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Constant

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Constants are real numbers or numerical values which are significantly interesting in some way^[1]. Constants recur in both mathematics and science (though especially in physics), therefore playing a fundamental role. They may represent universal facts -also called invariants-, link physical quantities within physical theories, check the overall consistency of physical models, simplify notation, or simply be ubiquitous. Constants are very versatile entities as they may be exact or approximate and experimentally determined, dimensionless or dimensionful with appropriate units of measurement applied, and even unspecified, when occurring for example in integrals.

When dealing with dimensionfull constants, a set of units must be chosen. Sometimes, one unit is defined in terms of another unit. For example, the metre is defined as $1/(299\ 792\ 458)$ of a light-second. This definition implies that, in metric units, the speed of light in vacuum is exactly $299\ 792\ 458$ metres per second^[2]. No increase in the precision of the measurement of the speed of light could alter this numerical value expressed in metres per second.

One always talks about definable, and almost always also, computable mathematical constants - Chaitin's constant being a notable exception. However, there are still some constants -mathematical or not-, like Landau's constants^[3], Erdos' reciprocal sum constants^[4] or the Fermi coupling constant for which only very rough estimates are known.

As noted by Steven Finch, "All numbers are not created equal." ^[5] We shall therefore make the following classification: 'super-universal' constants -those appearing in different parts of science-, universal constants in a certain domain, and the more specific constants: mathematical curiosities, specific physical facts and unspecified constants.

1 'Super-universal' constants - occurring in different parts of science
2 Universal constants - occurring in general mathematical settings or physical phenomena
3 Mathematical curiosities, specific physical facts and unspecified constants
4 Notation
5 Mystery and aesthetics behind constants
6 See also
7 References
8 Further reading
9 External links

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, Euclidian geometry, analysis and logistic maps respectively. However, mathematical constants such as Apéry's constant and the Golden ratio occur unexpectedly outside of mathematics.

The circumference of a circle with diameter 1 is

π

Pi, though having a natural definition in Euclidian geometry (the circumference of a circle of diameter 1), may be found in many different places in mathematics. 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, $F = \frac{1}{4\pi\varepsilon_0}\frac{\left|q_1 q_2\right|}{r^2}$ ^[6].

Exponential growth (green) describes many physical phenomena.

The expansion of the universe is described using exponentials.

The exponential growth constant 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. $e\,$ 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^[7]. 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 $p_n = 1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots+(-1)^n\frac{1}{n!}$ and as the number $n\,$ of guests tends to infinity, $p_n\,$ approaches $1/e\,$ .

Bifurcation diagram of the logistic map.

Iterations of continuous maps serve as the simplest examples of models for dynamical systems.^[8] 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^[9] 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 non-linear dynamical equations. The map was popularized in a seminal 1976 paper by the English biologist Robert May^[10], in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst. The difference equation is intended to capture the two effects or 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 third-order terms of the electron's gyromagnetic ratio, computed using quantum electrodynamics^[11]. Also, Pascal Wallish noted that $\sqrt{m_n/m_e}\approxeq\frac{3}{\sqrt{\varphi}-\zeta(3)}$ ^[12], where $m_n,m_e,\varphi$ are the neutron mass, the electron mass and the Golden ratio respectively.

Golden rectangles in an icosahedron

$F\left(n\right)=\frac{\varphi^n-(1-\varphi)^n}{\sqrt 5}$

An explicit formula for the n^th Fibonacci number involving the Golden ratio.

The number $\varphi$ turns up frequently in geometry, particularly in figures with pentagonal symmetry. Indeed, the length of a regular pentagon's diagonal is $\varphi$ 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^[13].

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.^[14] Zeising wrote in 1854:

[The Golden Ratio is a universal law] in which is contained the ground-principle 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.^[15]

In mathematics, these constants are invariants of wide classes of objects. For example, $π$ is the invariant ratio of any circle's circumference to its diameter^[16]. It is in specific mathematical settings that such natural definitions arise. For example, Catalan's constant in combinatorics, Euler-Mascheroni constant in analysis, etc. In physics, universal constants appear in the basic theoretical equations upon which the entire science rests or are the properties of the fundamental particles of physics of which all matter is constituted (the electron charge $e$ , the electron mass $m e$ and the fine-structure constant $α$ ).

