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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] 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 [2] | ||
Square root of 2,
Pythagoras constant.[3] |
1.41421 35623 73095 04880 [Mw 2][OEIS 2] | Positive root of | 1800 to 1600 BCE[4] | ||
Square root of 3,
Theodorus' constant[5] |
1.73205 08075 68877 29352 [Mw 3][OEIS 3] | Positive root of | 465 to 398 BCE | ||
Square root of 5[6] | 2.23606 79774 99789 69640[OEIS 4] | Positive root of | |||
Phi, Golden ratio[7] | 1.61803 39887 49894 84820 [Mw 4][OEIS 5] | ~300 BCE | |||
Silver ratio[8] | 2.41421 35623 73095 04880 [Mw 5][OEIS 6] | ~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 6][OEIS 7] | Real root of | 46-120 CE[10] | ||
Cube root of 3 | 1.44224 95703 07408 38232[OEIS 8] | Real root of | |||
Twelfth root of 2[11] | 1.05946 30943 59295 26456[OEIS 9] | Real root of | |||
Supergolden ratio[12] | 1.46557123187676802665[OEIS 10] |
Medieval and Early Modern
Name | Symbol | Decimal expansion | Formula | Year | Set |
---|---|---|---|---|---|
Imaginary unit[13] | 0 + 1i | Either of the two roots of [nb 2] | 1501 to 1576 | ||
Wallis Constant | 2.09455 14815 42326 59148 [Mw 7][OEIS 11] | 1616 to 1703 |
|||
Euler's number[14] | 2.71828 18284 59045 23536 [Mw 8][OEIS 12] | [nb 3] | 1618[15] | ||
Natural logarithm of 2[16] | 0.69314 71805 59945 30941 [Mw 9][OEIS 13] | 1619,[17] 1668[18] | |||
Sophomore's dream1 J.Bernoulli[19] |
0.78343 05107 12134 40705 [OEIS 14] | 1697 | |||
Sophomore's dream2 J.Bernoulli[20] |
1.29128 59970 62663 54040 [Mw 10][OEIS 15] | 1697 | |||
Lemniscate constant[21] | 2.62205 75542 92119 81046 [Mw 11][OEIS 16] | 1718 to 1798 | |||
Euler–Mascheroni constant | 0.57721 56649 01532 86060 [Mw 12][OEIS 17] | 1735 | ? | ||
Erdős–Borwein constant[22] | 1.60669 51524 15291 76378 [Mw 13][OEIS 18] | 1749[23] | |||
Omega constant | 0.56714329040978387299 [Mw 14][OEIS 19] |
Where W is the Lambert W function |
1758,1783 | ||
Laplace limit[24] | 0.66274 34193 49181 58097 [Mw 15][OEIS 20] | ~1782 | |||
Gauss's constant[25] | 0.83462 68416 74073 18628 [Mw 16][OEIS 21] |
where agm is the arithmetic–geometric mean |
1799[26] |
19th century
Name | Symbol | Decimal expansion | Formula | Year | Set |
---|---|---|---|---|---|
Ramanujan–Soldner constant[27][28] | 1.45136 92348 83381 05028 [Mw 17][OEIS 22] | ; root of the logarithmic integral function. | 1812[Mw 18] | ||
Hermite constant[29] | 1.15470 05383 79251 52901 [Mw 19] | 1822 to 1901 | |||
Liouville number[30] | 0.11000 10000 00000 00000 0001 [Mw 20][OEIS 23] | Before 1844 | |||
Hermite–Ramanujan constant[31] | 262 53741 26407 68743 .99999 99999 99250 073 [Mw 21][OEIS 24] |
1859 | |||
Glaisher–Kinkelin constant | 1.28242 71291 00622 63687[Mw 22][OEIS 25] | 1860 to 1894 | |||
Catalan's constant[32][33][34] | 0.91596 55941 77219 01505 [Mw 23][OEIS 26] | 1864 | ? | ||
Dottie number[35] | 0.73908 51332 15160 64165 [Mw 24][OEIS 27] | 1865[Mw 24] | |||
Meissel–Mertens constant[36] | 0.26149 72128 47642 78375 [Mw 25][OEIS 28] | where γ is Euler's constant and p is a prime | 1866 & 1873 |
? | |
Weierstrass constant [37] | 0.47494 93799 87920 65033 [Mw 26][OEIS 29] | 1872 ? | |||
Hafner–Sarnak–McCurley constant (2) [38] | 0.60792 71018 54026 62866 [Mw 27][OEIS 30] | where pn is a prime | 1883[Mw 27] | ||
Universal parabolic constant[39] | 2.29558 71493 92638 07403 [Mw 28][OEIS 31] | Before 1891[40] | |||
Cahen's constant[41] | 0.64341 05462 88338 02618 [Mw 29][OEIS 32] |
Where sk is the kth term of Sylvester's sequence 2, 3, 7, 43, 1807, ...
