OEIS 25]

Gallery[edit]

19th Century[edit]

Name Symbol Decimal Expansion Formula Year Set
The Triangular root of 2

Triangular root of 2.[37]

1.56155 28128 08830 27491[OEIS 26] 1810[37]
Ramanujan–Soldner constant[38][39] 1.45136 92348 83381 05028 [Mw 20] [OEIS 27] ; root of the logarithmic integral function. 1812[Mw 21]
Hermite constant [40] 1.15470 05383 79251 52901 [Mw 22] 1822 to 1901
Riemann Function Zeta(2) 1.64493 40668 48226 43647 [Mw 23][OEIS 28] 1826
to
1866
Liouville number [41] 0.11000 10000 00000 00000 0001 [Mw 24][OEIS 29] Before 1844
Cube root of 1 [42] -0.5
± 0.86602 54037 84438 64676 i [OEIS 30]
Before 1850[43]

Hermite–Ramanujan constant[44] 262 53741 26407 68743
.99999 99999 99250 073 [Mw 25][OEIS 31]
1859
Catalan's constant[45][46][47] 0.91596 55941 77219 01505 [Mw 26][OEIS 32] 1864 ?
Dottie number [48] 0.73908 51332 15160 64165 [Mw 27][OEIS 33] 1865[Mw 27]
Meissel–Mertens constant [49] 0.26149 72128 47642 78375 [Mw 28][OEIS 34] 1866
&
1873
?
Weierstrass constant [50] 0.47494 93799 87920 65033 [Mw 29][OEIS 35] 1872 ?
Kasner number 1.75793 27566 18004 53270 [Mw 30][OEIS 36] 1878
to
1955
Hafner–Sarnak–McCurley constant (2) [51] 0.60792 71018 54026 62866 [Mw 31][OEIS 37] 1883[Mw 31]
Cahen's constant [52] 0.64341 05462 88338 02618 [Mw 32][OEIS 38]

Where sk is the kth term of Sylvester's sequence 2, 3, 7, 43, 1807, ...
Defined as:

1891
Universal parabolic constant [53] 2.29558 71493 92638 07403 [Mw 33][OEIS 39] Before 1891[54]
Apéry's constant [55] 1.20205 69031 59594 28539 [Mw 34][OEIS 40]

1895[56]

?

Gelfond's constant [57] 23.14069 26327 79269 0057 [Mw 35][OEIS 41] 1900[58]

Gallery[edit]

Números triangulares.png

The Triangular root of 2

Triangular root of 2.[37]

1900-1949[edit]

Name Symbol Decimal Expansion Formula Year Set
Lebesgue constant (interpolation) [59][60] 1.43599 11241 76917 43235 [Mw 36][OEIS 42] ~1902
Favard constant [61] 1.23370 05501 36169 82735 [Mw 37][OEIS 43] 1902
to
1965
Golden angle [62] 2.39996 32297 28653 32223 [Mw 38] [OEIS 44] = 137.5077640500378546 ...° 1907
Sierpiński's constant [63] 2.58498 17595 79253 21706 [Mw 39][OEIS 45]

1907
NielsenRamanujan constant [64] 0.82246 70334 24113 21823 [Mw 40][OEIS 46] 1909
Lebesgue constant L2 [65] 1.64218 84352 22121 13687 [Mw 36][OEIS 47] 1910
Area of the Mandelbrot fractal [66] 1.5065918849 ± 0.0000000028 [Mw 41][OEIS 48] 1912
Gieseking constant [67] 1.01494 16064 09653 62502 [Mw 42][OEIS 49]

.