The area between the two curves (red) tends to a limit.

The Euler-Mascheroni constant is a recurring constant in number theory. The French mathematician De la Vallée Poussin 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^[17]. The Euler-Mascheroni 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 Euler-Mascheroni constant exhibits a close link between the discrete and the continuous (see curves on the right).

$\begin{matrix} 1 \\ 11 \\ 21 \\ 1211 \\ 111221 \\ 312211 \\ \vdots \end{matrix}$

Conway's look-and-say sequence

Conway's constant is the invariant growth rate of all derived strings similar to the look-and-say sequence (except two trivial ones)^[18]. It is given by the unique positive real root of a polynomial of degree 71 with integer coefficients^[19].

If a real number $r\,$ is written using simple continued fraction $r=a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\dots}}}$ , then, as Russian mathematician Khinchin proved in 1934, the limit as $n\,$ tends to infinity of the geometric mean $(a_1a_2\dots a_n)^{1/n}$ exists, and, except for a set of measure 0, this limit is a constant, Khinchin's constant^[20]^[21].

The velocity of light and Planck's constant are examples of quantities that occur naturally in the mathematical formulation of certain fundamental physical theories, the former in James Clerk Maxwell's theory of electric and magnetic fields and Albert Einstein's theories of relativity, and the latter in quantum theory. For example, in special relativity, mass and energy are equivalent: $E=mc^2\,$ ^[22] where $c^2\,$ is the constant of proportionality. In quantum mechanics, the energy and frequency of a photon are related by $E=h\nu\,$ .

The speed of light is also used to express other fundamental constants ^[23] such as the electric constant $\epsilon_0=(4\pi 10^{-7} c^2)^{-1}\,$ , Coulomb's constant $k=10^{-7} c^2\,$ and the characteristic impedance of vacuum $Z_0=4\pi10^{-7}c\,$ .

The electron charge and the electron mass are examples of constants that characterize the basic, or elementary, particles that constitute matter, such as the electron, alpha particle, proton, neutron, muon, and pion^[24]. Many constants can be expressed using the fundamental constants $h,\,c,\,e$ . For example. it is a property of a supercurrent (superconducting electrical current) that the magnetic flux passing through any area bounded by such a current is quantized. The magnetic flux quantum $\Phi_0=hc/(2e)\,$ is a physical constant, as it is independent of the underlying material as long as it is a superconductor. Also, the fundamental fine-structure constant $\alpha=\mu_0ce^2/(2h)\,$ where the permeability of free space $μ 0$ is just a numerical constant equal to $4\pi\times 10^{-7}$ .

This Babylonian clay tablet gives an approximation of in four sexagesimal figures, which is about six decimal figures .

This Babylonian clay tablet gives an approximation of $\sqrt{2}$ in four sexagesimal figures, which is about six decimal figures ^[25].

$c=\sum_{j=1}^\infty 10^{-j!}=0.\underbrace{\overbrace{110001}^{3!\,digits}000000000000000001}_{4!\,digits}000\dots\,$

Liouville's constant is a simple example of a transcendental number.

Some constants, such as the square root of two, Liouville's constant and Champernowne constant $C_{10} = \color{black}0.\color{blue}1\color{black}2\color{blue}3\color{black}4\color{blue}5\color{black}6\color{blue}7\color{black}8\color{blue}9\color{black}10\color{blue}11\color{black}12\color{blue}13\color{black}14\color{blue}15\color{black}16\dots$ are not important mathematical invariants but retain interest being simple representatives of special sets of numbers, the irrational numbers^[26], the transcendental numbers^[27] and the normal numbers (in base 10)^[28] respectively. The discovery of the irrational numbers is usually attributed to the Pythagorean Hippasus of Metapontum who proved, most likely geometrically, the irrationality of $\sqrt{2}$ and Liouville's constant, named after French mathematician Joseph Liouville, was the first transcendental number ever constructed^[29].

