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| {{Technical|date=March 2012}}
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| {{About|numerical curiosities|the technical mathematical concept of coincidence|Coincidence point}}
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| A '''mathematical coincidence''' can be said to occur when two expressions show a near-equality that lacks direct theoretical explanation. For example, there is a near-equality around the [[round number]] 1000 between powers of two and powers of ten: <math>2^{10} = 1024 \approx 1000 = 10^3</math>. Some of these coincidences are used in [[engineering]] when one expression is taken as an approximation of the other.
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| ==Introduction==
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| A mathematical coincidence often involves an [[integer]], and the surprising (or "coincidental") feature is the fact that a [[real number]] arising in some context is considered by some ill-defined standard as a "close" approximation to a small integer or to a multiple or power of ten, or more generally, to a [[rational number]] with a small [[denominator]]. Other kinds of mathematical coincidence than near-integer reals, such as integers simultaneously satisfying multiple seemingly unrelated criteria or coincidences regarding units of measurement, may also be considered. In the class of those coincidences that are of a purely mathematical sort, some simply result from sometimes very deep mathematical facts, while others appear to come 'out of the blue'.
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| Given the [[countably infinite]] number of ways of forming mathematical expressions using a finite number of symbols, the number of symbols used and the [[Arithmetic precision|precision]] of approximate equality might be the most obvious way to assess mathematical coincidences; but there is no standard, and the [[strong law of small numbers]] is the sort of thing one has to appeal to with no formal opposing mathematical guidance.{{Citation needed|date=May 2009}} Beyond this, some sense of [[Mathematical beauty|mathematical aesthetics]] could be invoked to adjudicate the value of a mathematical coincidence, and there are in fact exceptional cases of true mathematical significance (see [[Ramanujan's constant]] below, which made it into print some years ago as a scientific [[April Fools' Day|April Fools']] joke<ref name="gardner">Reprinted as {{Cite book|last1=Gardner|first1=Martin|authorlink1=Martin Gardner|title=The Colossal Book of Mathematics|year=2001|publisher=W. W. Norton & Company|location=New York|isbn=0-393-02023-1|pages=674–694|chapter=Six Sensational Discoveries|separator=,}}</ref>). All in all, though, they are generally to be considered for their curiosity value or, perhaps, to encourage new mathematical learners at an elementary level.
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| ==Some examples==
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| ===Rational approximants===
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| Sometimes simple rational approximations are exceptionally close to interesting irrational values. These are explainable in terms of large terms in the [[continued fraction]] representation of the irrational value, but further insight into why such improbably large terms occur is often not available.
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| Rational approximants (convergents of continued fractions) to ratios of logs of different numbers are often invoked as well, making coincidences between the powers of those numbers.<ref name=schroeder/>
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| Many other coincidences are combinations of numbers that put them into the form that such rational approximants provide close relationships.
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| ====Concerning π====
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| * The first [[convergent (continued fraction)|convergent]] of π, [3; 7] = 22/7 = 3.1428..., was known to [[Archimedes]],<ref name=beckmann/> and is correct to about 0.04%. The third convergent of π, [3; 7, 15, 1] = 355/113 = 3.1415929..., found by [[Zu Chongzhi]],<ref>
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| {{Cite book
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| | author = Yoshio Mikami
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| | title = Development of Mathematics in China and Japan
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| | publisher = B. G. Teubner
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| | page = 135
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| | year = 1913
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| | url = http://books.google.com/?id=4e9LAAAAMAAJ&q=intitle:Development+intitle:%22China+and+Japan%22+355&dq=intitle:Development+intitle:%22China+and+Japan%22+355 }}</ref> is correct to six decimal places;<ref name=beckmann>{{Cite book
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| | title = A History of Pi
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| | author = Petr Beckmann
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| | publisher = Macmillan
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| | year = 1971
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| | isbn = 978-0-312-38185-1
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| | pages = 101, 170
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| | url = http://books.google.com/?id=TB6jzz3ZDTEC&pg=PA101&dq=pi+113+355++digits
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| }}</ref> this high accuracy comes about because π has an unusually large next term in its continued fraction representation: π = [3; 7, 15, 1, 292, ...].<ref>{{Cite book
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| | title = CRC concise encyclopedia of mathematics
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| | author = Eric W. Weisstein
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| | publisher = CRC Press
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| | year = 2003
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| | isbn = 978-1-58488-347-0
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| | page = 2232
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| | url = http://books.google.com/?id=_8TyhSqHUiEC&pg=PA2232&dq=pi+113+355++292+convergent
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| }}</ref>
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| * A coincidence involving π and the [[golden ratio]] φ is given by <math>\pi \approx 4 / \sqrt{\varphi} = 3.1446\dots</math>. This is related to [[Kepler triangle#A mathematical coincidence|Kepler triangles]].
