Constants are real numbers or numerical values which are significantly interesting in some way[1]. In Mathematics, the real numbers may be described informally in several different ways The term "constant" is used both for mathematical constants and for physical constants, but with quite different meanings. A mathematical constant is a number usually a Real number, that arises naturally in Mathematics. A physical Constant is a Physical quantity that is generally believed to be both universal in nature and constant in time

One always talks about definable, and almost always also computable, mathematical constants — Chaitin's constant being a notable exception. A Real number a is first-order definable in the language of set theory without parameters, if there is a formula φ in the language of Set theory In Mathematics, Theoretical computer science and Mathematical logic, the computable numbers, also known as the recursive numbers or the In the Computer science subfield of Algorithmic information theory a Chaitin Constant or halting probability is a Real number that However for some computable mathematical constants only very rough numerical estimates are known.

When dealing with physical dimensionful constants, a set of units must be chosen. Sometimes, one unit is defined in terms of other units. For example, the metre is defined as $1/(299\ 792\ 458)$ of a light-second. The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International This definition implies that, in metric units, the speed of light in vacuum is exactly $299\ 792\ 458$ metres per second[2]. The metric system is a decimalised system of measurement. It exists in several variations with different choices of base units, though the choice of base units does No increase in the precision of the measurement of the speed of light could alter this numerical value expressed in metres per second.

## Mathematical constants

Main article: Mathematical constant

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. A mathematical constant is a number usually a Real number, that arises naturally in Mathematics. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line Note The term model has a different meaning in Model theory, a branch of Mathematical logic. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. Analysis has its beginnings in the rigorous formulation of Calculus. The logistic map is a Polynomial mapping of degree 2, often cited as an archetypal example of how complex chaotic behaviour can arise from very simple However, mathematical constants such as Apéry's constant and the Golden ratio occur unexpectedly outside of mathematics. In Mathematics, Apéry's Constant is a curious number that occurs in a variety of situations In Mathematics and the Arts two quantities are in the Golden ratio if the Ratio between the sum of those quantities and the larger one is the

### Archimedes' constant π

The circumference of a circle with diameter 1 is π.

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. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems A definition is a statement of the meaning of a Word or Phrase. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. The circumference is the distance around a closed Curve. Circumference is a kind of Perimeter. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the Geometry, a diameter of a Circle is any straight Line segment that passes through the center of the circle and whose Endpoints are on the Key examples include the Gaussian integral in complex analysis, nth roots of unity in number theory and Cauchy distributions in probability. The Gaussian integral, or probability integral, is the Improper integral of the Gaussian function e^ over the entire real line Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous Probability distribution. Probability is the likelihood or chance that something is the case or will happen 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. In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain In Physical cosmology, the cosmological constant (usually denoted by the Greek capital letter Lambda: Λ was proposed by Albert Einstein as a modification The presence of pi in physical principles, laws and formulas can have very simple explanations. A personal and cultural value is a Relative ethic value, an assumption upon which implementation can be extrapolated The laws of science are various established Scientific laws or Physical laws as they are sometimes called that are considered universal and invariable facts of the In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information 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, $F = \frac{1}{4\pi\varepsilon_0}\frac{\left|q_1 q_2\right|}{r^2}$[3]. ---- Bold text Coulomb's law', developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form The magnitude of a mathematical object is its size a property by which it can be larger or smaller than other objects of the same kind in technical terms an Ordering ---- Bold text Coulomb's law', developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction.

### The exponential growth – or Napier's – constant e

Exponential growth (green) describes many physical phenomena.

The exponential growth constant appears in many parts of applied mathematics. Exponential growth (including Exponential decay) occurs when the growth rate of a mathematical function is proportional to the function's current value For example, as the Swiss mathematician Jacob Bernoulli discovered, $e\,$ arises in compound interest. Switzerland (English pronunciation; Schweiz Swiss German: Schwyz or Schwiiz Suisse Svizzera Svizra officially the Swiss Confederation For other family members named Jacob see Bernoulli family. Jacob Bernoulli (also known as James or Jacques) ( Basel Compound interest is the concept of adding accumulated Interest back to the principal so that interest is earned on interest from that moment on 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. Probability theory is the branch of Mathematics concerned with analysis of random phenomena 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) $1/e\,$. Probability is the likelihood or chance that something is the case or will happen 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[4]. Legal residents and citizens To be French according to the first article of the Constitution is to be a citizen of France regardless of one's origin race or religion ( Pierre Raymond de Montmort, a French Mathematician, was born in Paris on 27 October 1678, and died there on 7 October In combinatorial Mathematics, a derangement is a Permutation in which none of the elements of the set appear in their original positions 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 $n\,$ tends to infinity, $p_n\,$ approaches $1/e\,$.

