A mathematical constant is a special number, usually a real number, that arises naturally in mathematics. Unlike physical constants, mathematical constants are defined independently of physical measurement.
Some mathematical constants, such as e and π, arise in many different contexts. Others, such as Graham's number or Skewes' number, only arise in a single specific context, but are notable because they are the earliest found, largest or smallest exemplar of a class of numbers. Many of the more interesting mathematical constants have a name, also when they can easily be specified by a short formula. What it means for a constant to arise "naturally", and what makes a constant "interesting", is ultimately a matter of taste, and some mathematical constants are notable more for historical reasons than for their intrinsic mathematical interest.
Mathematical constants are always definable numbers and are almost always also computable numbers (Chaitin's constant being a significant exception). However, computable constants need not be easily computed; the De BruijnNewman constant, for example, has no known digits of its decimal expansion.
Constants may be sorted by size but alternate classifications are used, such as using continued fractions.
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Ubiquitous in many different fields of science, such recurring constants include π, e and the Feigenbaum constants which are linked to the mathematical models used to describe physical phenomena, Euclidean geometry, analysis and logistic maps respectively. However, mathematical constants such as Apéry's constant and the Golden ratio occur unexpectedly outside of mathematics.
Pi, though having a natural definition in Euclidean geometry (the circumference of a circle of diameter 1), may be found in many different places in mathematics. Pi is also the circumference of any given circle divided by its diameter. Key examples include the Gaussian integral in complex analysis, n^{th} roots of unity in number theory and Cauchy distributions in probability. However, its universality is not limited to mathematics. Indeed, various formulas in physics, such as Heisenberg's uncertainty principle, and constants such as the cosmological constant bear the constant pi. The presence of pi in physical principles, laws and formulas can have very simple explanations. For example, Coulomb's law, describing the inverse square proportionality of the magnitude of the electrostatic force between two electric charges and their distance, states that, in SI units,
The exponential growth constant e appears in many parts of applied mathematics. For example, as the Swiss mathematician Jacob Bernoulli discovered, e arises in compound interest. Indeed, an account that starts at $1, and yields 1+R dollars at simple interest, will yield e^{R} dollars with continuous compounding. The constant e also has applications to probability theory, where it arises in a way not obviously related to exponential growth. Suppose that a gambler plays a slot machine with a one in n probability and plays it n times. Then, for large n (such as a million) the probability that the gambler will win nothing at all is (approximately) 1/e.
Another application of e, also discovered in part by Jacob Bernoulli along with French mathematician Pierre Raymond de Montmort is in the problem of derangements, also known as the hat check problem.^{[2]} Here n guests are invited to a party, and at the door each guest checks his hat with the butler who then places them into labelled boxes. But the butler does not know the name of the guests, and so must put them into boxes selected at random. The problem of de Montmort is: what is the probability that none of the hats gets put into the right box. The answer is
and as n tends to infinity, p_{n} approaches 1/e.
Iterations of continuous maps serve as the simplest examples of models for dynamical systems.^{[3]} Named after mathematical physicist Mitchell Feigenbaum, the two Feigenbaum constants appear in such iterative processes: they are mathematical invariants of logistic maps with quadratic maximum points^{[4]} and their bifurcation diagrams.
The logistic map is a polynomial mapping, often cited as an archetypal example of how chaotic behaviour can arise from very simple nonlinear dynamical equations. The map was popularized in a seminal 1976 paper by the English biologist Robert May,^{[5]} in part as a discretetime demographic model analogous to the logistic equation first created by Pierre François Verhulst. The difference equation is intended to capture the two effects of reproduction and starvation.
Despite being a special value of the Riemann zeta function, Apéry's constant arises naturally in a number of physical problems, including in the second and thirdorder terms of the electron's gyromagnetic ratio, computed using quantum electrodynamics.^{[6]} Also, the neuroscientist David Eagleman noted that
where are the neutron mass, the electron mass and the Golden ratio respectively.
The number turns up frequently in geometry, particularly in figures with pentagonal symmetry. Indeed, the length of a regular pentagon's diagonal is times its side. The vertices of a regular icosahedron are those of three mutually orthogonal golden rectangles. Also, it appears in the Fibonacci sequence, related to growth by recursion.^{[8]} It is approximately equal to 1.61803398874, or, more precisely
Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavours. In these phenomena he saw the golden ratio operating as a universal law.^{[9]} Zeising wrote in 1854:
[The Golden Ratio is a universal law] in which is contained the groundprinciple of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.^{[10]}