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| * The [[Feynman point]] is a sequence of six 9s that begins at the 762nd decimal place of the decimal representation of pi. For a randomly chosen [[normal number]], the probability of any chosen number sequence of six digits (including 6 of a number, 658 020, or the like) occurring this early in the decimal representation is only 0.08%. Pi is conjectured, but not known, to be a normal number.
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| ====Concerning base 2====
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| * The coincidence <math>2^{10} = 1024 \approx 1000 = 10^3</math>, correct to 2.4%, relates to the rational approximation <math>\textstyle\frac{\log10}{\log2} \approx 3.3219 \approx \frac{10}{3}</math>, or <math> 2 \approx 10^{3/10}</math> to within 0.3%. This relationship is used in engineering, for example to approximate a factor of two in [[power (mathematics)|power]] as 3 [[decibel|dB]] (actual is 3.0103 dB – see [[3dB-point]]), or to relate a [[kilobyte]] to a [[kibibyte]]; see [[binary prefix]].<ref>{{Cite book
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| | title = Matlab und Simulink
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| | author = Ottmar Beucher
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| | publisher = Pearson Education
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| | year = 2008
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| | isbn = 978-3-8273-7340-3
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| | page = 195
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| | url = http://books.google.com/?id=VgLCb7B3OtYC&pg=PA195&dq=3.0103+1024+1000
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| }}</ref><ref>{{Cite book
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| | title = Digital Filters in Hardware: A Practical Guide for Firmware Engineers
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| | author = K. Ayob
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| | publisher = Trafford Publishing
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| | year = 2008
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| | isbn = 978-1-4251-4246-9
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| | page = 278
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| | url = http://books.google.com/?id=6nmnbIxpY3MC&pg=PA278&dq=3.0103-db
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| }}</ref>
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| * This coincidence can also be expressed <math>5^3 = 125 \approx 128 = 2^7 </math>, and is invoked for instance in [[shutter speed]] settings on cameras, as approximations to powers of two (128, 256, 512) in the sequence of speeds 125, 250, 500, etc.<ref name=schroeder/>
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| ====Concerning musical intervals====
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| * The coincidence <math>2^{19} \approx 3^{12}</math>, from <math>\frac{\log3}{\log2} \approx 1.5849\dots \approx \frac{19}{12}</math> leads to the observation commonly used in [[music]] to relate the tuning of 7 [[semitone]]s of [[equal temperament]] to a [[perfect fifth]] of [[just intonation]]: <math>2^{7/12}\approx 3/2;</math>, correct to about 0.1%. The just fifth is the basis of [[Pythagorean tuning]] and most known systems of music. From the consequent approximation <math>{(3/2)}^{12}\approx 2^7,</math> it follows that the [[circle of fifths]] terminates seven [[octave]]s higher than the origin.<ref name=schroeder>{{Cite book
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| | title = Number theory in science and communication | |
| | author = Manfred Robert Schroeder
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| | publisher = Springer
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| | edition = 2nd
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| | year = 2008
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| | isbn = 978-3-540-85297-1
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| | pages = 26–28
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| | url = http://books.google.com/?id=2KV2rfP0yWEC&pg=PA27&dq=coincidence+circle-of-fifths+1024+7-octaves+%22one+part+in+a+thousand%22
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| }}</ref>
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| * The coincidence <math>\sqrt[12]{2}\sqrt[7]{5} = 1.33333319\ldots \approx \frac43</math> leads to the [[Schisma|rational version]] of [[12-TET]], as noted by [[Johann Kirnberger]].{{Citation needed|date=September 2010}}
