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| A '''mathematical constant''' is a special [[number]], usually a [[real number]], that is "significantly interesting in some way".<ref name="mathworld">{{cite web | url=http://mathworld.wolfram.com/Constant.html | title=Constant | publisher=MathWorld | accessdate=April 13, 2011 | author=Weisstein, Eric W.}}</ref> Constants arise in many different areas of [[mathematics]], with constants such as [[e (mathematical constant)|{{mvar|e}}]] and [[pi|{{pi}}]] occurring in such diverse contexts as [[geometry]], [[number theory]] and [[calculus]]. | | Filippo Inzaghi men were out to make amends after their 100 per cent start to the new Serie A season was halted by a 1 0 home defeat to champs Juventus on Saturday.<br><br>But Milan started poorly and found them selves 2 0 behind in the first quarter of the match thanks to goals from Lorenzo Tonelli and Manuel Pucciarelli.<br><br>Fernando Torres marked his full debut with a fine headed goal to bring Milan straight to it before half time, nevertheless,having said that, by contrast, Fellow Chelsea loanee Marco van Ginkel first start lasted just 31 minutes as he was withdrawn with one more injury.<br><br>Keisuke Honda pulled Milan level with a smart strike in the better half, But Milan wouldn't muster a winner, even though Empoli captain Mirko Valdifiori being shown a late red card.<br><br>or maybe handing debuts to the [http://www.Dict.cc/?s=Chelsea Chelsea] duo, Inzaghi remembered defender Daniele Bonera, in an effort to revitalise his side after Saturday setback.<br><br>But it was Empoli who made remember that assured start, Moving the ball well and manipulating the tempo.<br><br>And they got their reward after 13 minutes when Tonelli took benefit from some flat footed defending from Bonera to meet Valdifiori in swinging corner with a powerful header to open the scoring.<br><br>It has been much worse for Milan in the following five minutes, But Christian Abbiati twice got down smartly to deny Pucciarelli and Francesco Tavano after more slack counselling.<br><br>But Abbiati was left helpless after 21 minutes as Pucciarelli doubled the lead. A well performed well Empoli free kick from Valdifiori, Flicked low in the birthday box, Caught Bonera and Milan on their heels once more and Pucciarelli was allowed to slot in from close range.<br><br>Milan improved marginally going forward, But remained limited to shots from distance with a low drive from Giacomo Bonaventura who came on to replace the injured Van Ginkel on your own to trouble goalkeeper Luigi Sepe.<br><br>That was until Torres made his mark with a wonderful flicked header into the far corner from Ignazio Abate right wing cross two minutes before the break.<br><br>Milan defending continued to present Empoli with businesses, having said that, And Tavano wasted the best of them 10 minutes into the second half as he scuffed wide when through one on one.<br><br>Tavano was made to pay more for that in the 57th minute when Honda levelled. Abate was involved with again, Supplying the Japan external, Who showed quick feet to engineer a shot from 20 yards that found the actual corner.<br><br>The transformation looked to be complete when Torres teed up Jeremy Menez in an almost identical position, But his shot crashed keep your distance the crossbar.<br><br>Milan had the energy, But when Torres tired and was replaced with 10 minutes to play, The holidaymakers energy dropped.<br><br>The game swung Milan way once more when Valdifiori was sent off for a high challenge on Nigel de Jong that earned him a second yellow card in the space of less than 60 seconds on 86 minutes, But Inzaghi men failed to make use.<br><br>If you have any type of inquiries regarding where and the best ways to utilize [http://www.kochanek-art.com/◇Ugg◇レディース-ショートブーツ-japan-19_22.html ショートブーツ レディース], you can contact us at our website. |
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| What it means for a constant to arise "naturally", and what makes a constant "interesting", is ultimately a matter of taste, and some mathematical constants are notable more for historical reasons than for their intrinsic mathematical interest. The more popular constants have been studied throughout the ages and computed to many decimal places.
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| All mathematical constants are [[Definable real number|definable numbers]] and usually are also [[computable number]]s ([[Chaitin's constant]] being a significant exception).
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| == Common mathematical constants ==
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| These are constants which one is likely to encounter during pre-college education in many countries.
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| === Archimedes' constant {{pi}} ===
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| {{main|Pi}}
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| [[Image:Pi-unrolled-720.gif|thumb|220px|right|The circumference of a circle with diameter 1 is {{pi}}.]] | |
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| The constant [[pi|{{pi}}]] (pi) has a natural [[definition]] in [[Euclidean geometry]] (the ratio between the [[circumference]] and [[diameter]] of a circle), but may also be found in many different places in mathematics: for example the [[Gaussian integral]] in [[complex analysis]], the [[roots of unity]] in [[number theory]] and [[Cauchy distribution]]s in [[probability]]. However, its universality is not limited to pure mathematics. Indeed, various [[formula]]e in physics, such as [[Heisenberg's uncertainty principle]], and constants such as the [[cosmological constant]] include the constant {{pi}}. The presence of {{pi}} in physical principles, [[Laws of science|laws]] and formulae 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]],
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| :<math> F = \frac{1}{4\pi\varepsilon_0}\frac{\left|q_1 q_2\right|}{r^2}.</math><ref>{{MathWorld|urlname=Sphere.html|title=Sphere}}</ref>
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| Besides <math>{\varepsilon_0}</math> corresponding to the dielectric constant in vacuum, the <math>{4\pi r^2}</math> factor in the above denominator expresses directly the surface of a sphere with radius r, having thus a very concrete meaning.
