List of mathematical constants
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems.[1][2] 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 a decimal expansion and set containing each number, ordered by year of discovery.
Explanations of the symbols in the right hand column can be found by clicking on them.
Antiquity
Name | Symbol | Decimal Expansion | Formula | Year | Set |
---|---|---|---|---|---|
One | 1 | 1 | None[nb 1] | Prehistory | |
Two | 2 | 2 | Prehistory | ||
One half | 1/2 | 0.5 | Prehistory | ||
Pi | 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 | ||
Square root of 5[7] | 2.23606 79774 99789 69640[OEIS 4] | Positive root of | |||
Phi, Golden ratio[1][8] | 1.61803 39887 49894 84820 [Mw 4][OEIS 5] | Positive root of | ~300 BCE | ||
Zero | 0 | 0 | The additive identity: | 300-100 century BCE[9] | |
Negative one | -1 | -1 | 300-200 BCE | ||
Cube root of 2 (Delian Constant) | 1.25992 10498 94873 16476 [Mw 5][OEIS 6] | Real root of | 46 -120 CE | ||
Cube root of 3 | 1.44224 95703 07408 38232[OEIS 7] | Real root of |
Medieval and Early Modern
Name | Symbol | Decimal Expansion | Formula | Year | Set |
---|---|---|---|---|---|
Imaginary unit [1][11] | 0 + 1i | Either of the two roots of [nb 2] | 1501 to 1576 | ||
Wallis Constant | 2.09455 14815 42326 59148 [Mw 6][OEIS 8] | 1616 to 1703 |
|||
Euler's number[1][12] | 2.71828 18284 59045 23536 [Mw 7][OEIS 9] | [nb 3] | 1618[13] | ||
Natural logarithm of 2 [14] | 0.69314 71805 59945 30941 [Mw 8][OEIS 10] | 1619,[15]1668[16] | |||
Sophomore's dream1 J.Bernoulli [17] |
0.78343 05107 12134 40705 [OEIS 11] | 1697 | |||
Sophomore's dream2 J.Bernoulli [18] |
1.29128 59970 62663 54040 [Mw 9][OEIS 12] | 1697 | |||
Lemniscate constant[19] | 2.62205 75542 92119 81046 [Mw 10][OEIS 13] | 1718 to 1798 | |||
Euler–Mascheroni constant[20] | 0.57721 56649 01532 86060 [Mw 11][OEIS 14] | |
1735 | ? | |
Erdős–Borwein constant[21] | 1.60669 51524 15291 76378 [Mw 12][OEIS 15] | 1749[22] | |||
Laplace limit [23] | 0.66274 34193 49181 58097 [Mw 13][OEIS 16] | ~1782 | ? | ||
Gauss's constant [24] | 0.83462 68416 74073 18628 [Mw 14][OEIS 17] |
where agm = Arithmetic–geometric mean |
1799[25] | ? |
19th century
Name | Symbol | Decimal Expansion | Formula | Year | Set |
---|---|---|---|---|---|
Ramanujan–Soldner constant[26][27] | 1.45136 92348 83381 05028 [Mw 15][OEIS 18] | ; root of the logarithmic integral function. | 1812[Mw 16] | ||
Hermite constant [28] | 1.15470 05383 79251 52901 [Mw 17] | 1822 to 1901 | |||
Liouville number [29] | 0.11000 10000 00000 00000 0001 [Mw 18][OEIS 19] | Before 1844 | |||
Hermite–Ramanujan constant[30] | 262 53741 26407 68743 .99999 99999 99250 073 [Mw 19][OEIS 20] |
1859 | |||
Catalan's constant[31][32][33] | 0.91596 55941 77219 01505 [Mw 20][OEIS 21] | 1864 | ? | ||
Dottie number [34] | 0.73908 51332 15160 64165 [Mw 21][OEIS 22] | 1865[Mw 21] | |||
Meissel–Mertens constant [35] | 0.26149 72128 47642 78375 [Mw 22][OEIS 23] | 1866 & 1873 |
? | ||
Weierstrass constant [36] | 0.47494 93799 87920 65033 [Mw 23][OEIS 24] | 1872 ? | |||
Hafner–Sarnak–McCurley constant (2) [37] | 0.60792 71018 54026 62866 [Mw 24][OEIS 25] | 1883[Mw 24] | |||
Cahen's constant [38] | 0.64341 05462 88338 02618 [Mw 25][OEIS 26] |
Where sk is the kth term of Sylvester's sequence 2, 3, 7, 43, 1807, ...