The Euler–Mascheroni constant is a recurring constant in number theory. The French mathematician Charles Jean de la ValléePoussin proved in 1898 that when taking any positive integer n and dividing it by each positive integer m less than n, the average fraction by which the quotient n/m falls short of the next integer tends to γ as n tends to infinity. Surprisingly, this average doesn't tend to one half. The EulerMascheroni constant also appears in Merten's third theorem and has relations to the gamma function, the zeta function and many different integrals and series. The definition of the EulerMascheroni constant exhibits a close link between the discrete and the continuous (see curves on the left).
Conway's constant is the invariant growth rate of all derived strings similar to the lookandsay sequence (except two trivial ones).^{[11]} It is given by the unique positive real root of a polynomial of degree 71 with integer coefficients.^{[12]}
If a real number r is written as a simple continued fraction:
then, as Russian mathematician Aleksandr Khinchin proved in 1934, the limit as n tends to infinity of the geometric mean: (a_{1}a_{2}...a_{n})^{1/n} exists, and, except for a set of measure 0, this limit is a constant, Khinchin's constant.^{[13]}^{[14]}
Some constants, such as the square root of 2, Liouville's constant and Champernowne constant:
are not important mathematical invariants but retain interest being simple representatives of special sets of numbers, the irrational numbers,^{[16]} the transcendental numbers^{[17]} and the normal numbers (in base 10)^{[18]} respectively. The discovery of the irrational numbers is usually attributed to the Pythagorean Hippasus of Metapontum who proved, most likely geometrically, the irrationality of the square root of 2. As for Liouville's constant, named after French mathematician Joseph Liouville, it was the first number to be proven transcendental.^{[19]}
In the computer science subfield of algorithmic information theory, Chaitin's constant is the real number representing the probability that a randomlychosen Turing machine will halt, formed from a construction due to ArgentineAmerican mathematician and computer scientist Gregory Chaitin. Chaitin's constant, though not being computable, has been proven to be transcendental and normal. Chaitin's constant is not universal, depending heavily on the numerical encoding used for Turing machines; however, its interesting properties are independent of the encoding.
When unspecified, constants indicate classes of similar objects, commonly functions, all equal up to a constant—technically speaking, this is may be viewed as 'similarity up to a constant'. Such constants appear frequently when dealing with integrals and differential equations. Though unspecified, they have a specific value, which often isn't important.
Indefinite integrals are called indefinite because their solutions are only unique up to a constant. For example, when working over the field of real numbers
where C, the constant of integration, is an arbitrary fixed real number.^{[20]} In other words, whatever the value of C, differentiating sin x + C with respect to x always yields cos x.
In a similar fashion, constants appear in the solutions to differential equations where not enough initial values or boundary conditions are given. For example, the ordinary differential equation y^{'} = y(x) has solution Ce^{x} where C is an arbitrary constant.
When dealing with partial differential equations, the constants may be functions, constant with respect to some variables (but not necessarily all of them). For example, the PDE
has solutions f(x,y)=C(y), where C(y) is an arbitrary function in the variable y.
It is common to express the numerical value of a constant by giving its decimal representation (or just the first few digits of it). For two reasons this representation may cause problems. First, even though rational numbers all have a finite or everrepeating decimal expansion, irrational numbers don't have such an expression making them impossible to completely describe in this manner. Also, the decimal expansion of a number is not necessarily unique. For example, the two representations 0.999... and 1 are equivalent^{[21]}^{[22]} in the sense that they represent the same number.
Calculating digits of the decimal expansion of constants has been a common enterprise for many centuries. For example, German mathematician Ludolph van Ceulen of the 16th century spent a major part of his life calculating the first 35 digits of pi.^{[23]} Using computers and supercomputers, some of the mathematical constants, including π, e, and the square root of 2, have been computed to more than one hundred billion digits. Fast algorithms have been developed, some of which — as for Apéry's constant — are unexpectedly fast.
Some constants differ so much from the usual kind that a new notation has been invented to represent them reasonably. Graham's number illustrates this as Knuth's uparrow notation is used.^{[24]}^{[25]}
It may be of interest to represent them using continued fractions to perform various studies, including statistical analysis. Many mathematical constants have an analytic form, that is they can constructed using wellknown operations that lend themselves readily to calculation. Not all constants have known analytic forms, though; Grossman's constant^{[26]} and Foias' constant^{[27]} are examples.
Symbolizing constants with letters is a frequent means of making the notation more concise. A standard convention, instigated by Leonhard Euler in the 18th century, is to use lower case letters from the beginning of the Latin alphabet or the Greek alphabet when dealing with constants in general.
However, for more important constants, the symbols may be more complex and have an extra letter, an asterisk, a number, a lemniscate or use different alphabets such as Hebrew, Cyrillic or Gothic.^{[25]}
Sometimes, the symbol representing a constant is a whole word. For example, American mathematician Edward Kasner's 9yearold nephew coined the names googol and googolplex.^{[25]}^{[28]}
The names are either related to the meaning of the constant (parabolic constant, twin prime constant, ...) or to a specific person (Sierpiński's constant, Josephson constant, ...).
Abbreviations used:
Symbol  Value  Name  Field  N  First Described  # of Known Digits 