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| The numeric value of {{pi}} is approximately 3.14159. [[Piphilology|Memorizing increasingly precise digits]] of {{pi}} is a world record pursuit. | |
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| === Euler's number {{mvar|e}} ===
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| [[Image:Exponential.svg|thumb|180px|right|Exponential growth (green) describes many physical phenomena.]] | |
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| [[e (mathematical constant)|Euler's number]] {{mvar|e}}, also known as the [[exponential growth]] constant, appears in many areas of mathematics, and one possible definition of it is the value of the following expression:
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| :<math>e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n</math>
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| For example, the [[Switzerland|Swiss]] mathematician [[Jacob Bernoulli]] discovered that {{mvar|e}} arises in [[compound interest]]: An account that starts at $1, and yields interest at annual rate {{math|''R''}} with continuous compounding, will accumulate to {{math|''e''<sup>''R''</sup>}} dollars at the end of one year. The constant {{mvar|e}} also has applications to [[probability theory]], where it arises in a way not obviously related to exponential growth. Suppose that a gambler plays a slot machine with a one in {{math|''n''}} probability of winning, and plays it {{mvar|n}} times. Then, for large {{mvar|n}} (such as a million) the [[probability]] that the gambler will win nothing at all is (approximately) {{math|1/''e''}}.
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| Another application of {{mvar|e}}, 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|author=Grinstead, C.M.|coauthors=Snell, J.L.|page=85|accessdate=2007-12-09}}</ref> Here {{mvar|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
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| :<math>p_n = 1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots+(-1)^n\frac{1}{n!}</math>
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| and as {{mvar|n}} tends to infinity, {{math|''p''<sub>''n''</sub>}} approaches {{math|1/''e''}}.
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| The numeric value of {{mvar|e}} is approximately 2.71828.
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| === Pythagoras' constant {{math|{{sqrt|2}}}} ===
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| [[File:Square root of 2 triangle.svg|thumb|200px|The square root of 2 is equal to the length of the [[hypotenuse]] of a [[right triangle]] with legs of length 1.]]
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| The '''square root of 2''', often known as '''root 2''', '''radical 2''', or '''Pythagoras's constant''', and written as {{math|{{sqrt|2}}}}, is the positive [[algebraic number]] that, when multiplied by itself, gives the number [[2 (number)|2]]. It is more precisely called the '''principal square root of 2''', to distinguish it from the negative number with the same property.
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| Geometrically the [[square root]] of 2 is the length of a diagonal across a [[Unit square|square with sides of one unit of length]]; this follows from the [[Pythagorean theorem]]. It was probably the first number known to be [[irrational number|irrational]].
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| Its numerical value truncated to 65 [[decimal|decimal places]] is:
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| :{{gaps|1.41421|35623|73095|04880|16887|24209|69807|85696|71875|37694|80731|76679|73799...}} {{OEIS|id=A002193}}.
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| [[File:Dedekind cut sqrt 2.svg|thumb| right| 200px| The square root of 2.]]
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| The quick approximation 99/70 (≈ 1.41429) for the square root of two is frequently used. Despite having a [[denominator]] of only 70, it differs from the correct value by less than 1/10,000 (approx. 7.2 × 10<sup> −5</sup>).
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| === The imaginary unit {{mvar|i}} ===
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| {{main|Imaginary unit}}
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| [[File:ImaginaryUnit5.svg|thumb|right|'''{{mvar|i}}''' in the [[complex plane|complex]] or [[Cartesian plane|cartesian]] plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis]]
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| The '''imaginary unit''' or '''unit imaginary number''', denoted as '''{{mvar|i}}''', is a [[mathematics|mathematical]] concept which extends the [[real number]] system {{math|ℝ}} to the [[complex number]] system {{math|ℂ}}, which in turn provides at least one [[Root of a function|root]] for every [[polynomial]] {{math|''P''(''x'')}} (see [[algebraic closure]] and [[fundamental theorem of algebra]]). The imaginary unit's core property is that {{math|''i''<sup>2</sup> {{=}} −1}}. The term "[[imaginary number|imaginary]]" is used because there is no [[real number]] having a negative [[square (algebra)|square]].
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| There are in fact two complex square roots of −1, namely {{mvar|i}} and {{math|−''i''}}, just as there are two complex square roots of every other real number, except [[zero]], which has one double square root.