In the computer science subfield of algorithmic information theory, Chaitin's constant is the real number representing the probability that a randomly-chosen Turing machine will halt, formed from a construction due to Argentine-American mathematician and computer scientist Gregory Chaitin. Amusingly, Chaitin's constant, though not being computable, it has been proven transcendental and normal.

Physical properties
Density (near r.t.)	19.3 g·cm⁻³
Liquid density at m.p.	17.31 g·cm⁻³
Melting point	1337.33 K (1064.18 °C, 1947.52 °F)
Boiling point	3129 K (2856 °C, 5173 °F)
Heat of fusion	12.55 kJ·mol⁻¹
Heat of vaporization	324 kJ·mol⁻¹
Heat capacity	(25 °C) 25.418 J·mol⁻¹·K⁻¹
References

Such constants represents characteristics of certain physical objects such as the chemical elements. Examples include density, melting point and heat of fusion. Some of the properties of gold are listed in the box on the right.

When unspecified, constants indicate classes of similar objects, commonly functions, all equal up to a constant - or, technically speaking, 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.

Solutions with different constants of integration of $y'(x)=-2y+e^{-x}\,$ .

Indefinite integrals are called indefinite because their solutions are only unique only up to a constant. For example, when working over the field of real numbers $\int\cos x\ dx=\sin x+C$ where $C\,$ , the constant of integration, is an arbitrary fixed real number^[30]. 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 $y'(x)=y(x)\,$ 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 $\frac{\partial f(x,y)}{\partial x}=0$ has solutions $f(x,y)=C(y)\,$ where $C(y)\,$ is an arbitrary function in the variable $y\,$ .

Different symbols are used to represent and manipulate constants, such as $1\,$ , $\pi\,$ and $\epsilon_0\,$ . It is common, in both mathematics and physics, to express the numerical value of a constant by giving its decimal expansion. For two reasons this representation may cause problems. First, even though rational numbers all have a finite or ever-repeating decimal expansion, some numbers don't have such an expression and it would hence be impossible to completely give its value using this method. Also, the decimal expansion of a number is not necessarily unique. For example, the numbers 0.999... and 1 are equivalent^[31]^[32], that is they represent the same constant.

Note that finding the decimal expansions of constants has often been of interest to many people. Some of the mathematical constant, such as $\pi,\,e,\,\sqrt{2}$ have over $10^{11}\,$ (one hundred billion) known digits and very fast algorithms, as for Apéry's constant, have being constructed. In physics, it is important to know the numerical values of the fundamental constants with high accuracy for at least two reasons, the first to achieve accurate quantitative descriptions of the physical universe and the second to test the overall consistency and correctness of the basic theories of physics.

$G=\left . \begin{matrix} 3 \underbrace{ \uparrow \ldots \uparrow } 3 \\ \underbrace{\vdots } \\ 3 \uparrow\uparrow\uparrow\uparrow 3 \end{matrix} \right \} \text{64 layers}$

Graham's number defined using up-arrow notation.

It happens that some constants differ so much from the usual kind that a special notation must be invented to represent them. Graham's number is such an example when the Knuth up-arrow notation is used^[33]^[34].

Commonly, constants in the physical sciences are represented using the scientific notation, with, when appropriate, the inaccuracy - or measurement error - attached. When writing Planck's constant $h=6.626\ 068\ 96(33) \times 10^{-34}\ \mbox{J}\cdot\mbox{s}$ ^[35] it is meant that $h=(6.626\ 068\ 96 \plusmn 0.000\ 000\ 003\ 3)\times 10^{-34}\ \mbox{J}\cdot\mbox{s}\,$ . Only the significant figures are shown and a greater precision would be superfluous, extra figures coming from experimental inaccuracies. When writing Isaac Newton's gravitational constant $G = \left(6.67428 \plusmn 0.00067 \right) \times 10^{-11} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2} \,$ ^[36] only 6 significant figures are given.