|
1891 | ||
Apéry's constant[42] | 1.20205 69031 59594 28539 [Mw 30][OEIS 33] |
|
1895[43] | ||
Gelfond's constant[44] | 23.14069 26327 79269 0057 [Mw 31][OEIS 34] | 1900[45] |
1900–1949
Name | Symbol | Decimal expansion | Formula | Year | Set |
---|---|---|---|---|---|
Favard constant[46] | 1.23370 05501 36169 82735 [Mw 32][OEIS 35] | 1902 to 1965 |
|||
Golden angle[47] | 2.39996 32297 28653 32223 [Mw 33][OEIS 36] | = 137.5077640500378546 ...° | 1907 | ||
Sierpiński's constant[48] | 2.58498 17595 79253 21706 [Mw 34][OEIS 37] | 1907 | |||
Nielsen–Ramanujan constant [49] | 0.82246 70334 24113 21823 [Mw 35][OEIS 38] | 1909 | |||
Gieseking constant[50] | 1.01494 16064 09653 62502 [Mw 36][OEIS 39] | . |
1912 | ||
Bernstein's constant[51] | 0.28016 94990 23869 13303 [Mw 37][OEIS 40] | 1913 | |||
Twin Primes Constant | 0.66016 18158 46869 57392 [Mw 38][OEIS 41] | 1922 | |||
Bloch–Landau constant[52] | 0.54325 89653 42976 70695 [Mw 39][OEIS 42] | 1929 | |||
Plastic number[53] | 1.32471 79572 44746 02596 [Mw 40][OEIS 43] | 1929 | |||
Golomb–Dickman constant[54] | 0.62432 99885 43550 87099 [Mw 41][OEIS 44] | where Li is the logarithmic integral | 1930 & 1964 |
||
Feller–Tornier constant[55] | 0.66131 70494 69622 33528 [Mw 42][OEIS 45] | where pn is a prime | 1932 | ? | |
Base 10 Champernowne constant[56] | 0.12345 67891 01112 13141 [Mw 43][OEIS 46] | 1933 | |||
Gelfond–Schneider constant[57] | 2.66514 41426 90225 18865 [Mw 44][OEIS 47] | 1934 | |||
Khinchin's constant[58] | 2.68545 20010 65306 44530 [Mw 45][OEIS 48] | 1934 | ? | ||
Khinchin–Lévy constant (1)[59] | 1.18656 91104 15625 45282 [Mw 46][OEIS 49] | 1935 | |||
Khinchin-Lévy constant (2)[60] | 3.27582 29187 21811 15978 [Mw 47][OEIS 50] | 1936 | |||
Mills' constant[61] | 1.30637 78838 63080 69046 [Mw 48][OEIS 51] | is prime | 1947 | ||
Euler–Gompertz constant[62] | 0.59634 73623 23194 07434 [Mw 49][OEIS 52] | Before 1948[OEIS 52] |
1950–1999
Name | Symbol | Decimal expansion | Formula | Year | Set |
---|---|---|---|---|---|
Van der Pauw constant | 4.53236 01418 27193 80962[OEIS 53] | Before 1958[OEIS 54] | |||
Magic angle[63] | 0.95531 66181 245092 78163[OEIS 55] | Before 1959[64][63] | |||
Lochs constant[65] | 0.97027 01143 92033 92574 [Mw 50][OEIS 56] | 1964 | |||
Lieb's square ice constant[66] | 1.53960 07178 39002 03869 [Mw 51][OEIS 57] | 1967 | |||
Baker constant[67] | 0.83564 88482 64721 05333[OEIS 58] | Before 1969[67] | |||
Niven's constant[68] | 1.70521 11401 05367 76428 [Mw 52][OEIS 59] | 1969 | |||
Porter's constant[69] | 1.46707 80794 33975 47289 [Mw 53][OEIS 60] |
where γ (= 0.5772156649...) is the Euler–Mascheroni Constant
|
1974 | ||