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| In contexts where {{mvar|i}} is ambiguous or problematic, {{mvar|j}} or the Greek [[iota|{{math|ι}}]] (see [[#Alternative notations|alternative notations]]) is sometimes used. In the disciplines of [[electrical engineering]] and [[control systems engineering]], the imaginary unit is often denoted by {{mvar|j}} instead of {{mvar|i}}, because {{mvar|i}} is commonly used to denote [[electric current]] in these disciplines.
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| == Constants in advanced mathematics ==
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| These are constants which are encountered frequently in higher mathematics.
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| === The Feigenbaum constants α and δ ===
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| [[Image:LogisticMap BifurcationDiagram.png|thumb|200px|left|Bifurcation diagram of the logistic map.]]
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| 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|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|isbn=0-521-81805-2}}</ref> and their [[bifurcation diagram]]s.
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| 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 Australian biologist [[Robert May, Baron May of Oxford|Robert May]],<ref>{{cite book|first=Robert|last=May|authorlink=Robert May, Baron May of Oxford|year=1976|title=Theoretical Ecology: Principles and Applications|publisher=Blackwell Scientific Publishers|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.
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| === Apéry's constant ζ(3) ===
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| <div class="thumb tright"><math>\zeta(3) = 1 + \frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \cdots</math></div>
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| 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>{{MathWorld|urlname=AperysConstant.html|title=Apéry's constant|author=Steven Finch}}</ref> The numeric value of ''ζ''(3) is approximately 1.2020569.
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| === The golden ratio φ ===
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| [[Image:Icosahedron-golden-rectangles.svg|thumb|right|Golden rectangles in an icosahedron]]
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| <div class="thumb tleft">
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| <div class="thumbinner" style="width:260px;"><math>F\left(n\right)=\frac{\varphi^n-(1-\varphi)^n}{\sqrt 5}</math>
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| <div class="thumbcaption">An explicit formula for the ''n''th [[Fibonacci number]] involving the [[golden ratio]] φ.</div></div></div>
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| The number φ, also called the [[Golden ratio]], turns up frequently in [[geometry]], particularly in figures with pentagonal [[symmetry]]. Indeed, the length of a regular [[pentagon]]'s [[diagonal]] is φ times its side. The vertices of a regular [[icosahedron]] are those of three mutually [[orthogonal]] [[golden rectangle]]s. Also, it appears in the [[Fibonacci number|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|isbn=0-7679-0815-5}}</ref> The golden ratio has the slowest convergence of any irrational number. It is, for that reason, one of the [[continued fraction#A property of the golden ratio φ|worst cases]] of [[Lagrange's approximation theorem]] and it is an extremal case of the [[Hurwitz's theorem (number theory)|Hurwitz inequality]] for [[Diophantine approximation]]s. This may be why angles close to the golden ratio often show up in [[phyllotaxis]] (the growth of plants).<ref>[http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat2.html Fibonacci Numbers and Nature - Part 2 : Why is the Golden section the "best" arrangement?], from [http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/ Dr. Ron Knott's] [http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/ Fibonacci Numbers and the Golden Section], retrieved 2012-11-29.</ref> It is approximately equal to 1.61803398874, or, more precisely <math>\scriptstyle\frac{1+\sqrt{5}}{2}.</math>
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| === The Euler–Mascheroni constant γ ===
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| [[Image:Euler-Mascheroni.jpg|thumb|180px|left|The area between the two curves (red) tends to a limit.]]
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| 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 [[Extended real number line|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 [[Riemann zeta function|zeta function]] and many different [[integral]]s and [[Series (mathematics)|series]].
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| 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 left).
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| The numeric value of <math>\gamma</math> is approximately 0.57721.
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| === Conway's constant λ ===
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| <div class="thumb tright">
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| <div class="thumbinner" style="width:80px;"><math>\begin{matrix} 1 \\ 11 \\ 21 \\ 1211 \\ 111221 \\ 312211 \\ \vdots \end{matrix}</math>
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| <div class="thumbcaption">[[John Horton Conway|Conway]]'s [[look-and-say sequence]]</div></div></div>
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| [[Conway's constant]] is the invariant growth rate of all [[derived string]]s similar to the [[look-and-say sequence]] (except for one trivial one).<ref name="ReferenceA">{{MathWorld|urlname=ConwaysConstant|title=Conway's Constant|author=Steven Finch}}</ref>
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| It is given by the unique positive real root of a [[polynomial]] of degree 71 with integer coefficients.<ref name="ReferenceA"/>
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| The value of λ is approximately 1.30357.
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| === Khinchin's constant ''K'' ===
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| If a real number ''r'' is written as a [[simple continued fraction]]:
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| :<math>r=a_0+\dfrac{1}{a_1+\dfrac{1}{a_2+\dfrac{1}{a_3+\cdots}}},</math>
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| where ''a''<sub>''k''</sub> are [[natural number]]s for all ''k''
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| then, as the [[Russians|Russian]] mathematician [[Aleksandr Khinchin]] proved in 1934, the [[limit of a sequence|limit]] as ''n'' tends to [[Extended real number line|infinity]] of the [[geometric mean]]: (''a''<sub>1</sub>''a''<sub>2</sub>...''a''<sub>''n''</sub>)<sup>1/''n''</sup> exists and is a constant, [[Khinchin's constant]], except for a set of [[measure (mathematics)|measure]] 0.<ref>{{cite book|first=Kac|title=M. Statistical Independence in Probability, Analysis and Number Theory|publisher=Mathematical Association of America|year=1959}}</ref><ref>{{MathWorld|urlname=KhinchinsConstant.html|title=Khinchin's Constant|author=Steven Finch}}</ref>
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| The numeric value of ''K'' is approximately 2.6854520010.