For mathematical constants, 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 well-known operations that lend themselves readily to calculation. However, Grossman's constant has no known analytic form^[37].

Symbolizing constants with letters is a frequent means of making the notation more concise. A standard convention, instigated by Euler in the 18th century, is to use lowercase letters from the beginning of the Latin alphabet $a,b,c,\dots\,$ or the Greek alphabet $\alpha,\beta,\,\gamma,\dots\,$ when dealing with constants in general.

Erdős–Borwein constant $E_B\,$
Embree-Trefethen constant $\beta*\,$
Brun's constant for twin prime $B_2\,$
Rydberg constant $R_\infty$
cardinal number aleph naught $\aleph_0$

Different kinds of notation.

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^[34].

$googol=10^{100}\,\ ,\ googolplex=10^{googol}=10^{10^{100}}\,$

Sometimes, the symbol representing a constant is a whole word. For example, American mathematician Edward Kasner's 9-year-old nephew coined the names googol and googolplex^[38]^[34]

The parabolic constant is the ratio of the arc length of the parabolic segment formed by the latus rectum (red) to its focal parameter (green).

The names are either related to the meaning of the constant (parabolic constant, characteristic impedance of vacuum, twin prime constant, electric constant, conductance quantum, ...), to a specific person (Planck's constant, Sierpiński's constant, Dirac's constant, Josephson constant, ...) or both (Newtonian constant of gravitation, Bohr magneton, Fermi coupling constant^[39],...).

A common practice in physics is to lump constants, to simplify the equations and algebraic manipulations. For example, Coulomb's constant $\kappa =(4\pi\epsilon_0)^{-1}\,$ ^[40] is just $\epsilon_0\,$ , $\pi\,$ and $4\,$ lumped together. Also, combining old constants does not necessarily make the new one less fundamental. For example, the -notably- dimensionless fine-structure constant $\alpha=\mu_0ce^2/(2h)\,$ is a fundamental constant of quantum electrodynamics and in the quantum theory of the interaction among electrons, muons and photons.

The Avogadro constant is the number of entities in one mole, commonly used in chemistry, where the entities are often atoms or molecules. Its unit is inverse mole. However, the mole being a counting unit, we can consider the Avogadro constant dimensionless, and, contrary to the speed of light, the Avogadro constant doesn't convert units, but acts as a scaling factor for dealing practically with large numbers.

Srinivasa Ramanujan

The Parthenon's facade showing an interpretation of golden rectangles in its proportions.

$e^{i\pi}+1=0\,$

Euler's identity relating five of the most important mathematical constants.

For some authors, constants, either mathematical or physical may be mysterious, beautiful or fascinating. For example, English mathematician Glaisher (1915) writes ^[1]: "No doubt the desire to obtain the values of these quantities to a great many figures is also partly due to the fact that most of them are interesting in themselves; for $e,\,\pi,\,\gamma,\,\log2$ , and many other numerical quantities occupy a curious, and some of them almost a mysterious, place in mathematics, so that there is a natural tendency to do all that can be done towards their precise determination".

Indian mathematician Srinivasa Ramanujan discovered the following mysterious identity containing pi and Pythogoras' constant $\sqrt{2}$ : $\frac{1}{\pi}=\frac{2\sqrt{2}}{9801} \sum^\infty_{k=0}\frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}$ .

Steven Finch writes that "The fact that certain constants appear at all and then echo throughout mathematics, in seemingly independent ways, is a source of fascination."^[41]

During the 1920s until his death, British astrophysicist Eddington increasingly concentrated on what he called "fundamental theory" which was intended to be a unification of quantum theory, relativity and gravitation. At first he progressed along "traditional" lines, but turned increasingly to an almost numerological analysis of the dimensionless ratios of fundamental constants. In a similar fashion, British theoretical physicist Paul Dirac studied ratios of fundamental physical constant to build his large numbers hypothesis.