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. 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.
The column headings may be clicked to sort the table alphabetically, by decimal value, or by set. Explanations of the symbols in the right hand column can be found by clicking on them.
List
Name Symbol Decimal expansion Formula Year SetZero00Additive identity of 300 to 100 BCEKim Plofker (2009), Mathematics in India, Princeton University Press, , pp. 54–56.✓✓✓✓One11Multiplicative identity of Prehistory✓✓✓✓Two22Prehistory✓✓✓✓One half0.5Multiplicative inverse of 2Prehistory✓✓✓✓Pi3.14159 26535 89793 23846 Ratio of a circle's circumference to its diameter1900 to 1600 BCE ✗✗✓✓Tau6.28318 53071 79586 47692Ratio of a circle's circumference to its radius; equal to 1900 to 1600 BCE ✗✗✓✓Square root of 2, Pythagoras constant1.41421 35623 73095 04880 Positive root of 1800 to 1600 BCEFowler and Robson, p. 368.
Photograph, illustration, and description of the root(2) tablet from the Yale Babylonian Collection
High resolution photographs, descriptions, and analysis of the root(2) tablet (YBC 7289) from the Yale Babylonian Collection✗✓✓✓Square root of 3, Theodorus' constant1.73205 08075 68877 29352 Positive root of 465 to 398 BCE✗✓✓✓Square root of 52.23606 79774 99789 69640 Positive root of ✗✓✓✓Phi, Golden ratio or 1.61803 39887 49894 84820 ~300 BCE✗✓✓✓Silver ratio.2.41421 35623 73095 04880 ~300 BCE✗✓✓✓ Negative one −1 −1Additive inverse of 1300 to 200 BCE✓✓✓✓Cube root of 2, Delian constant1.25992 10498 94873 16476 Real root of 46 to 120 CE✗✓✓✓Cube root of 31.44224 95703 07408 38232 Real root of ✗✓✓✓Twelfth root of 21.05946 30943 59295 26456 Real positive root of ✗✓✓✓Supergolden ratio1.46557 12318 76768 02665 Real root of ✗✓✓✓Imaginary unitPrincipal root of 1501 to 1576✗✓✓✓Connective constant for the hexagonal lattice1.84775 90650 22573 51225 , as a root of the polynomial 1593 ✗✓✓✓Kepler–Bouwkamp constant0.11494 20448 53296 20070 1596 ????Wallis's constant2.09455 14815 42326 59148 Real root of 1616 to 1703✗✓✓✓Euler's number2.71828 18284 59045 23536 1618✗✗?✓Natural logarithm of 20.69314 71805 59945 30941 Real root of 1619 & 1668 ✗✗✓✓Lemniscate constant2.62205 75542 92119 81046 Ratio of the perimeter of Bernoulli's lemniscate to its diameter1718 to 1798✗✗✓✓Basel's number1.64493 40668 48226 43647 with the Riemann zeta function 1734✗?✓✓Euler's constant0.57721 56649 01532 86060 Limiting difference between the harmonic series and the natural logarithm1735???✓Erdős–Borwein constant1.60669 51524 15291 76378 1749✗???Square super-root of 21.55961 04694 62369 34997Real root of 1758 & 1783✗✗??Omega constant0.56714 32904 09783 87299 with the Lambert W function 1758 & 1783✗✗??Apéry's constant1.20205 69031 59594 28539 with the Riemann zeta function 1780 ✗?✓✓Laplace limit0.66274 34193 49181 58097 Real root of ~1782✗✗??Soldner constant1.45136 92348 83381 05028 ;
Root of the logarithmic integral function1792 ????Gauss's constant0.83462 68416 74073 18628 where agm is the arithmetic–geometric mean and is the lemniscate constant1799✗✗?✓Second Hermite constant1.15470 05383 79251 52901 1822 to 1901✗✓✓✓Liouville's constant0.11000 10000 00000 00000 0001 Before 1844✗✗??First continued fraction constant0.69777 46579 64007 98201 where Iα(x) are the modified Bessel functions of the first kind1855✗✗?Ramanujan's constant262 53741 26407 68743 .99999 99999 99250 073 1859✗✗?Glaisher–Kinkelin constant1.28242 71291 00622 63687 1860 ???Catalan's constant0.91596 55941 77219 01505 with the Dirichlet beta function 1864??