|
1891 | ||
Universal parabolic constant [39] | 2.29558 71493 92638 07403 [Mw 26][OEIS 27] | Before 1891[40] | |||
Apéry's constant [41] | 1.20205 69031 59594 28539 [Mw 27][OEIS 28] |
|
1895[42] | ||
Gelfond's constant [43] | 23.14069 26327 79269 0057 [Mw 28][OEIS 29] | 1900[44] |
1900–1949
Name | Symbol | Decimal Expansion | Formula | Year | Set |
---|---|---|---|---|---|
Favard constant [45] | 1.23370 05501 36169 82735 [Mw 29][OEIS 30] | 1902 to 1965 |
|||
Golden angle [46] | 2.39996 32297 28653 32223 [Mw 30][OEIS 31] | = 137.5077640500378546 ...° | 1907 | ||
Sierpiński's constant [47] | 2.58498 17595 79253 21706 [Mw 31][OEIS 32] |
|
1907 | ||
Nielsen–Ramanujan constant [48] | 0.82246 70334 24113 21823 [Mw 32][OEIS 33] | 1909 | |||
Area of the Mandelbrot fractal [49] | 1.5065918849 ± 0.0000000028 [Mw 33][OEIS 34] | 1912 | |||
Gieseking constant [50] | 1.01494 16064 09653 62502 [Mw 34][OEIS 35] | . |
1912 | ||
Bernstein's constant [51] | 0.28016 94990 23869 13303 [Mw 35][OEIS 36] | 1913 | |||
Twin Primes Constant [52] | 0.66016 18158 46869 57392 [Mw 36][OEIS 37] | 1922 | |||
Plastic number [53] | 1.32471 79572 44746 02596 [Mw 37][OEIS 38] | 1929 | |||
Bloch–Landau constant [54] | 0.54325 89653 42976 70695 [Mw 38][OEIS 39] | 1929 | |||
Golomb–Dickman constant [55] | 0.62432 99885 43550 87099 [Mw 39][OEIS 40] | 1930 & 1964 |
|||
Feller–Tornier constant [56] | 0.66131 70494 69622 33528 [Mw 40][OEIS 41] | 1932 | ? | ||
Base 10 Champernowne constant [57] | 0.12345 67891 01112 13141 [Mw 41][OEIS 42] | 1933 | |||
Gelfond–Schneider constant [58] | 2.66514 41426 90225 18865 [Mw 42][OEIS 43] | 1934 | |||
Khinchin's constant [59] | 2.68545 20010 65306 44530 [Mw 43][OEIS 44] | 1934 | ? | ||
Khinchin–Lévy constant[60] | 1.18656 91104 15625 45282 [Mw 44][OEIS 45] | 1935 | |||
Khinchin-Lévy constant [61] | 3.27582 29187 21811 15978 [Mw 45][OEIS 46] | 1936 | |||
Mills' constant [62] | 1.30637 78838 63080 69046 [Mw 46][OEIS 47] | is prime | 1947 | ||
Euler–Gompertz constant [63] | 0.59634 73623 23194 07434 [Mw 47][OEIS 48] | Before 1948[OEIS 48] |
1950–1999
Name | Symbol | Decimal Expansion | Formula | Year | Set |
---|---|---|---|---|---|
Van der Pauw constant | 4.53236 01418 27193 80962[OEIS 49] | Before 1958[OEIS 50] | |||
Lochs constant [64] | 0.97027 01143 92033 92574 [Mw 48][OEIS 51] | 1964 | |||
Lieb's square ice constant [65] | 1.53960 07178 39002 03869 [Mw 49][OEIS 52] | 1967 | |||
Niven's constant [66] | 1.70521 11401 05367 76428 [Mw 50][OEIS 53] | 1969 | |||
Baker constant [67] | 0.83564 88482 64721 05333[OEIS 54] | Before 1969[67] | |||
Porter's constant[68] | 1.46707 80794 33975 47289 [Mw 51][OEIS 55] |
|
1974 | ||
Feigenbaum constant δ [69] | 4.66920 16091 02990 67185 [Mw 52][OEIS 56] |
|
1975 | ||
Chaitin's constants [70] | In general they are uncomputable numbers. But one such number is 0.00787 49969 97812 3844 [Mw 53][OEIS 57] |
|
1975 | ||
Fransén–Robinson constant [71] | 2.80777 02420 28519 36522 [Mw 54][OEIS 58] | 1978 | |||
Robbins constant [72] | 0.66170 71822 67176 23515 [Mw 55][OEIS 59] | 1978 | |||
