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 some physical constants, mathematical constants are defined independently of any physical measurement.

Many particular numbers have special significance in mathematics, and arise in many different contexts. 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 that one can talk about are definable numbers, and almost always also computable. A notable non computable constant is Chaitin's constant. However, there are still some mathematical constants for which only very rough estimates are known.

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

Abbreviations used:

R - Rational number, I - Irrational number (may be algebraic or transcendental), A - Algebraic number (irrational), T - Transcendental number (irrational)
Gen - General, NuT - Number theory, ChT - Chaos theory, Com - Combinatorics, Inf - Information theory, Ana - Mathematical analysis
Symbol Value Name Field N First Described # of Known Digits
0
 = 0 Zero Gen R c. 7th-5th century BC N/A
1
 = 1 One, Unity Gen R N/A
i
 = \sqrt{-1} Imaginary unit Gen, Ana A 16th century N/A
π
 ≈ 3.14159 26535 89793 23846 26433 83279 50288 Pi, Archimedes' constant or Ludolph's number Gen, Ana T by c. 2000 BC 1,241,177,300,000
e
 ≈ 2.71828 18284 59045 23536 02874 71352 66249 Napier's constant, or Euler's number, base of Natural logarithm Gen, Ana T 1618 100,000,000,000
√2
 ≈ 1.41421 35623 73095 04880 16887 24209 69807 Pythagoras' constant, square root of two Gen A by c. 800 BC 137,438,953,444
√3
 ≈ 1.73205 08075 68877 29352 74463 41505 87236 Theodorus' constant, square root of three Gen A by c. 800 BC
γ
 ≈ 0.57721 56649 01532 86060 65120 90082 40243 Euler-Mascheroni constant Gen, NuT 1735 116,580,041
φ
 ≈ 1.61803 39887 49894 84820 45868 34365 63811 Golden ratio Gen A by 3rd century BC 3,141,000,000
ρ
 ≈ 1.32471 95724 47460 25960 90885 44780 97340 Plastic constant NuT A 1928
β*
 ≈ 0.70258 Embree-Trefethen constant NuT
δ
 ≈ 4.66920 16091 02990 67185 32038 20466 20161 Feigenbaum constant ChT 1975
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
Sh
 ≈ 0.05 09 89 Hughes constant Shs
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? none
K
 ≈ 0.91596 55941 77219 01505 46035 14932 38411 Catalan's constant Com 201,000,000
K
 ≈ 0.76422 36535 89220 66299 06987 31250 09232 Landau-Ramanujan constant NuT 30,010
K
 ≈ 1.13198 824 Viswanath's constant NuT 8
B´L
 = 1 Legendre's constant NuT R N/A
μ
 ≈ 1.45136 92348 83381 05028 39684 85892 02744 Ramanujan-Soldner constant NuT 75,500
EB
 ≈ 1.60669 51524 15291 76378 33015 23190 92458 Erdős–Borwein constant NuT I
β
 ≈ 0.28016 94990 23869 13303 Bernstein's constant Ana
λ
 ≈ 0.30366 30028 98732 65859 74481 21901 55623 Gauss-Kuzmin-Wirsing constant Com 1974 385
σ
 ≈ 0.35323 63718 54995 98454 Hafner-Sarnak-McCurley constant NuT 1993
λ, μ
 ≈ 0.62432 99885 43550 87099 29363 83100 83724 Golomb-Dickman constant Com, NuT 1930
1964
≈ 0.64341 05463 Cahen's constant T 1891 4000
≈ 0.66274 34193 49181 58097 47420 97109 25290 Laplace limit
≈ 0.80939 40205 Alladi-Grinstead constant NuT
Λ
 ≈ 1.09868 58055 Lengyel's constant Com 1992
≈ 1.18656 91104 15625 45282 17229 75947 23712 Khinchin-Lévy constant NuT
ζ(3)
 ≈ 1.20205 69031 59594 28539 97381 61511 44999 Apéry's constant I 1979 2,000,000,000
θ
 ≈ 1.30637 78838 63080 69046 86144 92602 60571 Mills' constant NuT 1947 6850
≈ 1.45607 49485 82689 67139 95953 51116 54356 Backhouse's constant
≈ 1.46707 80794 Porter's constant NuT 1975
≈ 1.53960 07178 Lieb's square ice constant Com 1967
≈ 1.70521 11401 05367 76428 85514 53434 50816 Niven's constant NuT 1969
K
 ≈ 2.58498 17595 79253 21706 58935 87383 17116 Sierpiński's constant
≈ 2.68545 20010 65306 44530 97148 35481 79569 Khinchin's constant NuT 1934 7350
F
 ≈ 2.80777 02420 28519 36522 15011 86557 77293 Fransén-Robinson constant Ana
L
 ≈ 0.5 Landau's constant Ana 1
J2
 ≈ 1.57891 32471 95724 47460 25960 90885 44780 973402 Jacevicius constant JjC A 1983
P2
 ≈ 2.29558 71493 92638 07403 42980 49189 49039 Universal parabolic constant Gen T
Ω
 ≈ 0.56714 32904 09783 87299 99686 62210 35555 Omega constant Ana T

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