### The Feigenbaum constants α and δ

Bifurcation diagram of the logistic map.

Iterations of continuous maps serve as the simplest examples of models for dynamical systems. The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position [5] 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[6] and their bifurcation diagrams. Mitchell Jay Feigenbaum (born December 19 1944) is a mathematical physicist whose pioneering studies in Chaos theory led to the discovery The logistic map is a Polynomial mapping of degree 2, often cited as an archetypal example of how complex chaotic behaviour can arise from very simple In Mathematics, particularly in Dynamical systems, a bifurcation diagram shows the possible long-term values (equilibria/fixed points or periodic orbits of a system

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. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that This article describes the use of the term nonlinearity in mathematics The map was popularized in a seminal 1976 paper by the English biologist Robert May[7], in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst. Year 1976 ( MCMLXXVI) was a Leap year starting on Thursday (link will display full calendar of the Gregorian calendar. The English people (from the adjective in Englisc) are a Nation and Ethnic group native to England who predominantly speak English Robert McCredie May Baron May of Oxford, OM, AC, FRS (born Australia, 8 January 1936) has been Chief Scientific Adviser Pierre François Verhulst ( October 28, 1804, Brussels, Belgium &ndash February 15, 1849, Brussels, Belgium The difference equation is intended to capture the two effects or reproduction and starvation.

### Apéry's constant ζ(3)

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[8]. In Mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in In Mathematics, Apéry's Constant is a curious number that occurs in a variety of situations The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J In Physics, the gyromagnetic ratio (also sometimes known as the magnetogyric ratio in other disciplines of a particle or system is the Ratio of its Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. Also, Pascal Wallisch noted that $\sqrt{m_n/m_e}\approxeq\frac{3}{\sqrt{\varphi}-\zeta(3)}$[9], where $m_n,m_e,\varphi$ are the neutron mass, the electron mass and the Golden ratio respectively. This article is a discussion of neutrons in general For the specific case of a neutron found outside the nucleus see Free neutron. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J In Mathematics and the Arts two quantities are in the Golden ratio if the Ratio between the sum of those quantities and the larger one is the

### The golden ratio φ

Golden rectangles in an icosahedron
$F\left(n\right)=\frac{\varphi^n-(1-\varphi)^n}{\sqrt 5}$
An explicit formula for the nth Fibonacci number involving the golden ratio. In Mathematics, the Fibonacci numbers are a Sequence of numbers named after Leonardo of Pisa, known as Fibonacci In Mathematics and the Arts two quantities are in the Golden ratio if the Ratio between the sum of those quantities and the larger one is the

The number $\varphi$ turns up frequently in geometry, particularly in figures with pentagonal symmetry. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or Indeed, the length of a regular pentagon's diagonal is $\varphi$ times its side. Regular pentagons The term pentagon is commonly used to mean a regular convex pentagon, where all sides are equal and all interior angles are equal (to A diagonal can refer to a line joining two nonconsecutive vertices of a Polygon or Polyhedron, or in contexts any upward or downward sloping line The vertices of a regular icosahedron are those of three mutually orthogonal golden rectangles. In Geometry, an icosahedron ( Greek: eikosaedron, from eikosi twenty + hedron seat /ˌaɪ In Mathematics, two Vectors are orthogonal if they are Perpendicular, i A golden rectangle is a Rectangle whose side lengths are in the Golden ratio, 1\varphi (one-to- phi) that is approximately 11 Also, it appears in the Fibonacci sequence, related to growth by recursion[10]. In Mathematics, the Fibonacci numbers are a Sequence of numbers named after Leonardo of Pisa, known as Fibonacci Recursion, in Mathematics and Computer science, is a method of defining functions in which the function being defined is applied within its own definition