Feigenbaum constant δ [70] | 4.66920 16091 02990 67185 [Mw 54][OEIS 61] |
|
1975 | ||
Chaitin's constants [71] | In general they are uncomputable numbers. But one such number is 0.00787 49969 97812 3844 [Mw 55][OEIS 62] |
|
1975 | ||
Fransén–Robinson constant[72] | 2.80777 02420 28519 36522 [Mw 56][OEIS 63] | 1978 | |||
Robbins constant[73] | 0.66170 71822 67176 23515 [Mw 57][OEIS 64] | 1978 | |||
Fractal dimension of the Cantor set[74] | 0.63092 97535 71457 43709 [Mw 58][OEIS 65] | Before 1979[OEIS 65] | |||
Feigenbaum constant α[75] | 2.50290 78750 95892 82228 [Mw 54][OEIS 66] | 1979 | ? | ||
Connective constant[76][77] | 1.84775 90650 22573 51225 [Mw 59][OEIS 67] |
as a root of the polynomial |
1982[78] | ||
Lehmer's conjecture constant[79] | 1.17628 08182 59917 50654 [Mw 60][OEIS 68] | 1983? | |||
Chebyshev constant[80] · [81] | 0.59017 02995 08048 11302 [Mw 61][OEIS 69] | Before 1987[Mw 61] | |||
Conway constant[82] | 1.30357 72690 34296 39125 [Mw 62][OEIS 70] | 1987 | |||
Prévost constant, Reciprocal Fibonacci constant[83] | 3.35988 56662 43177 55317 [Mw 63][OEIS 71] |
Fn: Fibonacci series |
Before 1988[OEIS 71] | ||
Brun 2 constant = Σ inverse of Twin primes [84] | 1.90216 05831 04 [Mw 64][OEIS 72] | where p is a prime such that p + 2 is also a prime | 1989[OEIS 72] | ||
Hafner–Sarnak–McCurley constant (1) [85] | 0.35323 63718 54995 98454 [Mw 65][OEIS 73] | where pk is a prime | 1993 | ||
Fractal dimension of the Apollonian packing of circles [86][87] |
1.30568 6729 ≈ by Thomas & Dhar 1.30568 8 ≈ by McMullen [Mw 66][OEIS 74] |
1994-1998 | |||
Backhouse's constant[88] | 1.45607 49485 82689 67139 [Mw 67][OEIS 75] |
|
1995 | ||
Viswanath constant[89] | 1.13198 82487 943 [Mw 68][OEIS 76] | where an = Fibonacci sequence | 1997 | ? | |
Regular paperfolding sequence[90][91] | 0.85073 61882 01867 26036 [Mw 69][OEIS 77] | Before 1998[91] | |||
Komornik–Loreti constant[92] | 1.78723 16501 82965 93301 [Mw 70][OEIS 78] |
tk = Thue–Morse sequence |
1998 | ||
Artin constant[93] | 0.37395 58136 19202 28805 [Mw 71][OEIS 79] | 1999 | |||
MRB constant[94][95][96] | 0.18785 96424 62067 12024 [Mw 72][Ow 1][OEIS 80] | 1999 | |||
Somos' quadratic recurrence constant[97] | 1.66168 79496 33594 12129 [Mw 73][OEIS 81] | 1999[Mw 73] | ? |
2000 onwards
Name | Symbol | Decimal expansion | Formula | Year | Set |
---|---|---|---|---|---|
Foias constant α [98] | 1.18745 23511 26501 05459 [Mw 74][OEIS 82] |
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 74][OEIS 83] | 2000 | |||
DeVicci's tesseract constant | 1.00743 47568 84279 37609[Mw 75][OEIS 84] | The largest cube that can pass through in an 4D hypercube.
