Constant
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{{otheruses|constant (disambiguation)}}
'''Constants''' are [[real number|real]] numbers or numerical values which are significantly interesting in some way<ref name="Site 2">{{cite web|url=http://mathworld.wolfram.com/Constant.html|title=Constant|publisher=[[MathWorld]]|accessdate=2007-12-09}}</ref>. The term "constant" is used both for [[mathematical constant]]s and for [[physical constant]]s, but with quite different meanings.
One always talks about [[Definable real number|definable]], and almost always also [[computable number|computable]], mathematical constants — [[Chaitin's constant]] being a notable exception. 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 <math>1/(299\ 792\ 458)</math> of a [[Speed of light|light-second]]. This definition implies that, in [[metric unit]]s, the speed of light in vacuum is exactly <math>299\ 792\ 458</math> [[metres per second]]<ref>{{cite web|url=http://physics.nist.gov/cgi-bin/cuu/Value?c |title=Speed of light in vacuum |accessdate=2007-08-08 |author=[[Committee on Data for Science and Technology|CODATA]]|work=CODATA recommended values |publisher=[[National Institute of Standards and Technology|NIST]]}}</ref>. No increase in the [[accuracy and precision|precision]] of the measurement of the speed of light could alter this numerical value expressed in metres per second.
== Mathematical constants ==
{{main|Mathematical constant}}
Ubiquitous in many different fields of science, such recurring constants include [[pi|<math>\pi</math>]], [[e (mathematical constant)|<math>e</math>]] and the [[Feigenbaum constants]] which are linked to the [[mathematical model]]s used to describe physical phenomena, [[Euclidean geometry]], [[mathematical analysis|analysis]] and [[logistic map]]s respectively. However, mathematical constants such as [[Apéry's constant]] and the [[Golden ratio]] occur unexpectedly outside of mathematics.
=== Archimedes' constant π ===
[[Image:Pi-unrolled-720.gif|thumb|220px|left|The circumference of a circle with diameter 1 is <math>\pi</math>.]]
[[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. Key examples include the [[Gaussian integral]] in [[complex analysis]], n<sup>th</sup> [[roots of unity]] in [[number theory]] and [[Cauchy distribution]]s 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 of science|laws]] and [[formula]]s can have very simple explanations. For example, [[Coulomb's law]], describing the inverse square proportionality of the [[Magnitude (mathematics)|magnitude]] of the [[electrostatic force]] between two [[electric charge]]s and their distance, states that, in [[International System of Units|SI units]], <math> F = \frac{1}{4\pi\varepsilon_0}\frac{\left|q_1 q_2\right|}{r^2}</math><ref>{{cite web|url=http://mathworld.wolfram.com/Sphere.html|title=Sphere|publisher=[[MathWorld]]}}</ref>.
=== The exponential growth – or Napier's – constant ''e'' ===
[[Image:Exponential.png|thumb|180px|right|Exponential growth (green) describes many physical phenomena.]]
The [[exponential growth]] constant appears in many parts of applied mathematics. For example, as the [[Switzerland|Swiss]] mathematician [[Jacob Bernoulli]] discovered, <math>e\,</math> arises in [[compound interest]]. Indeed, an account that starts at $1, and yields <math>1+R\,</math> dollars at simple interest, will yield <math>e^R\,</math> dollars with continuous compounding. <math>e\,</math> 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) <math>1/e\,</math>. Another application of <math>e\,</math>, also discovered in part by Jacob Bernoulli along with [[French people|French]] mathematician [[Pierre Raymond de Montmort]] is in the problem of [[derangement]]s, also known as the ''hat check problem''<ref>{{cite web|url=http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html|title=Introduction to probability theory|authors=Grinstead, C.M. and Snell, J.L.|licence=[[GFDL]]|page=85|accessdate=2007-12-09}}</ref>. 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 <math>p_n = 1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots+(-1)^n\frac{1}{n!}</math> and as <math>n\,</math> tends to infinity, <math>p_n\,</math> approaches <math>1/e\,</math>.