1912
Bernstein's constant [68] 0.28016 94990 23869 13303 [Mw 43][OEIS 50] 1913
Area bounded by the
eccentric rotation of
Reuleaux triangle [69]
0.98770 03907 36053 46013 [Mw 44][OEIS 51]    where a= side length of the square 1914
Twin Primes Constant [70] 0.66016 18158 46869 57392 [Mw 45][OEIS 52] 1922
Spiral of Theodorus[71] or constant of Theodorus (not to be confused with the square root of 3) 1.86002 50792 21190 30718[OEIS 53] 1923 to 2018. [nb 5][72]
Plastic number [73] 1.32471 79572 44746 02596 [Mw 46][OEIS 54] 1929
Bloch–Landau constant [74] 0.54325 89653 42976 70695 [Mw 47][OEIS 55] 1929
Golomb–Dickman constant [75] 0.62432 99885 43550 87099 [Mw 48][OEIS 56] 1930
&
1964
Feller–Tornier constant [76] 0.66131 70494 69622 33528 [Mw 49][OEIS 57] 1932 ?
Base 10 Champernowne constant [77] 0.12345 67891 01112 13141 [Mw 50][OEIS 58] 1933
Gelfond–Schneider constant [78] 2.66514 41426 90225 18865 [Mw 51][OEIS 59] 1934
Khinchin's constant [79] 2.68545 20010 65306 44530 [Mw 52][OEIS 60] 1934 ?
Khinchin–Lévy constant[80] 1.18656 91104 15625 45282 [Mw 53][OEIS 61] 1935
Khinchin-Lévy constant [81] 3.27582 29187 21811 15978 [Mw 54][OEIS 62] 1936
Lévy 2 constant[82] 2.37313 82208 31250 90564[OEIS 63] 1936
Disk Covering [83] 0.82699 33431 32688 07426 [Mw 55][OEIS 64] 1939
1949
Calabi triangle constant [84] 1.55138 75245 48320 39226 [Mw 56][OEIS 65] 1946 ~
Mills' constant [85] 1.30637 78838 63080 69046 [Mw 57][OEIS 66] is prime 1947
Euler–Gompertz constant [86] 0.59634 73623 23194 07434 [Mw 58][OEIS 67] Before 1948[OEIS 67]

Gallery[edit]

1950-1999[edit]

Name Symbol Decimal Expansion Formula Year Set
Van der Pauw constant 4.53236 01418 27193 80962[OEIS 68] Before 1958[OEIS 69]
Beta, Kneser-Mahler polynomial constant[87] 1.38135 64445 18497 79337[OEIS 70] 1963
Lochs constant [88] 0.97027 01143 92033 92574 [Mw 59][OEIS 71] 1964
Hyperbolic tangent of 1 [89] 0.76159 41559 55764 88811 [Mw 60][OEIS 72] Before 1965[90]
Lieb's square ice constant [91] 1.53960 07178 39002 03869 [Mw 61][OEIS 73] 1967
πe [92] 22.45915 77183 61045 47342[OEIS 74] Before 1968[93] ?[94]
Niven's constant [95] 1.70521 11401 05367 76428 [Mw 62][OEIS 75] 1969
Baker constant [96] 0.83564 88482 64721 05333[OEIS 76] Before 1969[96]
Baxter's
Four-coloring
constant [97]
1.46099 84862 06318 35815 [Mw 63][OEIS 77]
Γ() = Gamma function
1970
Gauss–Kuzmin–Wirsing constant [98] 0.30366 30028 98732 65859 [Mw 64][OEIS 78]

where is an analytic function with .

1973
Porter's constant[99] 1.46707 80794 33975 47289 [Mw 65][OEIS 79]

1974
Feigenbaum constant δ [100] 4.66920 16091 02990 67185 [Mw 66][OEIS 80]

1975
Chaitin's constants [101] In general they are uncomputable numbers but one such number is 0.00787 49969 97812 3844 [Mw 67][OEIS 81]
  • p: Halted program
  • |p|: Size in bits of program p
  • P: Domain of all programs that stop.
1975
Infinite product constant, with Alladi-Grinstead [102] 1.75874 36279 51184 82469[OEIS 82] 1977
Alladi–Grinstead constant [103] 0.80939 40205 40639 13071 [Mw 68][OEIS 83] 1977
Fransén–Robinson constant [104] 2.80777 02420 28519 36522 [Mw 69][OEIS 84] 1978
Robbins constant [105] 0.66170 71822 67176 23515 [Mw 70][OEIS 85] 1978
Feigenbaum constant α[106] 2.50290 78750 95892 82228 [Mw 66][OEIS 86] 1979 ?
Fractal dimension of the Cantor set [107] 0.63092 97535 71457 43709 [Mw 71][OEIS 87] Before 1979[OEIS 87]
Connective constant [108][109] 1.84775 90650 22573 51225 [Mw 72][OEIS 88]

as a root of the polynomial

1982[110]
Salem number,[111]