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| == Mathematical curiosities and unspecified constants ==
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| === Simple representatives of sets of numbers ===
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| [[Image:Ybc7289-bw.jpg|left|thumb|160px|This [[Babylonia]]n clay tablet gives an approximation of the square root of 2 in four [[sexagesimal]] figures, which is about six [[decimal]] figures.<ref>{{cite journal|last=Fowler|first=David|authorlink=David Fowler (mathematician)|coauthors=[[Eleanor Robson]]|date=November 1998|title=Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context|journal=Historia Mathematica|volume=25|issue=4 |page=368|url=http://www.hps.cam.ac.uk/dept/robson-fowler-square.pdf|accessdate=2007-12-09|doi=10.1006/hmat.1998.2209 |archiveurl = http://web.archive.org/web/20071128130320/http://www.hps.cam.ac.uk/dept/robson-fowler-square.pdf <!-- Bot retrieved archive --> |archivedate = 2007-11-28}}<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>]]
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| <div class="thumb tright">
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| <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>
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| <div class="thumbcaption">[[Liouville number#Liouville constant|Liouville's constant]] is a simple example of a [[transcendental number]].</div></div></div>
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| Some constants, such as the [[square root of 2]], [[Liouville's constant]] and [[Champernowne constant]]:
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| :<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>
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| 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|publisher=Transactions of the American Mathematical Society, Vol. 17, No. 4|jstor=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|issue=4}}</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 the square root of 2. As for Liouville's constant, named after [[French people|French]] mathematician [[Joseph Liouville]], it was the first number to be proven transcendental.<ref>{{MathWorld|urlname=LiouvillesConstant|title=Liouville's Constant}}</ref>
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| === Chaitin's constant Ω ===
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| 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 [[Argentina|Argentine]]-[[United States|American]] mathematician and [[computer scientist]] [[Gregory Chaitin]]. Chaitin's constant, though not being [[computable number|computable]], has been proven to be [[Transcendental number|transcendental]] and [[Normal number|normal]]. Chaitin's constant is not universal, depending heavily on the numerical encoding used for Turing machines; however, its interesting properties are independent of the encoding.
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| === Unspecified constants ===
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| 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.
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| [[Image:Different constants of integration.jpg|thumb|200px|left| Solutions with different constants of integration of <math>y'(x)=-2y+e^{-x}\,</math>.]]
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| ==== In integrals ====
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| [[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
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| :<math>\int\cos x\ dx=\sin x+C</math>
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| where ''C'', the [[constant of integration]], is an arbitrary fixed real number.<ref>{{cite book|title=Calculus with analytic geometry|first=Henry|last=Edwards|coauthors=David Penney|edition=4e|page=269|publisher=Prentice Hall|isbn= 0-13-300575-5|year=1994}}</ref> In other words, whatever the value of ''C'', [[Derivative|differentiating]] sin ''x'' + ''C'' with respect to ''x'' always yields cos ''x''.
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| ==== In differential equations ====
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| 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]] ''y''<sup>'</sup> = ''y''(''x'') has solution ''Ce''<sup>''x''</sup> where ''C'' is an arbitrary constant.
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| 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]]
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| :<math>\frac{\partial f(x,y)}{\partial x}=0</math>
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| has solutions ''f''(''x'',''y'') = ''C''(''y''), where ''C''(''y'') is an arbitrary function in the [[variable (mathematics)|variable]] ''y''.
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| == Notation ==
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| === Representing constants ===
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| It is common to express the numerical value of a constant by giving its [[decimal representation]] (or just the first few digits of it). For two reasons this representation may cause problems. First, even though rational numbers all have a finite or ever-repeating decimal expansion, irrational numbers don't have such an expression making them impossible to completely describe in this manner. Also, the decimal expansion of a number is not necessarily unique. For example, the two representations [[0.999...]] and 1 are equivalent<ref>{{cite book|last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |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|isbn=0-534-36298-2|page=706}}</ref> in the sense that they represent the same number.
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| Calculating digits of the decimal expansion of constants has been a common enterprise for many centuries. For example, [[Germans|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://www-groups.dcs.st-and.ac.uk/~history/Biographies/Van_Ceulen.html Ludolph van Ceulen] – biography at the MacTutor History of Mathematics archive.</ref> Using computers and [[supercomputer]]s, some of the mathematical constants, including π, ''e'', and the square root of 2, have been computed to more than one hundred billion digits. Fast [[algorithm]]s have been developed, some of which — as for [[Apéry's constant]] — are unexpectedly fast.