Positive root of |
Before 2001 | ||
Ramanujan nested radical[99] | 2.74723 82749 32304 33305 | Before 2001[99] | |||
Brun 4 constant = Σ inv.prime quadruplets [100] | 0.87058 83799 75 [Mw 64][OEIS 85] |
|
Before 2002[100] | ||
Hausdorff dimension, Sierpinski triangle[101] | 1.58496 25007 21156 18145 [Mw 76][OEIS 86] | Before 2002[101] | |||
Heath-Brown–Moroz constant[102] | 0.00131 76411 54853 17810 [Mw 77][OEIS 87] | Before 2002[102] | ? | ||
Lebesgue constant[103] | 0.98943 12738 31146 95174 [Mw 78][OEIS 88] | Before 2002[103] | |||
2nd du Bois-Reymond constant[104] | 0.19452 80494 65325 11361 [Mw 79][OEIS 89] | Before 2003[104] | |||
Landau–Ramanujan constant[105] | 0.76422 36535 89220 66299 [Mw 80][OEIS 90] | Before 2005[105] | ? | ||
Stephens constant[106] | 0.57595 99688 92945 43964 [Mw 81][OEIS 91] | Before 2005[106] | ? | ||
Taniguchi constant[106] | 0.67823 44919 17391 97803 [Mw 82][OEIS 92] |
|
Before 2005[106] | ? | |
Raabe's formula[107] | 0.91893 85332 04672 74178 [Mw 83][OEIS 93] | Before 2011[107] | |||
Copeland–Erdős constant[108] | 0.23571 11317 19232 93137 [Mw 84][OEIS 94] | Before 2012[108] | |||
Kepler–Bouwkamp constant[109] | 0.11494 20448 53296 20070 [Mw 85][OEIS 95] | Before 2013[109] | |||
Prouhet–Thue–Morse constant[110] | 0.41245 40336 40107 59778 [Mw 86][OEIS 96] | where is the Thue–Morse sequence and Where |
Before 2014[110] |
Mathematical constants sorted by their representations as continued fractions
Continued fractions with more than 20 known terms have been truncated, with an ellipsis to show that they continue. Rational numbers have two continued fractions; the version in this list is the shorter one. Decimal representations are rounded or padded to 10 places if the values are known.
Name | Symbol[α] | Member of | decimal | Continued fraction | Notes |
---|---|---|---|---|---|
0.00000 00000 | [0; ] | ||||
Golden ratio conjugate | 0.61803 39887 | [0; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …] | irrational | ||
Cahen's constant | 0.64341 05463 | [0; 1, 1, 1, 22, 32, 132, 1292, 252982, 4209841472, 2694251407415154862, …] | All terms are squares and truncated at 10 terms due to large size. | ||
First Hardy–Littlewood conjecture | 0.66016 18158 | [0; 1, 1, 1, 16, 2, 2, 2, 2, 1, 18, 2, 2, 11, 1, 1, 2, 4, 1, 16, 3, …] | Hardy–Littlewood's twin prime constant. Presumed irrational, but not proved. | ||
Euler-Mascheroni constant | 0.57721 56649[111] | [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, 1, …][111] | Presumed irrational, but not proved. | ||
Omega constant | 0.56714 32904 | [0; 1, 1, 3, 4, 2, 10, 4, 1, 1, 1, 1, 2, 7, 306, 1, 5, 1, 2, 1, 5, …] | |||
Embree–Trefethen constant | 0.70258 | [0; 1, 2, 2, 1, 3, 5, 1, 2, 6, 1, 1, 5, …] | Value only known to 5 decimal places. | ||
Continued fraction constant | Continued fraction constant | 0.69777 46579 | [0; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, …] | Equal to a ratio of modified Bessel functions of the first kind evaluated at 2 | |