Lehmer's conjecture

1.17628 08182 59917 50654 [Mw 73][OEIS 89] 1983?
Conway constant [112] 1.30357 72690 34296 39125 [Mw 74][OEIS 90] 1987
Prévost constant Reciprocal Fibonacci constant[113] 3.35988 56662 43177 55317 [Mw 75][OEIS 91]

Fn: Fibonacci series

Before 1988[OEIS 91]
Brun 2 constant = Σ inverse of Twin primes [114] 1.90216 05831 04 [Mw 76][OEIS 92] 1989[OEIS 92]
Vardi constant[115] 1.26408 47353 05301 11307 [Mw 77][OEIS 93] 1991
Flajolet and Richmond [116] 0.28878 80950 86602 42127 [Mw 78][OEIS 94] 1992
Hafner–Sarnak–McCurley constant (1) [117] 0.35323 63718 54995 98454 [Mw 79][OEIS 95] 1993
Fractal dimension of the Apollonian packing of circles
[118][119]

1.30568 6729 ≈ by Thomas & Dhar
1.30568 8 ≈ by McMullen [Mw 80][OEIS 96]
1994
1998
Backhouse's constant [120] 1.45607 49485 82689 67139 [Mw 81][OEIS 97]

1995
Grothendieck constant [121] 1.78221 39781 91369 11177 (postulated)[Mw 82][OEIS 98] Before 1995[121]
Murata Constant [122] 2.82641 99970 67591 57554 [Mw 83][OEIS 99] Before 1996[122]
Smarandache Constant[123] 1.09317 04591 95490 89396 [Mw 84][OEIS 100] where μ(n) is the Kempner function Before 1996[OEIS 100]
Viswanath constant[124] 1.13198 82487 943 [Mw 85][OEIS 101]      where an = Fibonacci sequence 1997 ?
Time constant [125] 0.63212 05588 28557 67840 [Mw 86][OEIS 102]

Before 1997[125]
Komornik–Loreti constant [126] 1.78723 16501 82965 93301 [Mw 87][OEIS 103]

tk = Thue–Morse sequence

1998
Khinchin harmonic mean [127] 1.74540 56624 07346 86349 [Mw 88][OEIS 104]

a1 ... an are elements of a continued fraction [a0; a1, a2, ..., an]

Before 1998[127]
Regular paperfolding sequence [128][129] 0.85073 61882 01867 26036 [Mw 89][OEIS 105] Before 1998[129]
Madelung Constant 2 [130] 5.97798 68121 78349 12266 [Mw 90][OEIS 106] Before 1998[130]
Artin constant [131] 0.37395 58136 19202 28805 [Mw 91][OEIS 107] 1999
MRB constant[132][133][134] 0.18785 96424 62067 12024 [Mw 92] [Ow 2][OEIS 108] 1999
Dimer constant 2D,
Domino tiling[135][136]

C=Catalan

0.29156 09040 30818 78013 [Mw 93][OEIS 109] Before 1999[135]
Hall-Montgomery Constant [137] 0.17150 04931 41536 06586 [Mw 94][OEIS 110] Before 1999[137]
Somos' quadratic recurrence constant [138] 1.66168 79496 33594 12129 [Mw 95][OEIS 111] 1999[Mw 95] ?
Steiner number, Iterated exponential Constant [139] 1.44466 78610 09766 13365 [Mw 96][OEIS 112] = Upper Limit of Tetration
Plouffe's A constant [140] ,

0.15915 49430 91895 33576 [Mw 97][OEIS 113]
Gauss' Lemniscate constant[141] 1.85407 46773 01371 91843 [Mw 98][OEIS 114]
1972[Mw 98]

Gallery[edit]

2000 onwards[edit]

Name Symbol Decimal Expansion Formula Year Set
Foias constant α [142] 1.18745 23511 26501 05459 [Mw 99][OEIS 115]