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| <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|year=1976|pmid=17797067|issue=4271|doi=10.1126/science.194.4271.1235}}</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>
| |
| | |
| 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 be constructed using well-known operations that lend themselves readily to calculation. Not all constants have known analytic forms, though; Grossman's constant<ref>{{MathWorld|urlname=GrossmansConstant|title=Grossman's constant}}</ref> and [[Foias constant|Foias' constant]]<ref>{{MathWorld|urlname=FoiasConstant|title=Foias' constant}}</ref> are examples.
| |
| | |
| === 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 />[[Champernowne constant]]s <math>C_b</math><br />[[cardinal number]] [[aleph naught|aleph naught <math>\aleph_0</math>]]
| |
| <div class="thumbcaption">Examples of different kinds of notation for constants.</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 script|Cyrillic]] or [[Blackletter|Gothic]].<ref name="Site 1"/>
| |
| | |
| <div class="thumb tleft">
| |
| <div class="thumbinner" style="width:420px;"><math>\mathrm{googol}=10^{100}\,\ ,\ \mathrm{googolplex}=10^\mathrm{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 name="Site 1"/><ref>{{cite book|title=Mathematics and the Imagination|publisher=[[Microsoft Press]]|year=1989|page=23|author=Edward Kasner and James R. Newman}}</ref>
| |
| | |
| [[Image:Parabolic constant illustration v4.svg|thumb|right|180px|The [[universal parabolic constant]] is the ratio, for any [[parabola]], of the [[arc length]] of the parabolic segment (red) formed by the [[latus rectum]] (blue) to the [[focal parameter]] (green).]]
| |
| | |
| The names are either related to the meaning of the constant ([[universal parabolic constant]], [[twin prime constant]], ...) or to a specific person ([[Sierpiński's constant]], [[Josephson constant]], ...).
| |
| | |
| == Table of selected mathematical constants ==
| |
| {{Main|List of mathematical constants}}
| |
| Abbreviations used:
| |
| : R – [[Rational number]], I – [[Irrational number]] (may be algebraic or transcendental), A – [[Algebraic number]] (irrational), T – [[Transcendental number]] (irrational)
| |
| : Gen – [[Mathematics|General]], NuT – [[Number theory]], ChT – [[Chaos theory]], Com – [[Combinatorics]], Inf – [[Information theory]], Ana – [[Mathematical analysis]]
| |
| | |
| {| class="wikitable sortable"
| |
| |- style="background:#a0e0a0;"
| |
| ! Symbol || Value || Name || Field|| ''N'' || First described || # of known digits
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">0</div>
| |
| || = 0
| |
| || [[0 (number)|Zero]]
| |
| || '''[[Mathematics|Gen]]'''
| |
| | style="text-align:center;"| ''[[rational number|R]]''
| |
| | align=right | c. 7th–5th century BC
| |
| | align=right | N/A
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">1</div>
| |
| || = 1
| |
| || [[1 (number)|One]], Unity
| |
| || '''[[Mathematics|Gen]]'''
| |
| | style="text-align:center;"| ''[[rational number|R]]''
| |
| | align=right |
| |
| | align=right | N/A
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">{{mvar|i}}</div>
| |
| || = {{math|{{sqrt|–1}}}}
| |
| || [[Imaginary unit]], unit imaginary number
| |
| || '''[[Mathematics|Gen]]''', '''[[Mathematical analysis|Ana]]'''
| |
| | style="text-align:center;"| ''[[algebraic number|A]]''
| |
| | align=right | 16th century
| |
| | align=right | N/A
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">{{pi}}</div>
| |
| || ≈ 3.14159 26535 89793 23846 26433 83279 50288
| |
| || [[Pi]], [[Archimedes]]' constant or [[Ludolph van Ceulen|Ludolph]]'s number
| |
| || '''[[Mathematics|Gen]]''', '''[[Mathematical analysis|Ana]]'''
| |
| | style="text-align:center;"| ''[[transcendental number|T]]''
| |
| | align=right | by c. 2000 BC
| |
| | align=right | 10,000,000,000,000<ref>[http://www.numberworld.org/misc_runs/pi-10t/details.html Pi Computation Record]</ref>
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">{{mvar|e}}</div>
| |
| || ≈ 2.71828 18284 59045 23536 02874 71352 66249
| |
| ||[[e (mathematical constant)|e]], Napier's constant, or Euler's number
| |