Landau–Ramanujan constant | 0.76422 36535 | [0; 1, 3, 4, 6, 1, 15, 1, 2, 2, 3, 1, 23, 3, 1, 1, 3, 1, 1, 7, 2, …] | May have been proven irrational. | ||
Gauss's constant | 0.83462 68417 | [0; 1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, 3, 8, 36, 1, 2, …] | Gauss's constant | ||
Brun's theorem | 0.87058 83800 | [0; 1, 6, 1, 2, 1, 2, 956, 8, 1, 1, 1, 23, …] | Brun's prime quadruplet constant. Estimated value; 99% confidence interval ± 0.00000 00005. | ||
Champernowne constant | 0.86224 01259 | [0; 1, 6, 3, 1, 6, 5, 3, 3, 1, 6, 4, 1, 3, 298, 1, 6, 1, 1, 3, 285, …] | Base 2 Champernowne constant. The binary expansion is | ||
Catalan's constant | 0.91596 55942[112] | [0; 1, 10, 1, 8, 1, 88, 4, 1, 1, 7, 22, 1, 2, 3, 26, 1, 11, 1, 10, 1, …][112] | Presumed irrational, but not proved. | ||
One half | 0.50000 00000 | [0; 2] | |||
Bernstein's constant | 0.28016 94990 | [0; 3, 1, 1, 3, 9, 6, 3, 1, 3, 13, 1, 16, 3, 3, 4, …] | Presumed irrational, but not proved. | ||
Meissel–Mertens constant | 0.26149 72128 | [0; 3, 1, 4, 1, 2, 5, 2, 1, 1, 1, 1, 13, 4, 2, 4, 2, 1, 33, 296, 2, …] | Presumed irrational, but not proved. | ||
MRB constant | 0.18785 96424 | [0; 5, 3, 10, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 12, 2, 17, 2, 2, 1, 1, …] | |||
Champernowne constant | 0.12345 67891 | [0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, , 6, 1, …] | Base 10 Champernowne constant. Champernowne constants in any base exhibit sporadic large numbers; the 40th term in has 2504 digits. | ||
1.00000 00000 | [1; ] | ||||
Golden ratio | 1.61803 39887[113] | [1; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …][114] | |||
Erdős–Borwein constant | 1.60669 51524 | [1; 1, 1, 1, 1, 5, 2, 1, 2, 29, 4, 1, 2, 2, 2, 2, 6, 1, 7, 1, 6, …] | Not known whether algebraic or transcendental. | ||
Brun's constant | 1.90216 05831 | [1; 1, 9, 4, 1, 1, 8, 3, 4, 7, 1, 3, 3, 1, 2, 1, 1, 12, 4, 2, 1, …] | Brun's twin prime constant. Estimated value; best bounds . | ||
Square root of 2 | 1.41421 35624 | [1; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, …] | |||
Ramanujan-Soldner constant | 1.45136 92349 | [1; 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, 4, 1, 12, 1, 1, 2, 2, 1, …] | Presumed irrational, but not proved. | ||
Backhouse's constant | 1.45607 49485 | [1; 2, 5, 5, 4, 1, 1, 18, 1, 1, 1, 1, 1, 2, 13, 3, 1, 2, 4, 16, 4, …] | |||
Plastic number | 1.32471 95724 | [1; 3, 12, 1, 1, 3, 2, 3, 2, 4, 2, 141, 80, 2, 5, 1, 2, 8, 2, 1, 1, …] | |||
Apéry's constant | 1.20205 69032[115] | [1; 4, 1, 18, 1, 1, 1, 4, 1, 9, 9, 2, 1, 1, 1, 2, 7, 1, 1, 7, 11, …][115] | |||
Random Fibonacci sequence | 1.13198 82488 | [1; 7, 1, 1, 2, 1, 3, 2, 1, 2, 1, 17, 1, 1, 2, 1, 2, 4, 1, 2, …] | Viswanath's constant. Apparently, Eric Weisstein calculated this constant to be approximately 1.13215 06911 with Mathematica. | ||
2.00000 00000 | [2; ] | ||||
Gelfond–Schneider constant | 2.66514 41426 | [2; 1, 1, 1, 72, 3, 4, 1, 3, 2, 1, 1, 1, 14, 1, 2, 1, 1, 3, 1, 3, …] | |||