Feigenbaum constant α[73] | 2.50290 78750 95892 82228 [Mw 52][OEIS 60] | 1979 | ? | ||
Fractal dimension of the Cantor set [74] | 0.63092 97535 71457 43709 [Mw 56][OEIS 61] | Before 1979[OEIS 61] | |||
Connective constant [75][76] | 1.84775 90650 22573 51225 [Mw 57][OEIS 62] |
as a root of the polynomial |
1982[77] | ||
Salem number,[78] | 1.17628 08182 59917 50654 [Mw 58][OEIS 63] | 1983? | |||
Chebyshev constant [79] · [80] | 0.59017 02995 08048 11302 [Mw 59][OEIS 64] | Before 1987[Mw 59] | |||
Conway constant [81] | 1.30357 72690 34296 39125 [Mw 60][OEIS 65] | 1987 | |||
Prévost constant, Reciprocal Fibonacci constant[82] | 3.35988 56662 43177 55317 [Mw 61][OEIS 66] |
Fn: Fibonacci series |
Before 1988[OEIS 66] | ||
Brun 2 constant = Σ inverse of Twin primes [83] | 1.90216 05831 04 [Mw 62][OEIS 67] | 1989[OEIS 67] | |||
Hafner–Sarnak–McCurley constant (1) [84] | 0.35323 63718 54995 98454 [Mw 63][OEIS 68] | 1993 | |||
Fractal dimension of the Apollonian packing of circles [85][86] |
1.30568 6729 ≈ by Thomas & Dhar 1.30568 8 ≈ by McMullen [Mw 64][OEIS 69] |
1994 1998 |
|||
Backhouse's constant [87] | 1.45607 49485 82689 67139 [Mw 65][OEIS 70] |
|
1995 | ||
Viswanath constant[88] | 1.13198 82487 943 [Mw 66][OEIS 71] | where an = Fibonacci sequence | 1997 | ? | |
Time constant [89] | 0.63212 05588 28557 67840 [Mw 67][OEIS 72] | |
Before 1997[89] | ||
Komornik–Loreti constant [90] | 1.78723 16501 82965 93301 [Mw 68][OEIS 73] |
tk = Thue–Morse sequence |
1998 | ||
Regular paperfolding sequence [91][92] | 0.85073 61882 01867 26036 [Mw 69][OEIS 74] | Before 1998[92] | |||
Artin constant [93] | 0.37395 58136 19202 28805 [Mw 70][OEIS 75] | 1999 | |||
MRB constant[94][95][96] | 0.18785 96424 62067 12024 [Mw 71][Ow 1][OEIS 76] | 1999 | |||
Somos' quadratic recurrence constant [97] | 1.66168 79496 33594 12129 [Mw 72][OEIS 77] | 1999[Mw 72] | ? |
2000 onwards
Name | Symbol | Decimal Expansion | Formula | Year | Set |
---|---|---|---|---|---|
Foias constant α [98] | 1.18745 23511 26501 05459 [Mw 73][OEIS 78] |
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 73][OEIS 79] | 2000 | |||
Raabe's formula [99] | 0.91893 85332 04672 74178 [Mw 74][OEIS 80] | Before 2011[99] | |||
Kepler–Bouwkamp constant [100] | 0.11494 20448 53296 20070 [Mw 75][OEIS 81] | Before 2013[100] | |||
Prouhet–Thue–Morse constant [101] | 0.41245 40336 40107 59778 [Mw 76][OEIS 82] | where is the Thue–Morse sequence and Where |
Before 2014[101] | ||
Heath-Brown–Moroz constant[102] | 0.00131 76411 54853 17810 [Mw 77][OEIS 83] | Before 2002[102] | ? | ||
Lebesgue constant [103] | 0.98943 12738 31146 95174 [Mw 78][OEIS 84] | Before 2002[103] | |||
2nd du Bois-Reymond constant [104] | 0.19452 80494 65325 11361 [Mw 79][OEIS 85] | Before 2003[104] | |||
Stephens constant [105] | 0.57595 99688 92945 43964 [Mw 80][OEIS 86] | Before 2005[105] | ? | ||
Taniguchi constant [105] | 0.67823 44919 17391 97803 [Mw 81][OEIS 87] |
|
Before 2005[105] | ? | |
Copeland–Erdős constant [106] | 0.23571 11317 19232 93137 [Mw 82][OEIS 88] | Before 2012[106] | |||
