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. 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 Set
One 1 1 Prehistory
Two 2 2 Prehistory
One half 1/2 0.5 Prehistory
Pi 3.14159 26535 89793 23846 Ratio of a circle's circumference to its diameter. 1900 to 1600 BCE
Tau (mathematical constant) 6.28318 53071 79586 47692 Ratio of a circle's circumference to its radius. Equivalent to 1900 to 1600 BCE
Square root of 2,

Pythagoras constant.

1.41421 35623 73095 04880 Positive root of 1800 to 1600 BCE
Square root of 3,

Theodorus' constant

1.73205 08075 68877 29352 Positive root of 465 to 398 BCE
Square root of 5 2.23606 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
Zero 0 0 300 to 100 BCE
Negative one −1 −1 300 to 200 BCE
Cube root of 2 1.25992 10498 94873 16476 Real root of 46 to 120 CE
Cube root of 3 1.44224 95703 07408 38232 Real root of
Twelfth root of 2 1.05946 30943 59295 26456 Real root of
Supergolden ratio 1.46557 12318 76768 02665

Real root of

Imaginary unit 0 + 1i Principal root of 1501 to 1576
Connective constant for the hexagonal lattice 1.84775 90650 22573 51225 , as a root of the polynomial 1593
Kepler–Bouwkamp constant 0.11494 20448 53296 20070 1596
Wallis's constant 2.09455 14815 42326 59148

Real root of

1616 to 1703
Euler's number 2.71828 18284 59045 23536 1618
Natural logarithm of 2 0.69314 71805 59945 30941 Real root of

1619 & 1668
Lemniscate constant 2.62205 75542 92119 81046

where is Gauss's constant

1718 to 1798
Euler's constant 0.57721 56649 01532 86060 1735
Erdős–Borwein constant 1.60669 51524 15291 76378 1749
Omega constant 0.56714 32904 09783 87299

where W is the Lambert W function

1758 & 1783
Apéry's constant 1.20205 69031 59594 28539 1780
Laplace limit 0.66274 34193 49181 58097 Real root of ~1782
Ramanujan–Soldner constant 1.45136 92348 83381 05028 ; root of the logarithmic integral function. 1792
Gauss's constant 0.83462 68416 74073 18628

where agm is the arithmetic–geometric mean

1799
Second Hermite constant 1.15470 05383 79251 52901 1822 to 1901
Liouville's constant 0.11000 10000 00000 00000 0001 Before 1844
First continued fraction constant 0.69777 46579 64007 98201

, where is the modified Bessel function

1855
Ramanujan's constant 262 53741 26407 68743
.99999 99999 99250 073 		1859
Glaisher–Kinkelin constant 1.28242 71291 00622 63687 1860
Catalan's constant 0.91596 55941 77219 01505 1864
Dottie number 0.73908 51332 15160 64165 Real root of 1865
Meissel–Mertens constant 0.26149 72128 47642 78375

where γ is the Euler–Mascheroni constant and p is prime

1866 & 1873
Universal parabolic constant 2.29558 71493 92638 07403 Before 1891
Cahen's constant 0.64341 05462 88338 02618

where sk is the kth term of Sylvester's sequence 2, 3, 7, 43, 1807, ...

1891
Gelfond's constant 23.14069 26327 79269 0057 1900
Gelfond–Schneider constant 2.66514 41426 90225 18865 Before 1902
Second Favard constant 1.23370 05501 36169 82735 1902 to 1965
Golden angle 2.39996 32297 28653 32223 or

in degrees

1907
Sierpiński's constant 2.58498 17595 79253 21706 1907
Landau–Ramanujan constant 0.76422 36535 89220 66299 1908
First Nielsen–Ramanujan constant 0.82246 70334 24113 21823 1909
Gieseking constant 1.01494 16064 09653 62502

.