Second Feigenbaum constant | 2.50290 78751 | [2; 1, 1, 85, 2, 8, 1, 10, 16, 3, 8, 9, 2, 1, 40, 1, 2, 3, 2, 2, 1, …] | |||
Base of the natural logarithm | 2.71828 18285[116] | [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, …][117] | |||
Khinchin's constant | 2.68545 20011[118] | [2; 1, 2, 5, 1, 1, 2, 1, 1, 3, 10, 2, 1, 3, 2, 24, 1, 3, 2, 3, 1, …][119] | |||
Fransén–Robinson constant | 2.80777 02420 | [2; 1, 4, 4, 1, 18, 5, 1, 3, 4, 1, 5, 3, 6, 1, 1, 1, 5, 1, 1, 1, …] | |||
Universal parabolic constant | 2.29558 71494 | [2; 3, 2, 1, 1, 1, 1, 3, 3, 1, 1, 4, 2, 3, 2, 7, 1, 6, 1, 8, 7, …] | |||
3.00000 00000 | [3; ] | ||||
Reciprocal Fibonacci constant | 3.35988 56662 | [3; 2, 1, 3, 1, 1, 13, 2, 3, 3, 2, 1, 1, 6, 3, 2, 4, 362, 2, 4, 8, …] | |||
3.14159 26536 | [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, …] | ||||
4.00000 00000 | [4; ] | ||||
First Feigenbaum constant | 4.66920 16091 | [4; 1, 2, 43, 2, 163, 2, 3, 1, 1, 2, 5, 1, 2, 3, 80, 2, 5, 2, 1, 1, …] | |||
5.00000 00000 | [5; ] | ||||
Gelfond's constant | 23.14069 26328 | [23; 7, 9, 3, 1, 1, 591, 2, 9, 1, 2, 34, 1, 16, 1, 30, 1, 1, 4, 1, 2, …] | Gelfond's constant. Can also be expressed as ; from this form, it is transcendental due to the Gelfond–Schneider theorem. |
See also
- Invariant (mathematics)
- List of mathematical symbols
- List of mathematical symbols by subject
- List of numbers
- List of physical constants
- Mathematical constants and functions
- Mathematical constants by continued fraction representation
- Particular values of the Riemann zeta function
- Physical constant
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
- ^ Although some of the symbols in the "Symbol" column are displayed in black due to math markup peculiarities, all are clickable and link to the respective constant's page.
References
- ^ Weisstein, Eric W. "Constant". mathworld.wolfram.com. Retrieved 2020-08-08.
- ^ Arndt & Haenel 2006, p. 167
- ^ Calvin C Clawson (2001). Mathematical sorcery: revealing the secrets of numbers. p. IV. ISBN 978 0 7382 0496-3.
- ^ Fowler and Robson, p. 368. Photograph, illustration, and description of the root(2) tablet from the Yale Babylonian Collection Archived 2012-08-13 at the Wayback Machine High resolution photographs, descriptions, and analysis of the root(2) tablet (YBC 7289) from the Yale Babylonian Collection
- ^ Vijaya AV (2007). Figuring Out Mathematics. Dorling Kindcrsley (India) Pvt. Lid. p. 15. ISBN 978-81-317-0359-5.
- ^ P A J Lewis (2008). Essential Mathematics 9. Ratna Sagar. p. 24. ISBN 9788183323673.
- ^ Timothy Gowers; June Barrow-Green; Imre Leade (2007). The Princeton Companion to Mathematics. Princeton University Press. p. 316. ISBN 978-0-691-11880-2.
- ^ Kapusta, Janos (2004), "The square, the circle, and the golden proportion: a new class of geometrical constructions" (PDF), Forma, 19: 293–313.
- ^ Kim Plofker (2009), Mathematics in India, Princeton University Press, ISBN 978-0-691-12067-6, pp. 54–56.
- ^ Plutarch. "718ef". Quaestiones convivales VIII.ii. Archived from the original on 2009-11-19. Retrieved 2019-05-24.