=== The Feigenbaum constants α and δ ===
[[Image:LogisticMap BifurcationDiagram.png|thumb|200px|left|Bifurcation diagram of the logistic map.]]
Iterations of continuous maps serve as the simplest examples of models for [[dynamical system]]s.<ref>{{cite book|author=Collet & Eckmann|year=1980|title=Iterated maps on the inerval as dynamical systems|publisher=Birkhauser|id=ISBN 3-7643-3026-0}}</ref> Named after mathematical physicist [[Mitchell Feigenbaum]], the two [[Feigenbaum constants]] appear in such iterative processes: they are mathematical invariants of [[logistic map]]s with quadratic maximum points<ref>{{cite book|last=Finch|first=Steven|year=2003|title=Mathematical constants|publisher=[[Cambridge University Press]]|page=67|id=ISBN 0-521-81805-3}}</ref> and their [[bifurcation diagram]]s.
The logistic map is a [[polynomial]] mapping, often cited as an archetypal example of how [[chaos theory|chaotic]] behaviour can arise from very simple [[non-linear]] dynamical equations. The map was popularized in a seminal [[1976]] paper by the [[English people|English]] biologist [[Robert May, Baron May of Oxford|Robert May]]<ref>{{cite book|first=Robert|last=May|authorlink=Robert May, Baron May of Oxford|date=1976|title=Theoretical Ecology: Principles and Applications|publisher=Blackwell Scientific Publishers|id=ISBN 0-632-00768-0}}</ref>, 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 of reproduction and starvation.
=== Apéry's constant ζ(3) ===
[[Image:Apéry's constant.png|right]]
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]]<ref>{{cite web|url=http://mathworld.wolfram.com/AperysConstant.html|title=Apéry's constant|first=Steven|last=Finch|accessdate=2007-12-08}}</ref>. Also, [[Pascal Wallisch]] noted that <math>\sqrt{m_n/m_e}\approxeq\frac{3}{\sqrt{\varphi}-\zeta(3)}</math><ref>{{cite web|url=http://www.eaglemanfoundation.org|title=The Eagleman Prize in Mathematics and Physics}}</ref>, where <math>m_n,m_e,\varphi</math> are the [[neutron]] mass, the [[electron]] mass and the [[Golden ratio]] respectively.
=== The golden ratio φ ===
[[Image:Icosahedron-golden-rectangles.svg|thumb|right|Golden rectangles in an icosahedron]]
<div class="thumb tleft">
<div class="thumbinner" style="width:260px;"><math>F\left(n\right)=\frac{\varphi^n-(1-\varphi)^n}{\sqrt 5}</math>
<div class="thumbcaption">An explicit formula for the ''n''<sup>th</sup> [[Fibonacci number]] involving the [[golden ratio]].</div></div></div>
The number <math>\varphi</math> turns up frequently in [[geometry]], particularly in figures with pentagonal [[symmetry]]. Indeed, the length of a regular [[pentagon]]'s [[diagonal]] is <math>\varphi</math> times its side. The vertices of a regular [[icosahedron]] are those of three mutually [[orthogonal]] [[golden rectangle]]s. Also, it appears in the [[Fibonacci sequence]], related to growth by [[recursion]]<ref>{{cite book|last=Livio|first=Mario|authorlink=Mario Livio|year=2002|title=The Golden Ratio: The Story of Phi, The World's Most Astonishing Number|publisher=Broadway Books|location=New York|id=ISBN 0-7679-0815-5}}</ref>.
Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the [[Plant stem|stem]]s of plants and of [[Leaf#Veins|vein]]s in leaves. He extended his research to the [[skeleton]]s of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of [[crystal]]s, even to the use of proportion in artistic endeavours. In these phenomena he saw the golden ratio operating as a universal law.<ref>{{cite journal|journal=Nexus Network Journal|first=Richard|last=Padovan|title=Proportion: Science, Philosophy, Architecture|volume=4|number=1|pages=113–122|doi=10.1007/s00004-001-0008-7 | year = 2002}}</ref> Zeising wrote in 1854:
<blockquote>[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 [[structure]]s, [[form]]s and [[Proportionality (mathematics)|proportions]], whether cosmic or individual, [[Organic chemistry|organic]] or [[inorganic]], [[Acoustics|acoustic]] or [[optical]]; which finds its fullest realization, however, in the human form.<ref>{{cite book|first=Adolf|last=Zeising|title=Neue Lehre van den Proportionen des meschlischen Körpers|date=1854|page=preface}}</ref></blockquote>
=== The Euler-Mascheroni constant γ ===
[[Image:Euler-Mascheroni.jpg|thumb|180px|left|The area between the two curves (red) tends to a limit.]]
The [[Euler–Mascheroni constant]] is a recurring constant in [[number theory]]. The [[French people|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 <math>\gamma</math> as n tends to [[infinity]]. Surprisingly, this average doesn't tend to one half.
The Euler-Mascheroni constant also appears in [[Mertens' theorems|Merten's third theorem]] and has relations to the [[gamma function]], the [[zeta function]] and many different [[integral]]s and [[Series (mathematics)|series]].
The definition of the Euler-Mascheroni constant exhibits a close link between the [[discrete mathematics|discrete]] and the [[Continuous function|continuous]] (see curves on the right).
=== Conway's constant λ ===
<div class="thumb tright">
<div class="thumbinner" style="width:80px;"><math>\begin{matrix} 1 \\ 11 \\ 21 \\ 1211 \\ 111221 \\ 312211 \\ \vdots \end{matrix}</math>
<div class="thumbcaption">[[John Horton Conway|Conway]]'s [[look-and-say sequence]]</div></div></div>
[[Conway's constant]] is the invariant growth rate of all [[derived string]]s similar to the [[look-and-say sequence]] (except two trivial ones)<ref>{{cite book|last=Finch|first=Steven|year=2003|title=Mathematical constants|publisher=[[Cambridge University Press]]|page=453|id=ISBN 0-521-81805-3}}</ref>.
It is given by the unique positive real root of a [[polynomial]] of degree 71 with integer coefficients<ref>{{cite web|url=http://mathworld.wolfram.com/ConwaysConstant.html|first=Steven|last=Finch|title=Conway's Constant|publisher=[[MathWorld]]|accessdate=2007-12-07}}</ref>.
=== Khinchin's constant ''K'' ===
If a real number <math>r\,</math> is written using [[simple continued fraction]]
: <math>r=a_0+\dfrac{1}{a_1+\dfrac{1}{a_2+\dfrac{1}{a_3+\cdots}}},</math>
then, as [[Russians|Russian]] mathematician [[Aleksandr Khinchin]] proved in 1934, the [[limit of a sequence|limit]] as <math>n\,</math> tends to [[infinity]] of the [[geometric mean]] <math>(a_1a_2\cdots a_n)^{1/n}</math> exists, and, except for a set of [[measure (mathematics)|measure]] 0, this limit is a constant, [[Khinchin's constant]]<ref>{{cite book|first=Kac|title=M. Statistical Independence in Probability, Analysis and Number Theory|publisher=Mathematical Association of America|date=1959}}</ref><ref>{{cite web|url=http://mathworld.wolfram.com/KhinchinsConstant.html|title=Khinchin's Constant|first=Steven|last=Finch|publisher=[[MathWorld]]|accessdate=2007-12-08}}</ref>.
== Physical constants ==
{{main|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]] <math>e</math>, the [[electron mass]] <math>m_e</math> and the [[fine-structure constant]] <math>\alpha</math>).