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. A stem is one of two main structural axes of a Vascular plant. In Botany, a leaf is an above-ground Plant organ specialized for Photosynthesis. 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 Biology, the skeleton is a strong and often a rigid framework that supports the body of an animal holding it upright and giving it shape and strength (Also skeletal In Materials science, a crystal is a Solid in which the constituent Atoms Molecules or Ions are packed in a regularly ordered repeating In these phenomena he saw the golden ratio operating as a universal law. [11] 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. Structure is a fundamental and sometimes Intangible notion covering the Recognition, Observation, nature, and Stability of This article is about proportionality the mathematical relation Organic chemistry is a discipline within Chemistry which involves the scientific study of the structure properties composition reactions, and preparation Traditionally inorganic compounds are considered to be of mineral not biological origin Acoustics is the interdisciplinary science that deals with the study of Sound, Ultrasound and Infrasound (all mechanical waves in gases liquids and solids [12]

### The Euler-Mascheroni constant γ

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

The Euler–Mascheroni constant is a recurring constant in number theory. The Euler–Mascheroni constant (also called the Euler constant) is a Mathematical constant recurring in analysis and Number theory, usually Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes The French mathematician Charles Jean 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. Legal residents and citizens To be French according to the first article of the Constitution is to be a citizen of France regardless of one's origin race or religion ( Charles-Jean Étienne Gustave Nicolas Baron de la Vallée Poussin ( August 14[[ 866]] - March 2[[ 962]] was a Belgian Mathematician. In Mathematics, an average, or central tendency of a Data set refers to a measure of the "middle" or " expected " value of Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness Surprisingly, this average doesn't tend to one half[13]. 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. In Mathematics, Mertens' theorems are three 1874 results in Number theory related to the density of Prime numbers and one result in analysis In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function A zeta function is a function which is composed of an infinite sum of powers that is which may be written as a Dirichlet series: \zeta(s = \sum_{k=1}^{\infty}f(k^s The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with The definition of the Euler-Mascheroni constant exhibits a close link between the discrete and the continuous (see curves on the right). Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete in the sense of not supporting or requiring the In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output

### Conway's constant λ

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

Conway's constant is the invariant growth rate of all derived strings similar to the look-and-say sequence (except two trivial ones)[14]. John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups In Mathematics, the look-and-say sequence is the sequence of integers beginning as follows 1 11 21 1211 111221 312211 13112221 1113213211 In Mathematics, the look-and-say sequence is the sequence of integers beginning as follows 1 11 21 1211 111221 312211 13112221 1113213211 In Mathematics, the look-and-say sequence is the sequence of integers beginning as follows 1 11 21 1211 111221 312211 13112221 1113213211 It is given by the unique positive real root of a polynomial of degree 71 with integer coefficients[15]. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations

### Khinchin's constant K

If a real number $r\,$ is written using simple continued fraction

$r=a_0+\dfrac{1}{a_1+\dfrac{1}{a_2+\dfrac{1}{a_3+\cdots}}},$

then, as Russian mathematician Aleksandr Khinchin proved in 1934, the limit as $n\,$ tends to infinity of the geometric mean $(a_1a_2\cdots a_n)^{1/n}$ exists, and, except for a set of measure 0, this limit is a constant, Khinchin's constant[16][17]. In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}} The Russian people (Русские— Russkie) are an East Slavic Ethnic group, primarily living in Russia and neighboring countries Aleksandr Yakovlevich Khinchin ( Russian Алекса́ндр Я́ковлевич Хи́нчин French Alexandre Khintchine ( July 19, 1894 The limit of a sequence is one of the oldest concepts in Mathematical analysis. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness The geometric mean in Mathematics, is a type of Mean or Average, which indicates the central tendency or typical value of a set of numbers In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with In Number theory, Aleksandr Yakovlevich Khinchin proved that for Almost all real numbers x, the infinitely many denominators a i

## Physical constants

Main article: Physical constant

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 me and the fine-structure constant α). A physical Constant is a Physical quantity that is generally believed to be both universal in nature and constant in time The elementary charge, usually denoted e, is the Electric charge carried by a single Proton, or equivalently the negative of the electric charge carried The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The fine-structure constant or Sommerfeld fine-structure constant, usually denoted \alpha \ is the Fundamental physical constant characterizing