Hausdorff dimension, Sierpinski triangle [107] | 1.58496 25007 21156 18145 [Mw 83][OEIS 89] | Before 2002[107] | |||
Magic angle [108] | 0.95531 66181 245092 78163[OEIS 90] | Before 2003[108] | |||
Landau–Ramanujan constant [109] | 0.76422 36535 89220 66299 [Mw 84][OEIS 91] | Before 2005[109] | ? | ||
Brun 4 constant = Σ inv.prime quadruplets [110] | 0.87058 83799 75 [Mw 62][OEIS 92] |
|
Before 2002[110] | ||
Ramanujan nested radical [111] | 2.74723 82749 32304 33305 | Before 2001[111] |
Other constants
Name | Symbol | Decimal Expansion | Formula | Year | Set |
---|---|---|---|---|---|
DeVicci's tesseract constant | 1.00743 47568 84279 37609[Mw 85][OEIS 93] | The largest cube that can pass through in an 4D hypercube.
Positive root of |
|||
Glaisher–Kinkelin constant | 1.28242 71291 00622 63687[Mw 86][OEIS 94] | ||||
See also
Notes
- 1 can be given as a primitive notion within Peano arithmetic. Alternatively, 0 can be a primitive notion in Peano arithmetic and 1 defined as the successor to 0. This article uses the former definition for pedagogical and chronological simplicity.
- Both i and -i are roots of this equation, though neither root is truly "positive" nor more fundamental than the other as they are algebraically equivalent. The distinction between signs of i and -i is in some ways arbitrary, but a useful notational device. See imaginary unit for more information.
- Can also be defined by the infinite series
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Site MathWorld Wolfram.com
- Weisstein, Eric W. "Pi Formulas". MathWorld.
- Weisstein, Eric W. "Pythagoras's Constant". MathWorld.
- Weisstein, Eric W. "Theodorus's Constant". MathWorld.
- Weisstein, Eric W. "Golden Ratio". MathWorld.
- Weisstein, Eric W. "Delian Constant". MathWorld.
- Weisstein, Eric W. "Wallis's Constant". MathWorld.
- Weisstein, Eric W. "e". MathWorld.
- Weisstein, Eric W. "Natural Logarithm of 2". MathWorld.
- Weisstein, Eric W. "Sophomore's Dream". MathWorld.
- Weisstein, Eric W. "Lemniscate Constant". MathWorld.
- Weisstein, Eric W. "Euler–Mascheroni Constant". MathWorld.
- Weisstein, Eric W. "Erdos-Borwein Constant". MathWorld.
- Weisstein, Eric W. "Laplace Limit". MathWorld.
- Weisstein, Eric W. "Gauss's Constant". MathWorld.
- Weisstein, Eric W. "Soldner's Constant". MathWorld.
- Weisstein, Eric W. "Soldner's Constant". MathWorld.
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- Weisstein, Eric W. "Liouville's Constant". MathWorld.
- Weisstein, Eric W. "Ramanujan Constant". MathWorld.
- Weisstein, Eric W. "Catalan's Constant". MathWorld.
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- Weisstein, Eric W. "Mertens Constant". MathWorld.
- Weisstein, Eric W. "Weierstrass Constant". MathWorld.
- Weisstein, Eric W. "Relatively Prime". MathWorld.
- Weisstein, Eric W. "Cahen's Constant". MathWorld.
- Weisstein, Eric W. "Universal Parabolic Constant". MathWorld.
- Weisstein, Eric W. "Apéry's Constant". MathWorld.
- Weisstein, Eric W. "Gelfonds Constant". MathWorld.
- Weisstein, Eric W. "Favard Constants". MathWorld.
- Weisstein, Eric W. "Golden Angle". MathWorld.
