Mathematical constant

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 Ludophs 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
B´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 constanthttp://mathworld.wolfram.com/BernsteinsConstant.html	Ana
λ	≈ 0.30366 30029	Gauss-Kuzmin-Wirsing constanthttp://mathworld.wolfram.com/Gauss-Kuzmin-WirsingConstant.html	Com		1974	385
D(1)	≈ 0.35323 63719	Hafner-Sarnak-McCurley constanthttp://mathworld.wolfram.com/Hafner-Sarnak-McCurleyConstant.html	NuT		1993
λ μ	≈ 0.62432 99885	Golomb-Dickman constanthttp://mathworld.wolfram.com/Golomb-DickmanConstant.html	Com NuT		1930 1964
	≈ 0.62946 50204	Cahen's constanthttp://www.tcaep.co.uk/science/constant/ctod.htm
	≈ 0.66274 34193	Laplace limithttp://mathworld.wolfram.com/LaplaceLimit.html
	≈ 0.80939 40205	Alladi-Grinstead constanthttp://mathworld.wolfram.com/Alladi-GrinsteadConstant.html	NuT
Λ	≈ 1.09868 58055	Lengyel's constanthttp://mathworld.wolfram.com/LengyelsConstant.html	Com		1992
	≈ 1.18656 91104	Khinchin-Lévy constanthttp://mathworld.wolfram.com/Khinchin-LevyConstant.html	NuT
	≈ 1.20205 69031 59594 28539 97381	Apéry's constanthttp://mathworld.wolfram.com/AperysConstant.html			1979	1,000,000,000
θ	≈ 1.30637 78838 63080 69046	Mills' constanthttp://mathworld.wolfram.com/MillsConstant.html	NuT	?	1947
	≈ 1.45607 49485 82689 67139 95953 51116 54356	Backhouse's constanthttp://mathworld.wolfram.com/BackhousesConstant.html
	≈ 1.46707 80794	Porter's constanthttp://mathworld.wolfram.com/PortersConstant.html	NuT		1975
	≈ 1.53960 07178	Lieb's square ice constanthttp://www.mathsoft.com/mathresources/constants/discretestructures/article/0,,2265,00.html	Com		1967
	≈ 1.70521 11401 05367	Niven's constanthttp://mathworld.wolfram.com/NivensConstant.html	NuT		1969
	≈ 2.58498 17596	Sierpinski's constanthttp://www.tcaep.co.uk/science/constant/detail/sierpinskisconstant.htm
	≈ 2.68545 2001	Khinchin's constanthttp://mathworld.wolfram.com/KhinchinsConstant.html	NuT	?	1934	7350
F	≈ 2.80777 02420	Fransén-Robinson constanthttp://mathworld.wolfram.com/Fransen-RobinsonConstant.html	Ana
L	≈ .5	Landau's constant	Ana			1

See also

• An alternative sorting based on the continued fraction representations
• Physical constants

External links

• Steven Finch's page of mathematical constants: http://pauillac.inria.fr/algo/bsolve/constant/constant.html
• Steven Finch's alternative indexing: http://pauillac.inria.fr/algo/bsolve/constant/table.html
• Xavier Gourdon and Pascal Sebah's page of numbers, mathematical constants and algorithms: http://numbers.computation.free.fr/Constants/constants.html
• Simon Plouffe's inverter: http://pi.lacim.uqam.ca/eng/
• CECM's Inverse symbolic calculator (ISC) (tells you how a given number can be constructed from mathematical constants): http://www.cecm.sfu.ca/projects/ISC/

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