### The speed of light c and Planck's constant h

The speed of light and the Planck 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. The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. Quantity is a kind of property which exists as magnitude or multitude James Clerk Maxwell (13 June 1831 &ndash 5 November 1879 was a Scottish mathematician and theoretical physicist. In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical For example, in special relativity, mass and energy are equivalent: E = mc2[18] where $c^2\,$ is the constant of proportionality. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial In Physics, mass–energy equivalence is the concept that for particles slower than light any Mass has an associated Energy and vice versa. In quantum mechanics, the energy and frequency of a photon are related by $E=h\nu\,$. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Frequency is a measure of the number of occurrences of a repeating event per unit Time.

The speed of light is also used to express other fundamental constants [19] 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 $Z_0=4\pi10^{-7}c\,$. Vacuum permittivity, referred to by international standards organizations as the electric constant, and denoted by the symbol ε0 is a fundamental Physical ---- Bold text Coulomb's law', developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form Impedance of Free Space (also known as Z_0 is a Canadian IDM and Electronica group

### The electron charge e and the electron mass me

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[20]. The elementary charge, usually denoted e, is the Electric charge carried by a single Proton, or equivalently the negative of the electric charge carried The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J In Particle physics, an elementary particle or fundamental particle is a particle not known to have substructure that is it is not known to be made The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J Alpha particles (named after and denoted by the first letter in the Greek alphabet, α consist of two Protons and two Neutrons bound together into a The proton ( Greek πρῶτον / proton "first" is a Subatomic particle with an Electric charge of one positive This article is a discussion of neutrons in general For the specific case of a neutron found outside the nucleus see Free neutron. The muon (from the letter mu (μ--used to represent it is an Elementary particle with negative Electric charge and a spin of 1/2 In Particle physics, pion (short for pi meson) is the collective name for three Subatomic particles, and. 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. A supercurrent is a Superconducting current that is electric current which flows without Dissipation. Magnetic flux, represented by the Greek letter Φ ( Phi) is a measure of quantity of Magnetism, taking into account the strength and the extent of a Magnetic 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. The magnetic flux quantum Φ0 is the Quantum of Magnetic flux passing through a Superconductor. Superconductivity is a phenomenon occurring in certain Materials generally at very low Temperatures characterized by exactly zero electrical resistance Also, the fundamental fine-structure constant $\alpha=\mu_0ce^2/(2h)\,$ where the permeability of free space μ0 is just a numerical constant equal to $4\pi\times 10^{-7}$. The fine-structure constant or Sommerfeld fine-structure constant, usually denoted \alpha \ is the Fundamental physical constant characterizing The vacuum permeability, referred to by international standards organizations as the magnetic constant, and denoted by the symbol μ 0 (also

## Mathematical curiosities, specific physical facts and unspecified constants

### Simple representatives of sets of numbers

This Babylonian clay tablet gives an approximation of $\sqrt{2}$ in four sexagesimal figures, which is about six decimal figures [21]. Babylonia was an Amorite state in lower Mesopotamia (modern southern Iraq) with Babylon as its capital Sexagesimal ( base-sixty) is a Numeral system with sixty as the base. The decimal ( base ten or occasionally denary) Numeral system has ten as its base.
$c=\sum_{j=1}^\infty 10^{-j!}=0.\underbrace{\overbrace{110001}^{3!\text{ digits}}000000000000000001}_{4!\text{ digits}}000\dots\,$
Liouville's constant is a simple example of a transcendental number. In Number theory, a Liouville number is a Real number x with the property that for any positive Integer n, there exist integers In Mathematics, a transcendental number is a Complex number that is not algebraic, that is not a solution of a non-zero Polynomial equation