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| * The coincidence <math>\sqrt[8]{5}\sqrt[3]{35} = 4.00000559\ldots \approx 4</math> leads to the rational version of [[quarter-comma meantone]] temperament.{{Citation needed|date=September 2010}}
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| * The coincidence <math>\sqrt[9]{0.6}\sqrt[28]{4.9} = 0.99999999754\ldots \approx 1</math> leads to the very tiny interval of <math>2^{9}3^{-28}5^{37}7^{-18}</math> (about a [[Cent (music)|millicent]] wide), which is the first [[Limit (music)|7-limit]] interval tempered out in [[103169-TET]].{{Citation needed|date=September 2010}}
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| ===Numerical expressions===
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| ====Concerning powers of pi====
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| * <math>\pi^2\approx10;</math> correct to about 1.3%.<ref name="Pi">Frank Rubin, [http://www.contestcen.com/pi.htm The Contest Center – Pi].</ref> This can be understood in terms of the formula for the [[Riemann zeta function|zeta function]] <math>\zeta(2)=\pi^2/6.</math><ref>[http://www.math.harvard.edu/~elkies/Misc/pi10.pdf Why is <math>\pi^2</math> so close to 10?], [[Noam Elkies]]</ref> This coincidence was used in the design of [[slide rule]]s, where the "folded" scales are folded on <math>\pi</math> rather than <math>\sqrt{10},</math> because it is a more useful number and has the effect of folding the scales in about the same place.{{Citation needed|date=May 2009}}
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| * <math>\pi^2\approx 227/23,</math> correct to 0.0004%.<ref name="Pi"/>
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| * <math>\pi^3\approx 31,</math> correct to 0.02%.
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| * <math>\sqrt[5]{\pi^3+1}\approx 2,</math> correct to 0.004%.
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| * <math>\pi\approx\left(9^2+\frac{19^2}{22}\right)^{1/4},</math> or <math>22\pi^4\approx 2143;</math><ref name=wolfram/> accurate to 8 decimal places (due to [[Srinivasa Aiyangar Ramanujan|Ramanujan]]: ''Quarterly Journal of Mathematics'', XLV, 1914, pp. 350–372). Ramanujan states that this "curious approximation" to <math>\pi</math> was "obtained empirically" and has no connection with the theory developed in the remainder of the paper.
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| Some plausible relations hold to a high degree of accuracy, but are nevertheless coincidental. One example is:
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| <math>
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| \int_0^\infty \cos(2x)\prod_{n=1}^\infty \cos\left(\frac{x}{n}\right)dx \approx \frac{\pi}{8}
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| </math>
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| The two sides of this expression only differ after the 42nd decimal place.<ref>http://crd.lbl.gov/~dhbailey/dhbpapers/math-future.pdf</ref>
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| ====Containing both pi and e====
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| * <math>\pi^4+\pi^5\approx e^6</math>, within 0.000 005%<ref name=wolfram/>
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| * <math>e^\pi - \pi\approx 19.99909998 </math> is very close to 20 (Conway, Sloane, Plouffe, 1988); this is equivalent to <math>(\pi+20)^i=-0.999 999 999 2\ldots -i\cdot 0.000 039\ldots \approx -1</math><ref name=wolfram>{{MathWorld|urlname=AlmostInteger|title=Almost Integer}}</ref>
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| * <math> \pi^{3^2}/e^{2^3}\approx 9.9998\approx 10</math><ref name=wolfram/>
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| ====Containing pi or e and 163====
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| * <math>{163}\cdot (\pi - e) \approx 69</math>, [http://www.google.com/search?q=100*(163*(pi-e)%2F69-1) within 0.0005%]<ref name=wolfram/>
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| * <math>\frac{163}{\ln 163} \approx 2^{5}</math>, [http://www.google.com/search?q=100*((163/ln(163))^(1/5)%2F2-1) within 0.000004%] <ref name=wolfram/>
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| * [[Ramanujan's constant]]: <math>e^{\pi\sqrt{163}} \approx (2^6\cdot 10005)^3+744</math>, within <math>2.9\cdot 10^{-28}%</math>, discovered in 1859 by [[Charles Hermite]].<ref>{{cite book | last = Barrow | first = John D | title = The Constants of Nature | publisher = Jonathan Cape | year = 2002 | location = London | isbn = 0-224-06135-6 }}</ref> This very close approximation is not a typical sort of ''accidental'' mathematical coincidence, where no mathematical explanation is known or expected to exist (as is the case for most others here). It is a consequence of the fact that 163 is a [[Heegner number]].