✓Dottie number0.73908 51332 15160 64165 Real root of 1865 ✗✗?Meissel–Mertens constant0.26149 72128 47642 78375 where γ is the Euler–Mascheroni constant and p is prime1866 & 1873???Universal parabolic constant2.29558 71493 92638 07403 Before 1891✗✗✓Cahen's constant0.64341 05462 88338 02618 where sk is the kth term of Sylvester's sequence (2, 3, 7, 43, 1807, ...)1891✗✗?Gelfond's constant23.14069 26327 79269 0057 1900✗✗?Totient summatory constant1.33978 41535 74347 24659 where is the set of prime numbers1900???Gelfond–Schneider constant2.66514 41426 90225 18865 Before 1902 ✗✗?Second Favard constant1.23370 05501 36169 82735 1902 to 1965✗✗✓Golden angle2.39996 32297 28653 32223 or
in degrees1907✗✗✓Sierpiński's constant2.58498 17595 79253 21706 1907???Landau–Ramanujan constant0.76422 36535 89220 66299 1908 ???First Nielsen–Ramanujan constant0.82246 70334 24113 21823 1909✗✗✓Gieseking constant1.01494 16064 09653 62502 with the trigamma function 1912??✓Bernstein's constant0.28016 94990 23869 13303 , where En(f) is the error of the best uniform approximation to a real function f(x) on the interval [−1, 1] by real polynomials of no more than degree n, and f(x) = |x|1913???Tribonacci constant1.83928 67552 14161 13255 Real root of 1914 to 1963✗✓✓Brun's constant≈ 1.90216 05831 04 where the sum ranges over all primes p such that p + 2 is also a prime1919 ???Twin primes constant0.66016 18158 46869 57392 1922???Plastic ratio1.32471 79572 44746 02596 Real root of 1924 ✗✓✓Bloch's constantThe best known bounds are 1925 ???Z score for the 97.5 percentile point1.95996 39845 40054 23552 where is the inverse error function
Real number such that 1925???Landau's constantThe best known bounds are .1929???Landau's third constant1929???Prouhet–Thue–Morse constant0.41245 40336 40107 59778 where is the nth term of the Thue–Morse sequence1929 ✗✗?Golomb–Dickman constant0.62432 99885 43550 87099 where li(t) is the logarithmic integral, and ρ(t) is the Dickman function1930 & 1964???Constant related to the asymptotic behavior of Lebesgue constants0.98943 12738 31146 95174 1930 ???Feller–Tornier constant0.66131 70494 69622 33528 1932???Base 10 Champernowne constant0.12345 67891 01112 13141 Defined by concatenating representations of successive integers: 0.1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...1933✗✗?Salem constant1.17628 08182 59917 50654 Largest real root of 1933 ✗✓✓Khinchin's constant 1934???Lévy's constant (1)1.18656 91104 15625 45282 1935???Lévy's constant (2)3.27582 29187 21811 15978 1936???Copeland–Erdős constant0.23571 11317 19232 93137 Defined by concatenating representations of successive prime numbers: 0.2 3 5 7 11 13 17 19 23 29 31 37 ...1946 ✗??Mills' constant1.30637 78838 63080 69046 Smallest positive real number A such that is prime for all positive integers n.1947✗??Gompertz constant0.59634 73623 23194 07434 Before 1948 ???de Bruijn–Newman constantThe number such that has real zeros if and only if λ ≥ Λ
where 1950???Van der Pauw constant4.53236 01418 27193 80962 Before 1958 ✗??Magic angle0.95531 66181 245092 78163 Before 1959 ✗✗✓Artin's constant0.37395 58136 19202 28805 Before 1961 ???Porter's constant1.46707 80794 33975 47289 where γ is the Euler–Mascheroni constant and is the derivative of the Riemann zeta function evaluated at s = 21961 ???Lochs constant0.97027 01143 92033 92574 1964???DeVicci's tesseract constant1.00743 47568 84279 37609 Positive root of