Some constants, such as the square root of 2, 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[22], the transcendental numbers[23] and the normal numbers (in base 10)[24] respectively. The square root of 2, also known as Pythagoras' Constant, often denoted by \sqrt{2}   or   √2 In Number theory, a Liouville number is a Real number x with the property that for any positive Integer n, there exist integers In Mathematics, the Champernowne Constant C10 is a transcendental real constant whose decimal expansion has important properties In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction In Mathematics, a transcendental number is a Complex number that is not algebraic, that is not a solution of a non-zero Polynomial equation A different topic is treated in the article titled Normal number (computing. The discovery of the irrational numbers is usually attributed to the Pythagorean Hippasus of Metapontum who proved, most likely geometrically, the irrationality of $\sqrt{2}$. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction Pythagoreanism is a term used for the Esoteric and metaphysical beliefs held by Pythagoras and his followers the Pythagoreans who were much influenced Hippasus of Metapontum (Ίππασος b c 500 BC in Magna Graecia, was a Greek Philosopher. As for Liouville's constant, named after French mathematician Joseph Liouville, it was the first transcendental number ever constructed[25]. Legal residents and citizens To be French according to the first article of the Constitution is to be a citizen of France regardless of one's origin race or religion ( Joseph Liouville ( March 24 1809 &ndash September 8 1882) was a French Mathematician.

### Chaitin's constant Ω

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. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their Algorithmic information theory is a subfield of Information theory and Computer science that concerns itself with the relationship between computation In the Computer science subfield of Algorithmic information theory a Chaitin Constant or halting probability is a Real number that Probability is the likelihood or chance that something is the case or will happen Turing machines are basic abstract symbol-manipulating devices which despite their simplicity can be adapted to simulate the logic of any Computer Algorithm For a topic outline on this subject see List of basic Argentina topics. The United States of America —commonly referred to as the A computer scientist is a person that has acquired knowledge of Computer science, the study of the theoretical foundations of information and computation and their application Gregory John Chaitin (born 1947 is an Argentine - American Mathematician and Computer scientist Amusingly, Chaitin's constant, though not being computable, has been proven transcendental and normal. In the Computer science subfield of Algorithmic information theory a Chaitin Constant or halting probability is a Real number that In Mathematics, Theoretical computer science and Mathematical logic, the computable numbers, also known as the recursive numbers or the In Mathematics, a transcendental number is a Complex number that is not algebraic, that is not a solution of a non-zero Polynomial equation A different topic is treated in the article titled Normal number (computing.

Physical properties
Density (near r.t.)19. The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different Room temperature (also referred to as ambient temperature) is a common term to denote a certain Temperature within enclosed space at which humans are accustomed 3  g·cm−3
Liquid density at m.p.17. The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different The melting point of a solid is the temperature range at which it changes state from solid to Liquid. 31  g·cm−3
Melting point1337. The melting point of a solid is the temperature range at which it changes state from solid to Liquid. 33 K
(1064. The kelvin (symbol K) is a unit increment of Temperature and is one of the seven SI base units The Kelvin scale is a thermodynamic 18 °C, 1947. The Celsius Temperature scale was previously known as the centigrade scale. 52 °F)
Boiling point3129 K
(2856 °C, 5173 °F)
Heat of fusion12. Fahrenheit is a temperature scale named after Daniel Gabriel Fahrenheit (1686–1736 a German Physicist who proposed it in 1724 The boiling point of a liquid is the temperature at which the Vapor pressure of the liquid equals the environmental pressure surrounding the liquid The kelvin (symbol K) is a unit increment of Temperature and is one of the seven SI base units The Kelvin scale is a thermodynamic The Celsius Temperature scale was previously known as the centigrade scale. Fahrenheit is a temperature scale named after Daniel Gabriel Fahrenheit (1686–1736 a German Physicist who proposed it in 1724 The standard Enthalpy of fusion (symbol \Delta{}H_{fus} also known as the heat of fusion or specific melting heat, is the amount of 55  kJ·mol−1
Heat of vaporization324  kJ·mol−1
Specific heat capacity(25 °C) 25. The joule per mole (symbol J·mol-1 is an SI derived unit of energy per amount of material The enthalpy of vaporization, (symbol \Delta{}_{v}H also known as the heat of vaporization or heat of evaporation, is the Energy required The joule per mole (symbol J·mol-1 is an SI derived unit of energy per amount of material Specific heat capacity, also known simply as specific heat, is the measure of the heat energy required to increase the Temperature of a unit quantity 418  J·mol−1·K−1
References