| || '''[[Mathematics|Gen]]''', '''[[Mathematical analysis|Ana]]'''
| |
| | style="text-align:center;"| ''[[transcendental number|T]]''
| |
| | align=right | 1618
| |
| | align=right | 100,000,000,000
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">{{math|{{sqrt|2}}}}</div>
| |
| || ≈ 1.41421 35623 73095 04880 16887 24209 69807
| |
| || [[Pythagoras]]' constant, [[square root of 2]]
| |
| || '''[[Mathematics|Gen]]'''
| |
| | style="text-align:center;"| ''[[algebraic number|A]]''
| |
| | align=right | by c. 800 BC
| |
| | style="text-align:right;"| 137,438,953,444
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">{{math|{{sqrt|3}}}}</div>
| |
| || ≈ 1.73205 08075 68877 29352 74463 41505 87236
| |
| || [[Theodorus of Cyrene|Theodorus]]' constant, [[square root of 3]]
| |
| || '''[[Mathematics|Gen]]'''
| |
| | style="text-align:center;"| ''[[algebraic number|A]]''
| |
| | align=right | by c. 800 BC
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">{{math|{{sqrt|5}}}}</div>
| |
| || ≈ 2.23606 79774 99789 69640 91736 68731 27623
| |
| || [[square root of 5]]
| |
| || '''[[Mathematics|Gen]]'''
| |
| | style="text-align:center;"| ''[[algebraic number|A]]''
| |
| | align=right | by c. 800 BC
| |
| | style="text-align:right;"|
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;"><math>\gamma</math></div>
| |
| || ≈ 0.57721 56649 01532 86060 65120 90082 40243
| |
| || [[Euler–Mascheroni constant]]
| |
| ||'''[[Mathematics|Gen]]''', '''[[Number theory|NuT]]'''
| |
| ||
| |
| | align=right | 1735
| |
| | style="text-align:right;"| 14,922,244,771
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;"><math>\phi</math></div>
| |
| || ≈ 1.61803 39887 49894 84820 45868 34365 63811
| |
| || [[Golden ratio]]
| |
| || '''[[Mathematics|Gen]]'''
| |
| | style="text-align:center;"| ''[[algebraic number|A]]''
| |
| | align=right | by 3rd century BC
| |
| | style="text-align:right;"| 100,000,000,000
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;"><math>\rho</math></div>
| |
| || ≈ 1.32471 79572 44746 02596 09088 54478 09734
| |
| || [[Plastic number|Plastic constant]]
| |
| || '''[[Number theory|NuT]]'''
| |
| | style="text-align:center;"| ''[[algebraic number|A]]''
| |
| | align=right | 1928
| |
| | style="text-align:right;"|
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;"><math>\beta</math><sup>*</sup></div>
| |
| || ≈ 0.70258
| |
| || [[Embree–Trefethen constant]]
| |
| ||'''[[Number theory|NuT]]'''
| |
| ||
| |
| ||
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;"><math>\delta</math></div>
| |
| || ≈ 4.66920 16091 02990 67185 32038 20466 20161
| |
| || [[Feigenbaum constant]]
| |
| || '''[[chaos theory|ChT]]'''
| |
| ||
| |
| |align=right | 1975
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;"><math>\alpha</math></div>
| |
| || ≈ 2.50290 78750 95892 82228 39028 73218 21578
| |
| || [[Feigenbaum constant]]
| |
| || '''[[chaos theory|ChT]]'''
| |
| ||
| |
| ||
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">''C''<sub>2</sub></div>
| |
| || ≈ 0.66016 18158 46869 57392 78121 10014 55577
| |
| || [[Twin prime conjecture|Twin prime constant]]
| |
| || '''[[Number theory|NuT]]'''
| |
| ||
| |
| ||
| |
| | style="text-align:right;"| 5,020
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">''M''<sub>1</sub></div>
| |
| || ≈ 0.26149 72128 47642 78375 54268 38608 69585
| |
| || [[Meissel–Mertens constant]]
| |
| || '''[[Number theory|NuT]]'''
| |
| ||
| |
| | style="text-align:right;"| 1866<br>1874
| |
| | style="text-align:right;"| 8,010
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">''B''<sub>2</sub></div>
| |
| || ≈ 1.90216 05823
| |
| || [[Brun's constant]] for twin primes
| |
| || '''[[Number theory|NuT]]'''
| |
| ||
| |
| | style="text-align:right;"| 1919
| |
| | style="text-align:right;"| 10
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">''B''<sub>4</sub></div>
| |
| || ≈ 0.87058 83800
| |
| || [[Brun's constant]] for prime quadruplets
| |
| || '''[[Number theory|NuT]]'''
| |
| ||
| |
| ||
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;"><math>\Lambda</math></div>
| |
| || ≥ –2.7 • 10<sup>−9</sup>
| |
| || [[de Bruijn–Newman constant]]
| |
| || '''[[Number theory|NuT]]'''
| |
| ||
| |
| | style="text-align:right;"| 1950?