Foias constant is the unique real number

such that if x1 = α then the sequence diverges to ∞. When x1 = α,

2000
Foias constant β 2.29316 62874 11861 03150 [Mw 99][OEIS 116] 2000
Infinite tetration of i [143] 0.43828 29367 27032 11162[OEIS 117]

+ 0.36059 24718 71385 485 i [Mw 100][OEIS 118]

Before 2002 [OEIS 119]
Goh-Schmutz constant [144] 1.11786 41511 89944 97314 [Mw 101][OEIS 120]
  • Ei: Exponential Integral
Before 2003[144]
Pell constant [145] 0.58057 75582 04892 40229 [Mw 102][OEIS 121] Before 2003[145] ?
Carefree constant 2 [146] 0.70444 22009 99165 59273[OEIS 122] Before 2004[146]
Fibonacci Factorial constant [147] 1.22674 20107 20353 24441 [Mw 103][OEIS 123] Before 2007[147]
Double factorial
constant
3.05940 74053 42576 14453 [Mw 104] [Ow 3][OEIS 124] Before 2008

[OEIS 125]

MKB constant[dubious ]
[148][149][150]
0.07077 60393 11528 80353 [OEIS 126]

-0.68400 03894 37932 129 i [Ow 4][OEIS 127]

2009
Fixed points
super-logarithm[151] · Tetration
0.31813 15052 04764 13531 [OEIS 128]

± 1.33723 57014 30689 40 i [Ow 5][OEIS 129]

For an initial value of x different to , etc.

Before 2009[151]
Volume of Reuleaux tetrahedron [152] 0.42215 77331 15826 62702 [Mw 105][OEIS 130] Before 2010[152]
Raabe's formula [153] 0.91893 85332 04672 74178 [Mw 106][OEIS 131] Before 2011[153]
Sum of the reciprocals of the averages of the twin prime pairs, JJGJJG 0.92883 58271 [Mw 107][OEIS 132] 2014
Silverman constant[154] 1.78657 64593 65922 46345 [Mw 108][OEIS 133]
ø() = Euler's totient function, σ1() = Divisor function.
Before 2007[154]
Kepler–Bouwkamp constant [155] 0.11494 20448 53296 20070 [Mw 109][OEIS 134] Before 2013[155]


Prouhet–Thue–Morse constant [156] 0.41245 40336 40107 59778 [Mw 110][OEIS 135]    where is the Thue–Morse sequence  and
Where
Before 2014[156]
Gamma(3/4) [157] 1.22541 67024 65177 64512 [Mw 111][OEIS 136] Before 2003[157]
Heath-Brown–Moroz constant[158] 0.00131 76411 54853 17810 [Mw 112][OEIS 137] Before 2002[158] ?
Kempner Serie(0) [159] 23.10344 79094 20541 6160 [Mw 113][OEIS 138]

(Excluding all denominators containing 0.)

Before 2003[159]
Lebesgue constant [160] 0.98943 12738 31146 95174 [Mw 36][OEIS 139] Before 2002[160]
2nd du Bois-Reymond constant [161] 0.19452 80494 65325 11361 [Mw 114][OEIS 140] Before 2003[161]
Lüroth constant[162] 0.78853 05659 11508 96106 [Mw 115][OEIS 141] Before 2007[162]
Stephens constant [163] 0.57595 99688 92945 43964 [Mw 116][OEIS 142] Before 2005[163] ?
Taniguchi constant [163] 0.67823 44919 17391 97803 [Mw 117][OEIS 143]
Before 2005[163] ?
Square root of i [164] 0.70710 67811 86547 52440
+0.70710 67811 86547 524 i [Mw 6][OEIS 144]
Before 2014[164]

Silver root
Tutte–Beraha constant [165]
3.24697 96037 17467 06105 [Mw 118][OEIS 145] Before 2005[165]
Area of Ford circle [166] 0.87228 40410 65627 97617 [Mw 119] Before 2011[166]
Cubic recurrence constant [167] [168] 1.15636 26843 32269 71685 [Mw 89][OEIS 146] Before 2007[168]
Froda constant[169] 6.58088 59910 17920 97085 Before 2008[169]
Masser–Gramain constant [170] 0.64624 54398 94813 30426 [Mw 120][OEIS 147]