- Weisstein, Eric W. "Sierpinski Constant". MathWorld.
- Weisstein, Eric W. "Nielsen-Ramanujan Constants". MathWorld.
- Weisstein, Eric W. "Mandelbrot Set". MathWorld.
- Weisstein, Eric W. "Gieseking's Constant". MathWorld.
- Weisstein, Eric W. "Bernstein's Constant". MathWorld.
- Weisstein, Eric W. "Twin Primes Constant". MathWorld.
- Weisstein, Eric W. "Plastic Constant". MathWorld.
- Weisstein, Eric W. "Landau Constant". MathWorld.
- Weisstein, Eric W. "Golomb-Dickman Constant". MathWorld.
- Weisstein, Eric W. "Feller-Tornier Constant". MathWorld.
- Weisstein, Eric W. "Champernowne Constant". MathWorld.
- Weisstein, Eric W. "Gelfond-Schneider Constant". MathWorld.
- Weisstein, Eric W. "Khinchin's Constant". MathWorld.
- Weisstein, Eric W. "Levy Constant". MathWorld.
- Weisstein, Eric W. "Levy Constant". MathWorld.
- Weisstein, Eric W. "Mills Constant". MathWorld.
- Weisstein, Eric W. "Gompertz Constant". MathWorld.
- Weisstein, Eric W. "Lochs' Constant". MathWorld.
- Weisstein, Eric W. "Liebs Square Ice Constant". MathWorld.
- Weisstein, Eric W. "Niven's Constant". MathWorld.
- Weisstein, Eric W. "Porter's Constant". MathWorld.
- Weisstein, Eric W. "Feigenbaum Constant". MathWorld.
- Weisstein, Eric W. "Chaitin's Constant". MathWorld.
- Weisstein, Eric W. "Fransen-Robinson Constant". MathWorld.
- Weisstein, Eric W. "Robbins Constant". MathWorld.
- Weisstein, Eric W. "Cantor Set". MathWorld.
- Weisstein, Eric W. "Self-Avoiding Walk Connective Constant". MathWorld.
- Weisstein, Eric W. "Salem Constants". MathWorld.
- Weisstein, Eric W. "Chebyshev Constants". MathWorld.
- Weisstein, Eric W. "Conway's Constant". MathWorld.
- Weisstein, Eric W. "Reciprocal Fibonacci Constant". MathWorld.
- Weisstein, Eric W. "Brun's Constant". MathWorld.
- Weisstein, Eric W. "Hafner-Sarnak-McCurley Constant". MathWorld.
- Weisstein, Eric W. "Apollonian Gasket". MathWorld.
- Weisstein, Eric W. "Backhouse's Constant". MathWorld.
- Weisstein, Eric W. "Random Fibonacci Sequence". MathWorld.
- Weisstein, Eric W. "e". MathWorld.
- Weisstein, Eric W. "Komornik-Loreti Constant". MathWorld.
- Weisstein, Eric W. "Paper Folding Constant". MathWorld.
- Weisstein, Eric W. "Artin's Constant". MathWorld.
- Weisstein, Eric W. "MRB Constant". MathWorld.
- Weisstein, Eric W. "SomossQuadraticRecurrence Constant". MathWorld.
- Weisstein, Eric W. "Foias Constant". MathWorld.
- Weisstein, Eric W. "Log Gamma Function". MathWorld.
- Weisstein, Eric W. "Polygon Inscribing". MathWorld.
- Weisstein, Eric W. "Thue-Morse Constant". MathWorld.
- Weisstein, Eric W. "Heath-Brown-Moroz Constant". MathWorld.
- Weisstein, Eric W. "Du Bois Reymond Constants". MathWorld.
- Weisstein, Eric W. "Stephen's Constant". MathWorld.
- Weisstein, Eric W. "Euler Product". MathWorld.
- Weisstein, Eric W. "Copeland-Erdos Constant". MathWorld.
- Weisstein, Eric W. "Pascal's Triangle". MathWorld.
- Weisstein, Eric W. "Landau-Ramanujan Constant". MathWorld.
- Weisstein, Eric W. "Prince Rupert's Cube". MathWorld.
- Weisstein, Eric W. "Glaisher-Kinkelin Constant". MathWorld.
Site OEIS.com
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Site OEIS Wiki
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