| |
| | style="text-align:right;"| none
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">K</div>
| |
| || ≈ 0.91596 55941 77219 01505 46035 14932 38411 </td>
| |
| || [[Catalan's constant]]
| |
| || '''[[combinatorics|Com]]'''
| |
| ||
| |
| ||
| |
| | style="text-align:right;"| 15,510,000,000
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">K</div>
| |
| || ≈ 0.76422 36535 89220 66299 06987 31250 09232
| |
| || [[Landau–Ramanujan constant]]
| |
| || '''[[Number theory|NuT]]'''
| |
| ||
| |
| ||
| |
| | style="text-align:right;"| 30,010
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">K</div>
| |
| || ≈ 1.13198 824
| |
| || [[Viswanath's constant]]
| |
| || '''[[Number theory|NuT]]'''
| |
| ||
| |
| ||
| |
| | style="text-align:right;"| 8
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">B´<sub>L</sub></div>
| |
| || = 1
| |
| || [[Legendre's constant]]
| |
| || '''[[Number theory|NuT]]'''
| |
| | style="text-align:center;"| ''[[Rational number|R]]''
| |
| ||
| |
| | style="text-align:right;"| N/A
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;"><math>\mu</math></div>
| |
| || ≈ 1.45136 92348 83381 05028 39684 85892 02744
| |
| || [[Ramanujan–Soldner constant]]
| |
| || '''[[Number theory|NuT]]'''
| |
| ||
| |
| ||
| |
| | style="text-align:right;"| 75,500
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">''E''<sub>B</sub></div>
| |
| || ≈ 1.60669 51524 15291 76378 33015 23190 92458
| |
| || [[Erdős–Borwein constant]]
| |
| || '''[[Number theory|NuT]]'''
| |
| | style="text-align:center;"| ''[[Irrational number|I]]''
| |
| ||
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;"><math>\beta</math></div>
| |
| || ≈ 0.28016 94990 23869 13303
| |
| || [[Bernstein's constant]]<ref>{{MathWorld|urlname=BernsteinsConstant|title=Bernstein's Constant}}</ref>
| |
| || '''[[Mathematical analysis|Ana]]'''
| |
| ||
| |
| ||
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;"><math>\lambda</math></div>
| |
| || ≈ 0.30366 30028 98732 65859 74481 21901 55623
| |
| || [[Gauss–Kuzmin–Wirsing constant]]
| |
| || '''[[combinatorics|Com]]'''
| |
| ||
| |
| | align=right | 1974
| |
| | align=right | 385
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;"><math>\sigma</math></div>
| |
| || ≈ 0.35323 63718 54995 98454
| |
| || [[Hafner–Sarnak–McCurley constant]]
| |
| || '''[[Number theory|NuT]]'''
| |
| ||
| |
| | align=right |1993
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;"><math>\lambda</math>, <math>\mu</math></div>
| |
| || ≈ 0.62432 99885 43550 87099 29363 83100 83724
| |
| || [[Golomb–Dickman constant]]
| |
| || '''[[combinatorics|Com]], [[Number theory|NuT]]'''
| |
| ||
| |
| | align=right | 1930<br> 1964
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|
| |
| || ≈ 0.64341 05463
| |
| || [[Cahen's constant]]
| |
| ||
| |
| | style="text-align:center;"| ''[[transcendental number|T]]''
| |
| | align=right | 1891
| |
| | align=right | 4000
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|
| |
| || ≈ 0.66274 34193 49181 58097 47420 97109 25290
| |
| ||[[Laplace limit]]
| |
| ||
| |
| ||
| |
| ||
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|
| |
| || ≈ 0.80939 40205
| |
| || [[Alladi–Grinstead constant]]<ref>{{MathWorld|urlname=Alladi-GrinsteadConstant|title=Alladi–Grinstead Constant}}</ref>
| |
| || '''[[Number theory|NuT]]'''
| |
| ||
| |
| ||
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;"><math>\Lambda</math></div>
| |
| || ≈ 1.09868 58055
| |
| || [[Lengyel's constant]]<ref>{{MathWorld|urlname=LengyelsConstant|title=Lengyel's Constant}}</ref>
| |
| || '''[[combinatorics|Com]]'''
| |
| ||
| |
| | align=right | 1992
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|
| |
| || ≈ 3.27582 29187 21811 15978 76818 82453 84386
| |
| || [[Lévy's constant]]
| |
| || '''[[Number theory|NuT]]'''
| |
| ||
| |
| ||
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"| <div style="font-size:125%;"><math>\zeta (3)</math></div>
| |
| || ≈ 1.20205 69031 59594 28539 97381 61511 44999
| |
| || [[Apéry's constant]]
| |
| ||
| |
| | style="text-align:center;"| ''[[irrational number|I]]''
| |
| | align=right | 1979
| |
| | align=right | 15,510,000,000
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;"><math>\theta</math></div>
| |
| || ≈ 1.30637 78838 63080 69046 86144 92602 60571
| |
| || [[Mills' constant]]
| |
| ||'''[[Number theory|NuT]]'''
| |
| ||
| |
| | align=right | 1947
| |
| | align=right | 6850
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|
| |
| || ≈ 1.45607 49485 82689 67139 95953 51116 54356
| |