0

= 0  Zero  Gen  R  c. 7th–5th century BCE  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 BCE  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 2  Gen  A  by c. 800 BCE  137,438,953,444 
√3

≈ 1.73205 08075 68877 29352 74463 41505 87236  Theodorus' constant, square root of 3  Gen  A  by c. 800 BCE  
√5

≈ 2.23606 79774 99789 69640 91736 68731 27623  square root of 5  Gen  A  by c. 800 BCE  1,000,000 
γ

≈ 0.57721 56649 01532 86060 65120 90082 40243  EulerMascheroni constant  Gen, NuT  1735  14,922,244,771  
φ

≈ 1.61803 39887 49894 84820 45868 34365 63811  Golden ratio  Gen  A  by 3rd century BCE  100,000,000,000 
ρ

≈ 1.32471 79572 44746 02596 09088 54478 09734  Plastic constant  NuT  A  1928  
β^{*}

≈ 0.70258  EmbreeTrefethen 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  
C_{2}

≈ 0.66016 18158 46869 57392 78121 10014 55577  Twin prime constant  NuT  5,020  
M_{1}

≈ 0.26149 72128 47642 78375 54268 38608 69585  MeisselMertens constant  NuT  1866 1874 
8,010  
B_{2}

≈ 1.90216 05823  Brun's constant for twin primes  NuT  1919  10  
B_{4}

≈ 0.87058 83800  Brun's constant for prime quadruplets  NuT  
Λ

≥ –2.7 • 10^{−9}  de BruijnNewman constant  NuT  1950?  none  
K

≈ 0.91596 55941 77219 01505 46035 14932 38411  Catalan's constant  Com  15,510,000,000  
K

≈ 0.76422 36535 89220 66299 06987 31250 09232  LandauRamanujan 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  RamanujanSoldner constant  NuT  75,500  
E_{B}

≈ 1.60669 51524 15291 76378 33015 23190 92458  Erdős–Borwein constant  NuT  I  
β

≈ 0.28016 94990 23869 13303  Bernstein's constant^{[29]}  Ana  
λ

≈ 0.30366 30028 98732 65859 74481 21901 55623  GaussKuzminWirsing constant  Com  1974  385  
σ

≈ 0.35323 63718 54995 98454  HafnerSarnakMcCurley 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  AlladiGrinstead constant^{[30]}  NuT  
Λ

≈ 1.09868 58055  Lengyel's constant^{[31]}  Com  1992  
≈ 3.27582 29187 21811 15978 76818 82453 84386  Lévy's constant  NuT  
ζ(3)

≈ 1.20205 69031 59594 28539 97381 61511 44999  Apéry's constant  I  1979  15,510,000,000  
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^{[32]}  
≈ 1.46707 80794  Porter's constant^{[33]}  NuT  1975  
≈ 1.53960 07178  Lieb's square ice constant^{[34]}  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  FransénRobinson constant  Ana  
L

≈ 0.5  Landau's constant  Ana  1  
P_{2}

≈ 2.29558 71493 92638 07403 42980 49189 49039  Parabolic constant  Gen  T  
Ω

≈ 0.56714 32904 09783 87299 99686 62210 35555  Omega constant  Ana  T  
≈ 0.187859  MRB constant  Gen  I  3:13GMT 1/11/1999^{[35]}  250,000 ^{[36]} 
A mathematical constant is a number, which has a special meaning for calculations. For example, the constant π (spoken as "pi") means the ratio of the length of a circle's circumference to its diameter. This value is always the same for any circle. In contrast to physical constants, mathematical constants do not come from physical measurements.
Here are some important mathematical constants:
Name  Symbol  Value  Meaning 

Pi, Archimedes' constant or Ludoph's number  π  ≈3.141592653589793  A transcendental number that is the ratio of the length of a circle's circumference to its diameter. It is also the area of the unit circle. 
E, Napier's constant  e  ≈2.718281828459045  A transcendental number that is the base of natural logarithms, sometimes called the "natural number". 
Golden ratio  φ  $\backslash frac\{\backslash sqrt\{5\}+1\}\{2\}\; \backslash approx\; 1.618$  It is the value of a larger value divided by a smaller value if this is equal to the value of the sum of the values divided by the larger value. 
Square root of 2, Pythagora's constant  $\backslash sqrt\{2\}$  $\backslash approx\; 1.414$  An irrational number that is the length of the diagonal of a square with sides of length 1. This number can not be written as a fraction. 