### Constants representing physical properties of elements

Such constants represents characteristics of certain physical objects such as the chemical elements. Recommended values for many properties of the elements together with various references are collected on these data pages A chemical element is a type of Atom that is distinguished by its Atomic number; that is by the number of Protons in its nucleus. Examples include density, melting point and heat of fusion. The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different The melting point of a solid is the temperature range at which it changes state from solid to Liquid. The standard Enthalpy of fusion (symbol \Delta{}H_{fus} also known as the heat of fusion or specific melting heat, is the amount of Some of the properties of gold are listed in the box on the right. Gold (ˈɡoʊld is a Chemical element with the symbol Au (from its Latin name aurum) and Atomic number 79

### Unspecified constants

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. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the Though unspecified, they have a specific value, which often isn't important.

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

#### In integrals

Indefinite integrals are called indefinite because their solutions are only unique up to a constant. In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative 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[26]. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Calculus, the Indefinite integral of a given function (ie the set of all Antiderivatives of the function is always written with a constant the constant In other words, whatever the value of $C\,$, differentiating $\sin x+C\,$ with respect to $x\,$ always yields $\cos x\,$. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change

#### In differential equations

In a similar fashion, constants appear in the solutions to differential equations where not enough initial values or boundary conditions are given. In Chemistry, a solution is a Homogeneous Mixture composed of two or more substances In Mathematics, in the field of Differential equations an initial value problem is an Ordinary differential equation together with specified value called In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints For example, the ordinary differential equation $y'(x)=y(x)\,$ has solution $Ce^x\,$ where $C\,$ is an arbitrary constant. In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its

When dealing with partial differential equations, the constants may be functions, constant with respect to some variables (but not necessarily all of them). In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i In Mathematics, a constant function is a function whose values do not vary and thus are Constant. 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 $y\,$. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i A variable (ˈvɛərɪəbl is an Attribute of a physical or an abstract System which may change its Value while it is under Observation.

## Notation

### Representing constants

Different symbols are used to represent and manipulate constants, such as $1\,$, $\pi\,$ and $\epsilon_0\,$. The musical instrument is spelled Cymbal. A symbol is something --- such as an object, Picture, written word a sound a piece It is common, both in mathematics and physics, to express the numerical value of a constant by giving its decimal representation (or just the first few digits of it). This article gives a mathematical definition For a more accessible article see Decimal. 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 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[27][28] 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[29]. Germany, officially the Federal Republic of Germany ( ˈbʊndəsʁepuˌbliːk ˈdɔʏtʃlant is a Country in Central Europe. Ludolph van Ceulen ( 28 January 1540 &ndash 31 December 1610) was a German Mathematician from Hildesheim. Nowadays, using computers and supercomputers, some of the mathematical constants, including $\{\pi,\,e,\,\sqrt{2}\}$, have been computed to more than one hundred billion — $10^{11}\,$ — digits. A supercomputer is a Computer that is at the frontline of processing capacity particularly speed of calculation (at the time of its introduction Fast algorithms have been developed, some of which — as for Apéry's constant — are unexpectedly fast. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation In Mathematics, Apéry's Constant is a curious number that occurs in a variety of situations In physics, the knowledge of the numerical values of the fundamental constants with high accuracy is crucial. First, it is necessary to achieve accurate quantitative descriptions of the physical universe. Also, it is helpful for testing 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 Knuth's up-arrow notation. Graham's number, named after Ronald Graham, is a large number that is an upper bound on the solution to a certain problem in Ramsey theory. In Mathematics, Knuth's up-arrow notation is a method of notation of very large Integers introduced by Donald Knuth in 1976

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 up-arrow notation is used[30][31]. Graham's number, named after Ronald Graham, is a large number that is an upper bound on the solution to a certain problem in Ramsey theory. In Mathematics, Knuth's up-arrow notation is a method of notation of very large Integers introduced by Donald Knuth in 1976