=== The speed of light ''c'' and Planck's constant ''h'' ===
The [[speed of light]] and the [[Planck constant]] are examples of [[quantity|quantities]] that occur naturally in the mathematical formulation of certain fundamental physical theories, the former in [[James Clerk Maxwell]]'s theory of [[electric field|electric]] and [[magnetic field]]s and [[Albert Einstein]]'s theories of relativity, and the latter in quantum theory. For example, in [[special relativity]], mass and energy are equivalent: [[Mass–energy equivalence|''E'' = ''mc''<sup>2</sup>]]<ref>{{Citation |first=Albert|last=Einstein|authorlink=Albert Einstein|year=1905|title=Ist die Trägheit eines Körpers von dessen Energieinhalt abhängig?|url=http://www.physik.uni-augsburg.de/annalen/history/papers/1905_18_639-641.pdf|journal=Annalen der Physik|volume =18|pages =639–643|accessdate=2007-12-09|doi=10.1002/andp.19053231314}} See also the [http://www.fourmilab.ch/etexts/einstein/specrel/www english translation].</ref> where <math>c^2\,</math> is the constant of proportionality. In [[quantum mechanics]], the energy and [[frequency]] of a photon are related by <math>E=h\nu\,</math>.
The speed of light is also used to express other fundamental constants <ref>{{cite web|url=http://physics.nist.gov/cgi-bin/cuu/Category?view=html&Universal.x=15&Universal.y=6 |title=Values of the Fundamental Constants|accessdate=2007-12-09|author=[[Committee on Data for Science and Technology|CODATA]]|work=CODATA recommended values|publisher=[[National Institute of Standards and Technology|NIST]]}}</ref> such as the [[electric constant]] <math>\epsilon_0=(4\pi 10^{-7} c^2)^{-1}\,</math>, [[Coulomb's constant]] <math>k=10^{-7} c^2\,</math> and the [[characteristic impedance of vacuum]] <math>Z_0=4\pi10^{-7}c\,</math>.
=== The electron charge <math>e</math> and the electron mass <math>m_e</math> ===
The [[electron charge]] and the [[electron mass]] are examples of constants that characterize the basic, or elementary, [[Elementary particle|particle]]s that constitute matter, such as the [[electron]], [[alpha particle]], [[proton]], [[neutron]], [[muon]], and [[pion]]<ref>{{cite book|title=[[Encyclopædia Britannica]]|article=Physical Constants|volume=5|page=75|edition=15|publisher=Helen Hemingway Benton|date=1974|ISBN=0-85229-290-2}}</ref>. Many constants can be expressed using the fundamental constants <math>h,\,c,\,e</math>. 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 [[quantum|quantized]]. The [[magnetic flux quantum]] <math>\Phi_0=hc/(2e)\,</math> 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]] <math>\alpha=\mu_0ce^2/(2h)\,</math> where the [[permeability of free space]] <math>\mu_0</math> is just a numerical constant equal to {{nowrap|<math>4\pi\times 10^{-7}</math>}}.
== Mathematical curiosities, specific physical facts and unspecified constants ==
=== Simple representatives of sets of numbers ===
[[Image:Ybc7289-bw.jpg|left|thumb|160px|This [[Babylonia]]n clay tablet gives an approximation of <math>\sqrt{2}</math> in four [[sexagesimal]] figures, which is about six [[decimal]] figures <ref>{{cite journal|last=Fowler|first=David|authorlink=David Fowler (mathematician)|coauthors=[[Eleanor Robson]]|year=1998|month=November|title=Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context|journal=Historia Mathematica|volume=25|issue=4 |pages=368|url=http://www.hps.cam.ac.uk/dept/robson-fowler-square.pdf|accessdate=2007-12-09|doi=10.1006/hmat.1998.2209}}<br>[http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html Photograph, illustration, and description of the ''root(2)'' tablet from the Yale Babylonian Collection]<br>[http://www.math.ubc.ca/%7Ecass/Euclid/ybc/ybc.html High resolution photographs, descriptions, and analysis of the ''root(2)'' tablet (YBC 7289) from the Yale Babylonian Collection]</ref>.]]