1912
Bernstein's constant 0.28016 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 constant 1.83928 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 prime

1919
Twin primes constant 0.66016 18158 46869 57392 1922
Plastic ratio 1.32471 79572 44746 02596

Real root of

1924
Bloch's constant The best known bounds are 1925
Z score for the 97.5 percentile point 1.95996 39845 40054 23552 where erf−1(x) is the inverse error function

Real number such that

1925
Landau's constant The best known bounds are 1929
Landau's third constant 1929
Prouhet–Thue–Morse constant 0.41245 40336 40107 59778

where is the nth term of the Thue–Morse sequence

1929
Golomb–Dickman constant 0.62432 99885 43550 87099

where Li(t) is the logarithmic integral, and ρ(t) is the Dickman function

1930 & 1964
Constant related to the asymptotic behavior of Lebesgue constants 0.98943 12738 31146 95174 1930
Feller–Tornier constant 0.66131 70494 69622 33528 1932
Base 10 Champernowne constant 0.12345 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 constant 1.17628 08182 59917 50654 Largest real root of 1933
Khinchin's constant 2.68545 20010 65306 44530  1934
Lévy's constant (1) 1.18656 91104 15625 45282 1935
Lévy's constant (2) 3.27582 29187 21811 15978 1936
Copeland–Erdős constant 0.23571 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' constant 1.30637 78838 63080 69046 Smallest positive real number A such that is prime for all positive integers n 1947
Gompertz constant 0.59634 73623 23194 07434 Before 1948
de Bruijn–Newman constant The number Λ where for where has real zeros if and only if λ ≥ Λ.

where .

1950
Van der Pauw constant 4.53236 01418 27193 80962 Before 1958
Magic angle 0.95531 66181 245092 78163 Before 1959
Artin's constant 0.37395 58136 19202 28805 Before 1961
Porter's constant 1.46707 80794 33975 47289

where γ is the Euler–Mascheroni constant and ζ '(2) is the derivative of the Riemann zeta function evaluated at s = 2

1961
Lochs constant 0.97027 01143 92033 92574 1964
DeVicci's tesseract constant 1.00743 47568 84279 37609 The largest cube that can pass through in an 4D hypercube.

Positive root of

1966
Lieb's square ice constant 1.53960 07178 39002 03869 1967
Niven's constant 1.70521 11401 05367 76428 1969
Stephens' constant 0.57595 99688 92945 43964 1969
Regular paperfolding sequence 0.85073 61882 01867 26036 1970
Reciprocal Fibonacci constant 3.35988 56662 43177 55317

where Fn is the nth Fibonacci number

1974
Chvátal–Sankoff constant for the binary alphabet

where E[λn,2] is the expected longest common subsequence of two random length-n binary strings

1975
Feigenbaum constant δ 4.66920 16091 02990 67185

where the sequence xn is given by

1975
Chaitin's constants In general they are uncomputable numbers.
But one such number is 0.00787 49969 97812 3844.
  • p: Halted program
  • |p|: Size in bits of program p
  • P: Domain of all programs that stop.
See also: Halting problem
1975
Robbins constant 0.66170 71822 67176 23515 1977
Weierstrass constant 0.47494 93799 87920 65033 Before 1978
Fransén–Robinson constant 2.80777 02420 28519 36522 1978
Feigenbaum constant α 2.50290 78750 95892 82228 Ratio between the width of a tine and the width of one of its two subtines in a bifurcation diagram 1979
Second du Bois-Reymond constant 0.19452 80494 65325 11361 1983
Erdős–Tenenbaum–Ford constant 0.08607 13320 55934 20688 1984
Conway's constant 1.30357 72690 34296 39125 Real root of the polynomial:

1987
Hafner–Sarnak–McCurley constant 0.35323 63718 54995 98454 1991
Backhouse's constant 1.45607 49485 82689 67139

where pk is the kth prime number

1995
Viswanath constant 1.13198 82487 943      where fn = fn−1 ± fn−2, where the signs + or − are chosen at random with equal probability 1/2 1997
Komornik–Loreti constant 1.78723 16501 82965 93301 Real number such that , or

where tk is the kth term of the Thue–Morse sequence

1998
Embree–Trefethen constant 0.70258 1999
Heath-Brown–Moroz constant 0.00131 76411 54853 17810 1999
MRB constant 0.18785 96424 62067 12024 1999
Prime constant 0.41468 25098 51111 66024 1999
Somos' quadratic recurrence constant 1.66168 79496 33594 12129 1999
Foias constant 1.18745 23511 26501 05459

Foias constant is the unique real number such that if x1 = α then the sequence diverges to infinity.

2000
Logarithmic capacity of the unit disk 0.59017 02995 08048 11302 Before 2003
Taniguchi constant 0.67823 44919 17391 97803 Before 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.