| || [[Backhouse's constant]]<ref>{{MathWorld|urlname=BackhousesConstant|title=Backhouse's Constant}}</ref>
| |
| ||
| |
| ||
| |
| ||
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|
| |
| || ≈ 1.46707 80794
| |
| || [[Porter's constant]]<ref>{{MathWorld|urlname=PortersConstant|title=Porter's Constant}}</ref>
| |
| || '''[[Number theory|NuT]]'''
| |
| ||
| |
| | align=right | 1975
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|
| |
| || ≈ 1.53960 07178
| |
| || [[Lieb's square ice constant]]<ref>{{MathWorld|urlname=LiebsSquareIceConstant|title=Lieb's Square Ice Constant}}</ref>
| |
| || '''[[combinatorics|Com]]'''
| |
| ||
| |
| | align=right | 1967
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|
| |
| || ≈ 1.70521 11401 05367 76428 85514 53434 50816
| |
| || [[Niven's constant]]
| |
| ||'''[[Number theory|NuT]]'''
| |
| ||
| |
| | align=right | 1969
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">''K''</div>
| |
| || ≈ 2.58498 17595 79253 21706 58935 87383 17116
| |
| || [[Sierpiński's constant]]
| |
| ||
| |
| ||
| |
| ||
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|
| |
| || ≈ 2.68545 20010 65306 44530 97148 35481 79569
| |
| || [[Khinchin's constant]]
| |
| ||'''[[Number theory|NuT]]'''
| |
| ||
| |
| | align=right | 1934
| |
| | align=right | 7350
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">''F''</div>
| |
| || ≈ 2.80777 02420 28519 36522 15011 86557 77293
| |
| || [[Fransén–Robinson constant]]
| |
| || '''[[Mathematical analysis|Ana]]'''
| |
| ||
| |
| ||
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">''L''</div>
| |
| || ≈ 0.5
| |
| || [[Landau's constants|Landau's constant]]
| |
| || '''[[Mathematical analysis|Ana]]'''
| |
| ||
| |
| ||
| |
| | align=right | 1
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">''P''<sub>2</sub></div>
| |
| || ≈ 2.29558 71493 92638 07403 42980 49189 49039
| |
| || [[Universal parabolic constant]]
| |
| || '''[[Mathematics|Gen]]'''
| |
| | style="text-align:center;"| ''[[transcendental number|T]]''
| |
| ||
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;">Ω</div>
| |
| || ≈ 0.56714 32904 09783 87299 99686 62210 35555
| |
| || [[Omega constant]]
| |
| || '''[[Mathematical analysis|Ana]]'''
| |
| | style="text-align:center;"| ''[[transcendental number|T]]''
| |
| ||
| |
| ||
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;"><math> C_{{}_{MRB}}</math></div>
| |
| || ≈ 0.187859
| |
| || [[MRB constant]]
| |
| || '''[[Number theory|NuT]]'''
| |
| |
| |
| |align="right"| 3:13GMT 1/11/1999<ref>{{cite web | last=Burns | first=Marvin R. | coauthors= etal. | title=Original Post | url=http://marvinrayburns.com/Original_MRB_Post.html | date=1999-01-11 | work= | publisher=[http://math2.org/ math2.org] | accessdate=2009-06-10 }}</ref>
| |
| |align="right"| 2,000,000 <ref>{{cite web | last=Burns | first=Marvin R. | coauthors= [[Eric W. Weisstein]] | title=A037077 | url=http://oeis.org/A037077 | date=1999-01-23 | work= | publisher=[[Sloane's OEIS]] | accessdate=2011-11-12 }}</ref>
| |
| |-
| |
| | style="background:#d0f0d0; text-align:center;"|<div style="font-size:125%;"><math>\psi</math></div>
| |
| || ≈ 3.35988 56662 43177 55317 20113 02918 ....
| |
| || [[Reciprocal Fibonacci constant]] [http://mathworld.wolfram.com/ReciprocalFibonacciConstant.html] [http://oeis.org/A079586]
| |
| ||
| |
| | style="text-align:center;"|
| |
| | align=right |
| |
| | align=right |
| |
| |}
| |
| | |
| ==See also==
| |
| *[[Invariant (mathematics)]]
| |
| *[[List of numbers]]
| |
| *[[Mathematical constants and functions]]
| |
| | |
| ==Notes==
| |
| {{reflist|colwidth=30em}}
| |
| | |
| == External links ==
| |
| * [http://mathworld.wolfram.com/topics/Constants.html Constants – from Wolfram MathWorld]
| |
| * [http://oldweb.cecm.sfu.ca/projects/ISC/ Inverse symbolic calculator (CECM, ISC)] (tells you how a given number can be constructed from mathematical constants)
| |
| * [http://oeis.org/wiki/Index_to_OEIS On-Line Encyclopedia of Integer Sequences (OEIS)]
| |
| * [http://pi.lacim.uqam.ca/eng/ Simon Plouffe's inverter]
| |
| * [http://www.people.fas.harvard.edu/~sfinch/ Steven Finch's page of mathematical constants]
| |
| * [http://numbers.computation.free.fr/Constants/constants.html Xavier Gourdon and Pascal Sebah's page of numbers, mathematical constants and algorithms]
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| | |
| {{DEFAULTSORT:Mathematical Constant}}
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| [[Category:Mathematical constants|*]]
| |
| [[Category:Mathematical tables|Constants]]
| |