Before 2003[170]
Gibbs constant [171]
Sin integral
1.85193 70519 82466 17036 [Mw 121][OEIS 148]

Before 2006[171]
Copeland–Erdős constant [172] 0.23571 11317 19232 93137 [Mw 122][OEIS 149] Before 2012[172]
Hausdorff dimension, Sierpinski triangle [173] 1.58496 25007 21156 18145 [Mw 123][OEIS 150] Before 2002[173]
Figure eight knot hyperbolic volume [174] 2.02988 32128 19307 25004 [Mw 124][OEIS 151]

Before 2008[174]
Carlson–Levin constant [175] 1.77245 38509 05516 02729 [Mw 125][OEIS 152] Before 2000[175]
Strongly Carefree constant [176] 0.28674 74284 34478 73410 [Mw 126][OEIS 153] Before 2005[176]
Magic angle [177] 0.95531 66181 245092 78163[OEIS 154] Before 2003[177]
Pólya Random walk constant [178] 0.34053 73295 50999 14282 [Mw 127][OEIS 155]

Before 2003[178]
Landau–Ramanujan constant [179] 0.76422 36535 89220 66299 [Mw 128][OEIS 156] Before 2005[179] ?
Upper iterated exponential [180] 0.69034 71261 14964 31946[OEIS 157] Before 2005[180]
Lower límit iterated exponential [181] 0.65836 55992 ... Before 2003[181]
Wright constant [182] 1.92878 00... [Mw 129][OEIS 158] Before 2000[182]
Brun 4 constant = Σ inv.prime quadruplets [183] 0.87058 83799 75 [Mw 76][OEIS 159]

Before 2002[183]
Ramanujan nested radical [184] 2.74723 82749 32304 33305 Before 2001[184]
Chebyshev constant [185] · [186] 0.59017 02995 08048 11302 [Mw 130][OEIS 160] Before 2014[185]

Gallery[edit]

Other constants[edit]

Name Symbol Decimal Expansion Formula Year Set Figure
Bronze ratio [204] 3.30277 56377 31994 64655 [Mw 131][OEIS 161]


Square root of Tau·e 4.13273 13541 22492 93846[OEIS 162]

Tetranacci constant 1.92756 19754 82925 30426 [Mw 132][OEIS 163] Positive root of

DeVicci's tesseract constant 1.00743 47568 84279 37609 [Mw 133][OEIS 164] The largest cube that can pass through in an 4D hypercube.

Positive root of

8-cell-orig.gif
Module of
Infinite
Tetration of i
0.56755 51633 06957 82538[OEIS 165]
Primorial constant
Sum of the product of inverse of primes [205]
0.70523 01717 91800 96514 [Mw 134][OEIS 166]
Plouffe's gamma constant [206] 0.14758 36176 50433 27417 [Mw 97][OEIS 167]
Trigo-arctan-animation.gif
Factorial(i)[207] 0.49801 56681 18356 04271 [OEIS 168]
+ 0.15494 98283 01810 68512 i [OEIS 169]
Sarnak constant 0.72364 84022 98200 00940 [Mw 135][OEIS 170] ?
Exp.gamma,
Barnes G-function [208]
1.78107 24179 90197 98523 [Mw 136][OEIS 171]

Rényi's Parking Constant [209] 0.74759 79202 53411 43517 [Mw 137][OEIS 172] Random car parking problem.svg ParallelParkingAnimation2.gif
Generalized continued fraction
of i
0.62481 05338 43826 58687 [OEIS 173]
+ 1.30024 25902 20120 419 i [OEIS 174]

Paris Constant 1.09864 19643 94156 48573 [Mw 138][OEIS 175]  with    and  
Exponential factorial constant
1.61111 49258 08376 736
111···111 27224 36828 [Mw 139] [OEIS 176]
183213 ones
Glaisher–Kinkelin constant 1.28242 71291 00622 63687 [Mw 140][OEIS 177] ?


Trott constant [210] 0.10841 01512 23111 36151 [Mw 141][OEIS 178]

Fractal dimension of the boundary of the dragon curve [211] 1.52362 70862 02492 10627 [Mw 142]