Name Symbol Set Decimal expansion Continued fraction Notes
Zero 0 0.00000 00000 [0; ]
Golomb–Dickman constant 0.62432 99885 [0; 1, 1, 1, 1, 1, 22, 1, 2, 3, 1, 1, 11, 1, 1, 2, 22, 2, 6, 1, 1, …] E. Weisstein noted that the continued fraction has an unusually large number of 1s.
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. Davison and Shallit used the continued fraction expansion to prove that the constant is transcendental.
Euler–Mascheroni constant 0.57721 56649 [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, 1, …] Using the continued fraction expansion, it was shown that if γ is rational, its denominator must exceed 10244663.
First 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 the ratio of modified Bessel functions of the first kind evaluated at 2.
Catalan's constant 0.91596 55942 [0; 1, 10, 1, 8, 1, 88, 4, 1, 1, 7, 22, 1, 2, 3, 26, 1, 11, 1, 10, 1, …] Computed up to 4851389025 terms by E. Weisstein.
One half 1/2 0.50000 00000 [0; 2]
Prouhet–Thue–Morse constant 0.41245 40336 [0; 2, 2, 2, 1, 4, 3, 5, 2, 1, 4, 2, 1, 5, 44, 1, 4, 1, 2, 4, 1, …] Infinitely many partial quotients are 4 or 5, and infinitely many partial quotients are greater than or equal to 50.
Copeland–Erdős constant 0.23571 11317 [0; 4, 4, 8, 16, 18, 5, 1, 1, 1, 1, 7, 1, 1, 6, 2, 9, 58, 1, 3, 4, …] Computed up to 1011597392 terms by E. Weisstein. He also noted that while the Champernowne constant continued fraction contains sporadic large terms, the continued fraction of the Copeland–Erdős Constant do not exhibit this property.
Base 10 Champernowne constant 0.12345 67891 [0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, 4.57540×10165, 6, 1, …] Champernowne constants in any base exhibit sporadic large numbers; the 40th term in has 2504 digits.
One 1 1.00000 00000 [1; ]
Phi, Golden ratio 1.61803 39887 [1; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …] The convergents are ratios of successive Fibonacci numbers.
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, …] The nth roots of the denominators of the nth convergents are close to Khinchin's constant, suggesting that is irrational. If true, this will prove the twin prime conjecture.
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, …] The convergents are ratios of successive Pell numbers.
Two 2 2.00000 00000 [2; ]
Euler's number 2.71828 18285 [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, …] The continued fraction expansion has the pattern [2; 1, 2, 1, 1, 4, 1, ..., 1, 2n, 1, ...].
Khinchin's constant 2.68545 20011 [2; 1, 2, 5, 1, 1, 2, 1, 1, 3, 10, 2, 1, 3, 2, 24, 1, 3, 2, 3, 1, …] For almost all real numbers x, the coefficients of the continued fraction of x have a finite geometric mean known as Khinchin's constant.
Three 3 3.00000 00000 [3; ]
Pi 3.14159 26536 [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, …] The first few convergents (3, 22/7, 333/106, 355/113, ...) are among the best-known and most widely used historical approximations of π.

Sequences of constants

Name Symbol Formula Year Set
Harmonic number Antiquity
Gregory coefficients 1670
Bernoulli number 1689
Hermite constants For a lattice L in Euclidean space Rn with unit covolume, i.e. vol(Rn/L) = 1, let λ1(L) denote the least length of a nonzero element of L. Then √γnn is the maximum of λ1(L) over all such lattices L. 1822 to 1901
Hafner–Sarnak–McCurley constant 1883
Stieltjes constants before 1894
Favard constants 1902 to 1965
Generalized Brun's Constant where the sum ranges over all primes p such that p + n is also a prime 1919
Champernowne constants Defined by concatenating representations of successive integers in base b.

1933
Lagrange number where is the nth smallest number such that has positive (x,y). before 1957
Feller's coin-tossing constants is the smallest positive real root of 1968
Stoneham number where b,c are coprime integers. 1973
Beraha constants 1974
Chvátal–Sankoff constants 1975
Hyperharmonic number and 1995
Gregory number for rational x greater than one. before 1996
Metallic mean before 1998

See also