Commonly, constants in the physical sciences are represented using the scientific notation, with, when appropriate, the inaccuracy - or measurement error - attached. Scientific notation, also sometimes known as standard form or as exponential notation, is a way of writing numbers that accommodates values too large or small to be Observational error is the difference between a measured value of quantity and its true value When writing the Planck constant $h=6.626\ 068\ 96(33) \times 10^{-34}\ \mbox{J}\cdot\mbox{s}$[32] it is meant that $h=(6.626\ 068\ 96 \plusmn 0.000\ 000\ 003\ 3)\times 10^{-34}\ \mbox{J}\cdot\mbox{s}\,$. The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. Only the significant figures are shown and a greater precision would be superfluous, extra figures coming from experimental inaccuracies. The significant figures (also called significant digits and abbreviated sig figs) of a number are those digits that carry meaning contributing to its accuracy When writing Isaac Newton's gravitational constant $G = \left(6.67428 \plusmn 0.00067 \right) \times 10^{-11} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2} \,$[33] only 6 significant figures are given. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements The gravitational constant, denoted G, is a Physical constant involved in the calculation of the gravitational attraction between objects with mass

For mathematical constants, it may be of interest to represent them using continued fractions to perform various studies, including statistical analysis. This is a list of Mathematical constants sorted by their representations as Continued fractions (Constants known to be irrational have infinite continued fractions Many mathematical constants have an analytic form, that is they can constructed using well-known operations that lend themselves readily to calculation. In Mathematics, an analytical expression (or expression in analytical form) is a Mathematical expression, constructed using well-known operations that lend However, Grossman's constant has no known analytic form[34].

### Symbolizing and naming of constants

Symbolizing constants with letters is a frequent means of making the notation more concise. See also Table of mathematical symbols Mathematical notation is used in Mathematics, and throughout the Physical sciences, Engineering A standard convention, instigated by Leonhard Euler in the 18th century, is to use lower case 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. A convention is a set of agreed, stipulated or generally accepted Standards norms social norms or criteria, often taking the form of Lower case (also lower-case or lowercase) minuscule, or small letters are the smaller form of letters as opposed to upper The Greek alphabet (Ελληνικό αλφάβητο is a set of twenty-four letters that has been used to write the Greek language since the late 9th or early

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. The Erdős–Borwein Constant is the sum of the reciprocals of the Mersenne numbers It is named after Paul Erdős and Peter Borwein In Number theory, the Embree-Trefethen constant is a threshold value labelled β*. In 1919 Viggo Brun showed that the sum of the reciprocals of the Twin primes (pairs of Prime numbers which differ by 2 converges to a Mathematical constant A twin prime is a Prime number that differs from another prime number by Two. The Rydberg Constant, named after the Swedish Physicist Johannes Rydberg, is a Physical constant relating to atomic spectra in the This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English.

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[31]. An asterisk ( *) (Latin asteriscum "little star" from Greek ἀστερίσκος) is a Typographical symbol or Glyph In Mathematics, the lemniscate of Bernoulli is an algebraic curve described by a Cartesian Equation of the form (x^2 The Hebrew alphabet (אָלֶף-בֵּית עִבְרִי alephbet ’ivri) consists of 22 letters used for writing the Hebrew language. The Cyrillic alphabet (səˈrɪlɪk also called azbuka, from the old name of the first two letters is actually a family of Alphabets, subsets of which are used by This article is about the 4th century alphabet of the Gothic bible

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

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[35][31]

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 United States of America —commonly referred to as the Edward Kasner (1878&ndash1955 ( City College of New York 1897 Columbia University M A googol is the Large number 10100 that is the digit 1 followed by one hundred zeros (in Decimal representation In Mathematics, the ratio of the Arc length of the parabolic segment formed by the Latus rectum of any Parabola to its Focal parameter is a Determining the length of an irregular arc segment — also called Rectification of a Curve — was historically difficult In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface