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| ===Other numerical curiosities===
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| * <math>10! = 6! \cdot 7! = 1! \cdot 3! \cdot 5! \cdot 7!</math>.<ref>Harvey Heinz, [http://www.magic-squares.net/narciss.htm#Factorial%20Products ''Narcissistic Numbers''].</ref>
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| * <math>\, 2^3=8</math> and <math>3^2=9\,</math> are the only non-trivial (i.e. at least square) consecutive powers of positive integers ([[Catalan's conjecture]]).
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| * <math>\,4^2 = 2^4</math> is the only positive integer solution of <math>a^b = b^a, a\neq b</math><ref>Ask Dr. Math, [http://mathforum.org/library/drmath/view/66166.html "Solving the Equation x^y = y^x"].</ref> (see [[Lambert's W function]] for a formal solution method)
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| * The [[Fibonacci number]] ''F''<sub>296182</sub> is (probably) a [[semiprime]], since ''F''<sub>296182</sub> = ''F''<sub>148091</sub> × ''L''<sub>148091</sub> where ''F''<sub>148091</sub> (30949 digits) and the [[Lucas number]] ''L''<sub>148091</sub> (30950 digits) are simultaneously [[probable prime]]s.<ref>David Broadhurst, [http://primes.utm.edu/curios/page.php?number_id=3994 "Prime Curios!: 10660...49391 (61899-digits)"].</ref>
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| * In a discussion of the [[birthday problem]], the number <math>\lambda=\frac{1}{365}{23\choose 2}</math> occurs, which is "amusingly" equal to <math>\ln(2)</math> to 4 digits.<ref>{{cite journal
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| | last1 = Arratia | first1 = Richard | author1-link = Richard Arratia
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| | last2 = Goldstein | first2 = Larry
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| | last3 = Gordon | first3 = Louis
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| | issue = 4
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| | journal = [[Statistical Science]]
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| | jstor = 2245366
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| | mr = 1092983
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| | pages = 403–434
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| | title = Poisson approximation and the Chen-Stein method
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| | volume = 5
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| | year = 1990}}</ref>
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| ===Decimal coincidences===
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| * <math>2^5 \cdot 9^2 = 2592</math>. This makes 2592 a nice [[Friedman number]].<ref name="Friedman">Erich Friedman, [http://www.stetson.edu/~efriedma/mathmagic/0800.html Problem of the Month (August 2000)].</ref>
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| * <math>\,1! + 4! + 5! = 145</math>. The only such [[factorion]]s (in base 10) are 1, 2, 145, 40585.<ref>{{OEIS|A014080}}</ref>
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| * <math>\frac {16} {64} = \frac {1\!\!\!\not6} {\not6 4} = \frac {1} {4}</math>, <math>\frac {26} {65} = \frac {2\!\!\!\not6} {\not6 5} = \frac {2} {5}</math>, <math>\frac {19} {95} = \frac {1\!\!\!\not9} {\not9 5} = \frac {1} {5}</math>, <math>\frac{49}{98}=\frac{4\!\!\!\not9}{\not98}=\frac{4}{8}</math> ([[anomalous cancellation]]<ref>{{Mathworld|title=Anomalous Cancellation|urlname=AnomalousCancellation}}</ref>). Also, the product of these four fractions reduces to exactly 1/100.
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| * <math>\,(4 + 9 + 1 + 3)^3 = 4{,}913</math> and <math>\,(1 + 9 + 6 + 8 + 3)^3=19{,}683</math>.<ref>{{OEIS|A061209}}</ref>
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| * <math>\,2^7 - 1 = 127</math>. This can also be written <math>\,127 = -1 + 2^7</math>, making 127 the smallest nice Friedman number.<ref name="Friedman"/>
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| * <math>\,1^3 + 5^3 + 3^3 = 153</math> ; <math>\,3^3 + 7^3 + 0^3 = 370</math> ; <math>\,3^3 + 7^3 +1^3 = 371</math> ; <math>\,4^3 + 0^3 +7^3 = 407</math> — all [[narcissistic numbers]]<ref>{{OEIS|A005188}}</ref>
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| * <math>\,(3 + 4)^3 = 343 </math><ref>[http://primes.utm.edu/curios/page.php/343.html Prime Curios!: 343].</ref>
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| * <math>\,588^2+2353^2 = 5882353 </math> and also <math>\, 1/17 = 0.0588235294117647\ldots</math> when rounded to 8 digits is 0.05882353. Mentioned by Gilbert Labelle in ~1980.{{Citation needed|date=September 2009}} 5882353 also happens to be prime.