And therefore Plato himself dislikes Eudoxus, Archytas, and Menaechmus for endeavoring to bring down the doubling the cube to mechanical operations
- ^ Christensen, Thomas (2002), The Cambridge History of Western Music Theory, p. 205, ISBN 978-0521686983
- ^ Koshy, Thomas (2017). Fibonacci and Lucas Numbers with Applications (2 ed.). John Wiley & Sons. ISBN 9781118742174. Retrieved 14 August 2018.
- ^ Keith J. Devlin (1999). Mathematics: The New Golden Age. Columbia University Press. p. 66. ISBN 978-0-231-11638-1.
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Site MathWorld Wolfram.com
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- ^ Weisstein, Eric W. "Silver Ratio". MathWorld.
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- ^ Weisstein, Eric W. "e". MathWorld.
- ^ Weisstein, Eric W. "Natural Logarithm of 2". MathWorld.
- ^ Weisstein, Eric W. "Sophomore's Dream". MathWorld.
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- ^ Weisstein, Eric W. "Erdos-Borwein Constant". MathWorld.
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- ^ Weisstein, Eric W. "Hermite Constants". MathWorld.
- ^ Weisstein, Eric W. "Liouville's Constant". MathWorld.
- ^ Weisstein, Eric W. "Ramanujan Constant". MathWorld.
- ^ Weisstein, Eric W. "Glaisher-Kinkelin Constant". MathWorld.
- ^ Weisstein, Eric W. "Catalan's Constant". MathWorld.
- ^ a b Weisstein, Eric W. "Dottie Number". MathWorld.
- ^ Weisstein, Eric W. "Mertens Constant". MathWorld.
- ^ Weisstein, Eric W. "Weierstrass Constant". MathWorld.
- ^ a b Weisstein, Eric W. "Relatively Prime". MathWorld.
- ^ Weisstein, Eric W. "Universal Parabolic Constant". MathWorld.
- ^ Weisstein, Eric W. "Cahen's 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. "Gieseking's Constant". MathWorld.
- ^ Weisstein, Eric W. "Bernstein's Constant". MathWorld.
- ^ Weisstein, Eric W. "Twin Primes Constant". MathWorld.
- ^ Weisstein, Eric W. "Landau Constant". MathWorld.
- ^ Weisstein, Eric W. "Plastic 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.
- ^ a b 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.
- ^ a b Weisstein, Eric W. "Chebyshev Constants". MathWorld.
- ^ Weisstein, Eric W. "Conway's Constant". MathWorld.
- ^ Weisstein, Eric W. "Reciprocal Fibonacci Constant". MathWorld.
- ^ a b 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. "Paper Folding Constant". MathWorld.
- ^ Weisstein, Eric W. "Komornik-Loreti Constant". MathWorld.
- ^ Weisstein, Eric W. "Artin's Constant". MathWorld.
- ^ Weisstein, Eric W. "MRB Constant". MathWorld.
- ^ a b Weisstein, Eric W. "SomossQuadraticRecurrence Constant". MathWorld.
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Site OEIS.com
- ^ OEIS: A000796
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Bibliography
- Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 2013-06-05. English translation by Catriona and David Lischka.
- Jensen, Johan Ludwig William Valdemar (1895), "Note numéro 245. Deuxième réponse. Remarques relatives aux réponses du MM. Franel et Kluyver", L'Intermédiaire des Mathématiciens, II: 346–347
- Cuyt, Annie A.M.; Petersen, Vigdis; Verdonk, Brigitte; Waadeland, Haakon; Jones, William B. (2008). "Mathematical constants". Handbook of Continued Fractions for Special Functions. Dordrecht, Netherlands: Springer Science + Business Media. ISBN 9781402069499.
- Borwein, Jonathan; van der Poorten, Alf; Shallit, Jeffrey; Zudilin, Wadim (2014). Neverending Fractions: An Introduction to Continued Fractions. Australian Mathematical Society Lecture Series. Vol. 23. Cambridge, United Kingdom: Cambridge University Press. ISBN 9780521186490. ISSN 0950-2815.
Further reading
- Wolfram, Stephen. "4: Systems Based on Numbers". Section 5: Mathematical Constants — Continued fractions. A New Kind of Science.
External links
- Mathematical constants
- Mathematics-related lists
- Mathematical tables
- Number-related lists
- Continued fractions