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.
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 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 constants (sorted by continued fraction representation).
Table of selected mathematical constants
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
|
=  |
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 | ||
|
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 | 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 | 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 | 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 | Mills' constant | NuT | 1947 | ||
| ≈ 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 | ||
|
P2
|
≈ 2.29558 71493 92638 07403 42980 | Universal parabolic constant | Gen | T |
External links
- Steven Finch's page of mathematical constants: http://pauillac.inria.fr/algo/bsolve/
- 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://oldweb.cecm.sfu.ca/projects/ISC/
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