<div class="thumb tright">
<div class="thumbinner" style="width:460px;"><math>c=\sum_{j=1}^\infty 10^{-j!}=0.\underbrace{\overbrace{110001}^{3!\text{ digits}}000000000000000001}_{4!\text{ digits}}000\dots\,</math>
<div class="thumbcaption">[[Liouville's constant]] is a simple example of a [[transcendental number]].</div></div></div>
Some constants, such as the [[square root of 2]], [[Liouville's constant|Liouville's constant]] and [[Champernowne constant|Champernowne constant <math>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</math>]] are not important mathematical invariants but retain interest being simple representatives of special sets of numbers, the [[irrational number]]s<ref>{{cite web|url=http://www.cut-the-knot.org/proofs/sq_root.shtml|title=Square root of 2 is irrational|first=Alexander|last=Bogomolny}}</ref>, the [[transcendental number]]s<ref>{{cite journal|title=On Transcendental Numbers|author=Aubrey J. Kempner|journal=Transactions of the American Mathematical Society|volume=17|issue=4|year=Oct 1916|pages=476–482|doi=10.2307/1988833}}</ref> and the [[normal number]]s (in base 10)<ref>{{cite journal|title=The onstruction of decimals normal in the scale of ten|first=david|last=Champernowne|authorlink=D. G. Champernowne|journal=Journal of the London Mathematical Society|volume=8|year=1933|pages=254–260|doi=10.1112/jlms/s1-8.4.254}}</ref> respectively. The discovery of the [[irrational number]]s is usually attributed to the [[Pythagoreanism|Pythagorean]] [[Hippasus of Metapontum]] who proved, most likely geometrically, the irrationality of <math>\sqrt{2}</math>. As for Liouville's constant, named after [[French people|French]] mathematician [[Joseph Liouville]], it was the first transcendental number ever constructed<ref>{{cite web|url=http://mathworld.wolfram.com/LiouvillesConstant.html|title=Liouville's Constant|publisher=[[MathWorld]]|accessdate=2007-12-09}}</ref>.
=== 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]]-[[United States|American]] mathematician and [[computer scientist]] [[Gregory Chaitin]]. Amusingly, [[Chaitin's constant]], though not being [[computable number|computable]], has been proven [[Transcendental number|transcendental]] and [[Normal number|normal]].
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=== Constants representing physical properties of elements ===
Such constants represents characteristics of certain physical objects such as the [[chemical element]]s. Examples include [[density]], [[melting point]] and [[heat of fusion]]. Some of the properties of [[gold]] are listed in the box on the right.
=== 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 [[integral]]s and [[differential equation]]s. Though unspecified, they have a specific value, which often isn't important.
[[Image:Different constants of integration.jpg|thumb|200px|left| Solutions with different constants of integration of <math>y'(x)=-2y+e^{-x}\,</math>.]]
==== In integrals ====
[[Indefinite integral]]s are called indefinite because their solutions are only unique up to a constant. For example, when working over the [[field (mathematics)|field]] of real numbers <math>\int\cos x\ dx=\sin x+C</math> where <math>C\,</math>, the [[constant of integration]], is an arbitrary fixed real number<ref>{{cite book|title=Calculus with analytic geometry|first=Henry|last=Edwards|author=Henry Edwards|coauthors=David Penney|edition=4e|page=269|publisher=Prentice Hall|id= ISBN 0-13-300575-5}}</ref>. In other words, whatever the value of <math>C\,</math>, [[Derivative|differentiating]] <math>\sin x+C\,</math> with respect to <math>x\,</math> always yields <math>\cos x\,</math>.