The edge length of the largest cube that can pass through a 4D hypercube1966 ✗✓✓Lieb's square ice constant1.53960 07178 39002 03869 1967✗✓✓Niven's constant1.70521 11401 05367 76428 1969???Stephens' constant0.57595 99688 92945 43964 1969 ???Regular paperfolding sequence0.85073 61882 01867 26036 1970 ✗✗?Reciprocal Fibonacci constant3.35988 56662 43177 55317 where Fn is the nth Fibonacci number1974 ✗??Chvátal–Sankoff constant for the binary alphabetwhere is the expected longest common subsequence of two random length-n binary strings1975???Feigenbaum constant δ4.66920 16091 02990 67185 where the sequence an is given by n-th period-doubling bifurcation of logistic map or any other one-dimensional map with a single quadratic maximum1975???Chaitin's constantsIn general they are uncomputables; one such number is 0.00787 49969 97812 3844.where is a halted program, is its size in bits, and is the domain of all programs that stop1975✗✗✗Robbins constant0.66170 71822 67176 23515 1977 ✗✗✓Weierstrass constant0.47494 93799 87920 65033 σ(1|1,i)/2, where σ(z|ω1,ω2) is the Weierstrass sigma function with half-periods ω1 and ω2Before 1978Waldschmidt, M. "Nombres transcendants et fonctions sigma de Weierstrass." C. R. Math. Rep. Acad. Sci. Canada 1, 111-114, 1978/79.✗✗?Fransén–Robinson constant2.80777 02420 28519 36522 1978???Feigenbaum constant α2.50290 78750 95892 82228 Ratio between the width of a tine and the width of one of its two subtines in a bifurcation diagram1979???Beam detection constant3.1415926535 ≤ ≤ 4.7998Length of the shortest opaque set for the unit circleBefore 1980???Second du Bois-Reymond constant0.19452 80494 65325 11361 1983 ✗✗?Erdős–Tenenbaum–Ford constant0.08607 13320 55934 20688 1984???Conway's constant1.30357 72690 34296 39125 Real root of the polynomial1987✗✓✓Hafner–Sarnak–McCurley constant0.35323 63718 54995 98454 1991 ???Backhouse's constant1.45607 49485 82689 67139 where pk is the kth prime number1995???Viswanath constant1.13198 82487 943 where fn = fn−1 ± fn−2, where the signs + or − are chosen at random with equal probability 1/21997???Komornik–Loreti constant1.78723 16501 82965 93301 Real number such that , or where tk is the kth term of the Thue–Morse sequence1998✗✗?Natural density constant of abundant numbers0.2476171 < < 0.2476475 where is the number of abundant numbers less than or equal to nBefore 1998???Embree–Trefethen constant0.70258The smallest value β such that solutions to the random sequence do not decay 1999???Heath-Brown–Moroz constant0.00131 76411 54853 17810 1999 ???MRB constant0.18785 96424 62067 12024 MRB constant1999???Prime constant0.41468 25098 51111 66024 1999 ✗??Somos' quadratic recurrence constant1.66168 79496 33594 12129 1999 ???Foias constant1.18745 23511 26501 05459 The unique real number α such that if x1 = α then the sequence diverges to infinity2000???Logarithmic capacity of the unit disk0.59017 02995 08048 11302 where is the lemniscate constantBefore 2003 ✗✗?Taniguchi constant0.67823 44919 17391 97803Before 2005???
Mathematical constants sorted by their representations as continued fractions The following list includes the continued fractions of some constants and is sorted by their representations. 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.