The names are either related to the meaning of the constant (parabolic constant, characteristic impedance of vacuum, twin prime constant, electric constant, conductance quantum, . In Mathematics, the ratio of the Arc length of the parabolic segment formed by the Latus rectum of any Parabola to its Focal parameter is a Impedance of Free Space (also known as Z_0 is a Canadian IDM and Electronica group The twin prime conjecture is a famous unsolved problem in Number theory that involves Prime numbers It states There are infinitely many primes Vacuum permittivity, referred to by international standards organizations as the electric constant, and denoted by the symbol ε0 is a fundamental Physical The conductance quantum is the quantized unit of conductance. . . ), to a specific person (Planck's constant, Sierpiński's constant, Dirac's constant, Josephson constant, . The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. Sierpiński's constant is a Mathematical constant usually denoted as K. The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. The magnetic flux quantum Φ0 is the Quantum of Magnetic flux passing through a Superconductor. . . ) or both (Newtonian constant of gravitation, Bohr magneton, Fermi coupling constant[36],. The gravitational constant, denoted G, is a Physical constant involved in the calculation of the gravitational attraction between objects with mass In Atomic physics, the Bohr magneton (symbol \mu_\mathrm{B} is named after the Physicist Niels Bohr. . . ).

### Lumping constants

A common practice in physics is to lump constants to simplify the equations and algebraic manipulations. For example, Coulomb's constant $\kappa =(4\pi\epsilon_0)^{-1}\,$[37] is just $\epsilon_0\,$, $\pi\,$ and $4\,$ lumped together. ---- Bold text Coulomb's law', developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form Also, combining old constants does not necessarily make the new one less fundamental. For example, the 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 fine-structure constant or Sommerfeld fine-structure constant, usually denoted \alpha \ is the Fundamental physical constant characterizing The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The muon (from the letter mu (μ--used to represent it is an Elementary particle with negative Electric charge and a spin of 1/2 In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena

### A notation simplifier : the Avogadro constant Na

The Avogadro constant is the number of entities in one mole, commonly used in chemistry, where the entities are often atoms or molecules. The Avogadro constant (symbols L, N A also called Avogadro's number, is the number of "elementary entities" (usually Atoms The mole (symbol mol) is a unit of Amount of substance: it is an SI base unit, and almost the only unit to be used to measure this Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny In Chemistry, a molecule is defined as a sufficiently stable electrically neutral group of at least two Atoms in a definite arrangement held together by 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. The Avogadro constant (symbols L, N A also called Avogadro's number, is the number of "elementary entities" (usually Atoms

## Mystery and aesthetics behind constants

$e^{i\pi}+1=0\,$
Euler's identity relating five of the most important mathematical constants. In Mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation e^{i \pi} + 1 = 0 \\! where

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". England is a Country which is part of the United Kingdom. Its inhabitants account for more than 83% of the total UK population whilst its mainland A mathematician is a person whose primary area of study and research is the field of Mathematics. James Whitbread Lee Glaisher ( 5 November 1848 - 7 December 1928) son of James Glaisher, the meteorologist was a prolific English

Indian mathematician Srinivasa Ramanujan discovered the following mysterious identity containing pi and Pythagoras' constant $\sqrt{2}$:$\frac{1}{\pi}=\frac{2\sqrt{2}}{9801} \sum^\infty_{k=0}\frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}$. India, officially the Republic of India (भारत गणराज्य inc-Latn Bhārat Gaṇarājya; see also other Indian languages) is a country

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. "[38]

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. British people, or Britons, are the native inhabitants of Great Britain and their descendants or citizens of the United Kingdom, of the Astrophysics is the branch of Astronomy that deals with the Physics of the Universe, including the physical properties ( Luminosity, Sir Arthur Stanley Eddington, OM (28 December 1882 – 22 November 1944 was an English Astrophysicist of the early 20th century A theory of everything ( TOE) is a putative Theory of Theoretical physics that fully explains and links together all known physical phenomena Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism. Gravitation is a natural Phenomenon by which objects with Mass attract one another At first he progressed along "traditional" lines, but turned increasingly to an almost numerological analysis of the dimensionless ratios of fundamental constants. Numerology is any of many Systems Traditions or Beliefs in a mystical or Esoteric relationship between Numbers and physical In a similar fashion, British theoretical physicist Paul Dirac studied ratios of fundamental physical constant to build his large numbers hypothesis. British people, or Britons, are the native inhabitants of Great Britain and their descendants or citizens of the United Kingdom, of the Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world The Dirac large numbers hypothesis (LNH refers to an observation made by Paul Dirac in 1937 relating ratios of size scales in the Universe to that of force scales

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Photograph, illustration, and description of the root(2) tablet from the Yale Babylonian Collection
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