==== In differential equations ====
In a similar fashion, constants appear in the [[solution]]s to differential equations where not enough [[initial value]]s or [[boundary condition]]s are given. For example, the [[ordinary differential equation]] <math>y'(x)=y(x)\,</math> has solution <math>Ce^x\,</math> where <math>C\,</math> is an arbitrary constant.
When dealing with [[partial differential equation]]s, the constants may be [[constant function|functions]], '''constant with respect to''' some variables (but not necessarily all of them). For example, the [[Partial differential equation|PDE]] <math>\frac{\partial f(x,y)}{\partial x}=0</math> has solutions <math>f(x,y)=C(y)\,</math> where <math>C(y)\,</math> is an arbitrary function in the [[variable]] <math>y\,</math>.
== Notation ==
=== Representing constants ===
Different [[symbol]]s are used to represent and manipulate constants, such as <math>1\,</math>, <math>\pi\,</math> and <math>\epsilon_0\,</math>. 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). 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<ref>{{cite book|last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |id=ISBN 0-07-054235-X|page=61 |theorem=3.26}}</ref><ref>{{cite book|last=Stewart|first=James|authorlink=James Stewart (mathematician)|title=Calculus: Early transcendentals|edition=4e|year=1999|publisher=Brooks/Cole|id=ISBN 0-534-36298-2|page=706}}</ref> 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, [[Germany|german]] mathematician [[Ludolph van Ceulen]] of the 16th century spent a major part of his life calculating the first 35 digits of pi<ref>[http://mathworld.wolfram.com/PiDigits.html Pi Digits - from Wolfram MathWorld<!-- Bot generated title -->]</ref>. Nowadays, using computers and [[supercomputer]]s, some of the mathematical constants, including <math>\{\pi,\,e,\,\sqrt{2}\}</math>, have been computed to more than one hundred billion — <math>10^{11}\,</math> — digits. Fast [[algorithm]]s have been developed, some of which — as for [[Apéry's constant]] — are unexpectedly fast. 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.
<div class="thumb tright">
<div class="thumbinner" style="width:260px;"><math>G=\left . \begin{matrix} 3 \underbrace{ \uparrow \ldots \uparrow } 3 \\ \underbrace{\vdots } \\ 3 \uparrow\uparrow\uparrow\uparrow 3 \end{matrix} \right \} \text{64 layers}</math>
<div class="thumbcaption">[[Graham's number]] defined using [[Knuth's up-arrow notation]].</div></div></div>
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<ref>{{cite journal|journal=Science|last=Knuth|first=Donald|authorlink=Donald Knuth|title=Mathematics and Computer Science: Coping with Finiteness. Advances in Our Ability to Compute are Bringing Us Substantially Closer to Ultimate Limitations.|volume=194|pages=1235–1242|date=1976}}</ref><ref name="Site 1">{{cite web|url=http://www.po28.dial.pipex.com/maths/constant.htm|title=mathematical constants|accessdate=2007-11-27}}</ref>.
Commonly, constants in the physical sciences are represented using the [[scientific notation]], with, when appropriate, the inaccuracy - or [[measurement error]] - attached. When writing the [[Planck constant]] <math>h=6.626\ 068\ 96(33) \times 10^{-34}\ \mbox{J}\cdot\mbox{s}</math><ref>{{cite web|url=http://physics.nist.gov/cgi-bin/cuu/Value?h |title=Planck constant|accessdate=2007-12-09|author=[[Committee on Data for Science and Technology|CODATA]]|work=CODATA recommended values|publisher=[[National Institute of Standards and Technology|NIST]]}}</ref> it is meant that <math>h=(6.626\ 068\ 96 \plusmn 0.000\ 000\ 003\ 3)\times 10^{-34}\ \mbox{J}\cdot\mbox{s}\,</math>. Only the [[significant figures]] are shown and a greater [[accuracy and precision|precision]] would be superfluous, extra figures coming from experimental inaccuracies. When writing [[Isaac Newton]]'s [[gravitational constant]] <math>G = \left(6.67428 \plusmn 0.00067 \right) \times 10^{-11} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2} \,</math><ref>{{cite web|url=http://www.physics.nist.gov/cgi-bin/cuu/Value?bg |title=Newtonian constant of gravitation|accessdate=2007-12-09 |author=[[Committee on Data for Science and Technology|CODATA]]|work=CODATA recommended values|publisher=[[National Institute of Standards and Technology|NIST]]}}</ref> only 6 significant figures are given.
