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Mathematical constant

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A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. Unlike physical constants, mathematical constants are defined independently of any physical measurement.

For example, up to multiplication with nonzero complex numbers, there is a unique holomorphic function f with f' = f. Therefore, f(1)/f(0) is a mathematical constant, the constant e. f is also a periodic function, and the absolute value of its period is another mathematical constant, 2π.

Mathematical constants are typically elements of the field of real numbers or complex numbers. Mathematical constants that one can talk about are definable numbers (and almost always also computable).

However, there are still some mathematical constants for which only very rough estimates are known.

An alternate sorting may be found at Mathematical constant (sorted by continued fraction representation)

Table of selected mathematical constants

Abbreviations used:

I – irrational number, A – algebraic number, T – transcendental number, ? – unknown
Gen – General, NuT – Number theory, ChT – Chaos theory, Com – Combinatorics, Inf – Information theory, Ana – Mathematical analysis
Symbol Approximate Value Name Field N First described # of known digits
π ≈ 3.14159 26535 89793 23846 26433 83279 50288 Pi, Archimedes' constant or Ludoph's number Gen, Ana T  ? 1,241,100,000,000
e ≈ 2.71828 18284 59045 23536 02874 71352 66249 Napier's constant, base of Natural logarithm Gen, Ana T   50,100,000,000
√2 ≈ 1.41421 35623 73095 04880 16887 24209 69807 Pythagoras' constant, square root of two Gen I,A   137,438,953,444
√3 ≈ 1.73205 08075 68877 29352 74463 41505 Theodorus' constant, square root of three Gen I,A  
γ ≈ 0.57721 56649 01532 86060 65120 90082 40243 Euler-Mascheroni constant Gen, NuT ?   108,000,000
φ ≈ 1.61803 39887 49894 84820 45868 34365 63811 Golden ratio Gen A   3,141,000,000
β* ≈ 0.70258 Embree-Trefethen constant NuT      
δ ≈ 4.66920 16091 02990 67185 32038 20466 20161 Feigenbaum constant ChT      
α ≈ 2.50290 78750 95892 82228 39028 73218 21578 Feigenbaum constant ChT      
C2 ≈ 0.66016 18158 46869 57392 78121 10014 55577 Twin prime constant NuT     5,020
M1 ≈ 0.26149 72128 47642 78375 54268 38608 69585 Meissel-Mertens constant NuT   1866
1874
8,010
B2 ≈ 1.90216 05823 Brun's constant for twin prime NuT   1919 10
B4 ≈ 0.87058 83800 Brun's constant for prime quadruplets NuT      
Λ > – 2.7 · 10-9 de Bruijn-Newman constant NuT   1950?  
K ≈ 0.91596 55941 77219 01505 46035 14932 38411 Catalan's constant Com     201,000,000
K ≈ 0.76422 36535 89220 66 Landau-Ramanujan constant NuT I (?)   30,010
K ≈ 1.13198 824 Viswanath's constant 1 NuT     8
L ≈ 1.08366 Legendre's constant NuT      
μ ≈ 1.45136 92348 83381 05028 39684 85892 027 Ramanujan-Soldner constant NuT     75,500
EB ≈ 1.60669 51524 15291 763 Erdös-Borwein constant NuT I    
Ω  ? Chaitin's constant Inf T    
β ≈ 0.28016 94990 Bernstein's constant[1] Ana      
λ ≈ 0.30366 30029 Gauss-Kuzmin-Wirsing constant[2] Com   1974 385
D(1) ≈ 0.35323 63719 Hafner-Sarnak-McCurley constant[3] NuT   1993  
λ μ ≈ 0.62432 99885 Golomb-Dickman constant[4] Com NuT   1930 1964  
  ≈ 0.62946 50204 Cahen's constant[5]        
  ≈ 0.66274 34193 Laplace limit[6]        
  ≈ 0.80939 40205 Alladi-Grinstead constant[7] NuT      
Λ ≈ 1.09868 58055 Lengyel's constant[8] Com   1992  
  ≈ 1.18656 91104 Khinchin-Lévy constant[9] NuT      
  ≈ 1.20205 69031 59594 28539 97381 Apéry's constant[10]     1979 1,000,000,000
θ ≈ 1.30637 78838 63080 69046 Mills' constant[11] NuT ? 1947  
  ≈ 1.45607 49485 82689 67139 95953 51116 54356 Backhouse's constant[12]        
  ≈ 1.46707 80794 Porter's constant[13] NuT   1975  
  ≈ 1.53960 07178 Lieb's square ice constant[14] Com   1967  
  ≈ 1.70521 11401 05367 Niven's constant[15] NuT   1969  
  ≈ 2.58498 17596 Sierpinski's constant[16]        
  ≈ 2.68545 2001 Khinchin's constant[17] NuT ? 1934 7350
F ≈ 2.80777 02420 Fransén-Robinson constant[18] Ana      
L ≈ .5 Landau's constant Ana     1

See also

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