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Encyclopedia > 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. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In physics, a physical constant is a physical quantity of a value that is generally believed to be both universal in nature and not believed to change in time. ...

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. Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... e is the unique number such that the value of the derivative (slope of a tangent line) of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...

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). In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... A real number a is first-order definable in the language of set theory, without parameters, if there is a formula Ï† in the language of set theory, with one free variable, such that a is the unique real number such that Ï†(a) holds (in the von Neumann universe V). ... In mathematics, theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. ...

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). This is a list of mathematical constants sorted by their representations as continued fractions: (Constants known to be irrational have infinite continued fractions: their last term is . ...

## Table of selected mathematical constants GA_googleFillSlot("encyclopedia_square");

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
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 Landau-Ramanujan constant NuT 30,010
K
≈ 1.13198 824 Viswanath's constant NuT 8
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 Erdős–Borwein constant NuT I
β
≈ 0.28016 94990 23869 13303 Bernstein's constant Ana
λ
≈ 0.30366 30029 Gauss-Kuzmin-Wirsing constant Com 1974 385
σ
≈ 0.35323 63718 54995 98454 Hafner-Sarnak-McCurley constant NuT 1993
λ, μ
≈ 0.62432 99885 Golomb-Dickman constant Com, NuT 1930
1964
≈ 0.64341 05463 Cahen's constant T 1891 4000
≈ 0.66274 34193 Laplace limit
Λ
≈ 1.09868 58055 Lengyel's constant Com 1992
≈ 1.18656 91104 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 Niven's constant NuT 1969
K
≈ 2.58498 17596 Sierpiński's constant
≈ 2.68545 20010 65306 44530 Khinchin's constant NuT 1934 7350
F
≈ 2.80777 02420 Fransén-Robinson constant Ana
L
≈ 0.5 Landau's constant Ana 1

This is a list of mathematical constants sorted by their representations as continued fractions: (Constants known to be irrational have infinite continued fractions: their last term is . ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... In physics, a physical constant is a physical quantity of a value that is generally believed to be both universal in nature and not believed to change in time. ... An astronomical constant is a physical constant used in astronomy. ...

Results from FactBites:

 Mathematical constant - Wikipedia, the free encyclopedia (336 words) A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. Therefore, f(1)/f(0) is a mathematical constant, the constant e. Mathematical constants are typically elements of the field of real numbers or complex numbers.
 Encyclopedia: Mathematical constant (3463 words) The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm: Intriguingly, the constant is also given by the integral: Its value is approximately γ ≈ 0. The Meissel-Mertens constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series summed only over the primes and the natural logarithm of the natural logarithm: Its value is approximately M ≈ 0. The SierpiÅ„skis constant is a mathematical constant usually denoted as K. One way of defining it is by limiting the expression: where r2(k) is a number of representations of k as a sum of the form a2 + b2 for natural a and b.
More results at FactBites »

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