For mathematical constants, it may be of interest to represent them using [[Mathematical constants (sorted by continued fraction representation)|continued fraction]]s 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<ref>{{cite web|url=http://mathworld.wolfram.com/GrossmansConstant.html|title=Grossman's constant|first=Steven|last=Finch|accessdate=2007-12-09}}</ref>.
=== Symbolizing and naming of constants ===
Symbolizing constants with letters is a frequent means of making the [[Mathematical notation|notation]] more concise. A standard [[Convention (norm)|convention]], instigated by [[Leonhard Euler]] in the 18th century, is to use [[lower case]] letters from the beginning of the [[Latin alphabet]] <math>a,b,c,\dots\,</math> or the [[Greek alphabet]] <math>\alpha,\beta,\,\gamma,\dots\,</math> when dealing with constants in general.
<div class="thumb tright">
<div class="thumbinner" style="width:220px;">[[Erdős–Borwein constant]] <math>E_B\,</math><br />[[Embree-Trefethen constant]] <math>\beta*\,</math><br />[[Brun's constant]] for [[twin prime]] <math>B_2\,</math><br />[[Rydberg constant]] <math>R_\infty</math><br />[[cardinal number]] [[aleph naught|aleph naught <math>\aleph_0</math>]]
<div class="thumbcaption">Different kinds of notation.</div></div></div>
However, for more important constants, the symbols may be more complex and have an extra letter, an [[asterisk]], a number, a [[Lemniscate of Bernoulli|lemniscate]] or use different alphabets such as [[Hebrew alphabet|Hebrew]], [[Cyrillic alphabet|Cyrillic]] or [[Gothic alphabet|Gothic]]<ref name="Site 1"/>.
<div class="thumb tleft">
<div class="thumbinner" style="width:420px;"><math>googol=10^{100}\,\ ,\ googolplex=10^{googol}=10^{10^{100}}\,</math>
</div></div>
Sometimes, the symbol representing a constant is a whole word. For example, [[United States|American]] mathematician [[Edward Kasner]]'s 9-year-old nephew coined the names [[googol]] and [[googolplex]]<ref>{{cite book|authors=Edward Kasner and James R. Newman|title=Mathematics and the Imagination|publisher=[[Microsoft Press]]|year=1989|page=23}}</ref><ref name="Site 1"/>
[[Image:Parabolic constant illustration v4.svg|thumb|right|180px|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 ([[Gravitational constant|Newtonian constant of gravitation]], [[Bohr magneton]], [[Fermi coupling constant]]<ref>{{cite web|url=http://physics.nist.gov/cgi-bin/cuu/Value?gf |title=Fermi coupling constant|accessdate=2007-12-09 |author=[[Committee on Data for Science and Technology|CODATA]]|work=CODATA recommended values|publisher=[[National Institute of Standards and Technology|NIST]]}}</ref>,...).
=== Lumping constants ===
A common practice in physics is to lump constants to simplify the equations and algebraic manipulations. For example, [[Coulomb's constant]] <math>\kappa =(4\pi\epsilon_0)^{-1}\,</math><ref>{{cite web|url=http://scienceworld.wolfram.com/physics/CoulombsConstant.html|title=Coulomb's constant|publisher=Wolfram World of Physics|accessdate=2007-12-09}}</ref> is just <math>\epsilon_0\,</math>, <math>\pi\,</math> and <math>4\,</math> lumped together. Also, combining old constants does not necessarily make the new one less fundamental. For example, the dimensionless [[fine-structure constant]] <math>\alpha=\mu_0ce^2/(2h)\,</math> is a fundamental constant of quantum electrodynamics and in the quantum theory of the interaction among [[electron]]s, [[muon]]s and [[photon]]s.
=== A notation simplifier : the Avogadro constant <math>N_a</math> ===
The [[Avogadro constant]] is the number of entities in one [[mole (unit)|mole]], commonly used in [[chemistry]], where the entities are often [[atom]]s or [[molecule]]s. 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 number]]s.
== Mystery and aesthetics behind constants ==
<div class="thumb tright">
<div class="thumbinner" style="width:150px;"><math>e^{i\pi}+1=0\,</math>
<div class="thumbcaption">[[Euler's identity]] relating five of the most important mathematical constants.</div></div></div>
For some authors, constants, either mathematical or physical may be mysterious, beautiful or fascinating. For example, [[England|English]] [[mathematician]] [[James Whitbread Lee Glaisher|Glaisher]] (1915) writes <ref name="Site 2" />:
"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 <math>e,\,\pi,\,\gamma,\,\log2</math>, 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".
[[India]]n mathematician [[Srinivasa Ramanujan]] discovered the following mysterious identity containing pi and Pythagoras' constant <math>\sqrt{2}</math>:<math>\frac{1}{\pi}=\frac{2\sqrt{2}}{9801} \sum^\infty_{k=0}\frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}</math>.
[[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."<ref>{{cite book|last=Finch|first=Steven|year=2003|title=Mathematical constants|publisher=[[Cambridge University Press]]|id=ISBN 0-521-81805-3}}</ref>
During the 1920s until his death, [[British people|British]] [[astrophysicist]] [[Arthur Stanley Eddington|Eddington]] increasingly concentrated on what he called "[[Theory of everything|fundamental theory]]" which was intended to be a unification of [[Quantum mechanics|quantum theory]], [[theory of relativity|relativity]] and [[gravitation]]. At first he progressed along "traditional" lines, but turned increasingly to an almost [[numerology|numerological]] analysis of the dimensionless ratios of fundamental constants. In a similar fashion, [[British people|British]] [[theoretical physicist]] [[Paul Dirac]] studied ratios of fundamental physical constant to build his [[Dirac large numbers hypothesis|large numbers hypothesis]].
== See also ==
{{Col-begin|class=references}}
{{Col-3}}
[[Mathematical constant]]<br />[[Physical constant]]<br />[[Astronomical constant]]
{{Col-3}}
[[scalar (mathematics)|Scalar]]<br />[[Coefficient]]<br />[[Number]]
{{Col-3}}
[[Constant function]]<br />[[Constant of integration]]<br />[[Cosmological constant]]
{{Col-end}}
==References==
{{reflist|2}}
==Further reading==
{{Wiktionarypar|constant}}
* {{cite book
| last = Finch
| first = Steven
| author = Steven Finch
| title = Mathematical Constants
| publisher = [[Cambridge University Press]]
| date = 2003
| isbn = 0-521-81805-2}}
* {{cite web
| last = Miller
| first = Jeff
| title = Earliest Uses of Symbols for Constants
| date = 2007
| url = http://members.aol.com/jeff570/constants.html}}
* {{cite book
| author = [[John Horton Conway]] & [[Richard K. Guy]]
| title = The book of numbers
| publisher = Springer-Verlag New York
| date = 1996
| isbn = 0-387-97993-X}}
==External links==
* [http://www.research.att.com/~njas/sequences/ On-Line Encyclopedia of Integer Sequences]
* [http://pi.lacim.uqam.ca/eng/ Plouffe's Inverter]
* [http://physics.nist.gov/cuu/Constants/index.html Fundamental Physical Constants from NIST]
* [http://www.codata.org/ CODATA]
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