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| * <math>\,2646798 = 2^1+6^2+4^3+6^4+7^5+9^6+8^7</math>. The largest such number is 12157692622039623539.<ref>{{OEIS|A032799}}</ref>
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| * <math>\sin(666^\circ) = \cos(6\cdot6\cdot6^\circ) = - \varphi/2</math>, where <math>\varphi</math> is the [[golden ratio]]<ref name=BeastNumber>{{MathWorld|title=Beast Number|urlname=BeastNumber}}</ref> (an amusing equality with an angle expressed in degrees) (see [[Number of the Beast]])
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| * <math>\,\phi(666)=6\cdot6\cdot6</math>, where <math>\phi</math> is [[Euler's totient function]]<ref name=BeastNumber/>
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| ===Numerical coincidences in numbers from the physical world===
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| ====Speed of light====
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| The [[speed of light]] is (by definition) exactly 299,792,458 m/s, very close to 300,000 km/s. This is a pure coincidence.<ref name="Miracles">{{cite web|last=Michon|first=Gérard P.|title=Numerical Coincidences in Man-Made Numbers|url=http://www.numericana.com/answer/miracles.htm|work=Mathematical Miracles|accessdate=29 April 2011}}</ref> It also roughly equals to one foot per nanosecond (the actual number is 0.9836 ft/ns).
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| ====Earth's diameter====
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| The polar diameter of the Earth is equal to half a billion inches, to within 0.1%.<ref>{{cite book |title=Our Inheritance in the Great Pyramid |first1=Charles |last1=Smythe |page=39 |url= http://books.google.co.uk/books?id=xB0WSOokTg0C&pg=PA39 |year=2004 |publisher=Kessinger Publishing |isbn=1-4179-7429-X}}</ref>
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| ====Gravitational acceleration====
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| While not constant but varying depending on [[latitude]] and [[altitude]], the [[gravitational acceleration|acceleration caused by Earth's gravity]] on the surface lies between 9.74 and 9.87, which is quite close to 10. This means that as a result of [[Newton's laws of motion|Newton's second law]], the weight of a kilogram of mass on Earth's surface corresponds roughly to 10 newtons of force exerted on an object.<ref>{{cite book |title=Cracking the AP Physics B & C Exam, 2004–2005 Edition |page=25 |url=http://books.google.co.uk/books?id=XcX_TvhNjK0C&pg=PA25&dq=approximation |isbn=0-375-76387-2 |year=2003 |publisher=Princeton Review Publishing}}</ref>
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| This is actually related to the aforementioned coincidence that the square of pi is close to 10. Originally, the meter was defined as the length of a pendulum whose half swing had a period equal to one second. Since the period of the full swing of a pendulum is approximated by the equation below, algebra shows that if this definition was maintained, gravitational acceleration measured in meters per second per second would be exactly equal to the square of pi.{{citation needed|date=April 2013}}
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| :<math>T \approx 2\pi \sqrt\frac{L}{g}</math>
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| When it was discovered that the circumference of the earth was very close to 40,000,000 times this value, the meter was [[Meter#Timeline of definition|redefined]] to reflect this, as it was a more objective standard. This had the effect of increasing the length of the meter by less than 1%, which was within the experimental error of the time.{{citation needed|date=April 2013}}
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| ====Rydberg constant====
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| The [[Rydberg constant]], when multiplied by the speed of light and expressed as a frequency, is close to <math>\frac{\pi^2}{3}\times 10^{15} \text{Hz}</math>:<ref name="Miracles"/>
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| :<math>\underline{3.2898}41960364(17) \times 10^{15} \text{Hz} = R_\infty c</math><ref>{{cite web|title=Rydberg constant times c in Hz|url=http://physics.nist.gov/cgi-bin/cuu/Value?rydchz|work=Fundamental physical constants|publisher=NIST|accessdate=25 July 2011}}</ref>
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| :<math>\underline{3.2898}68133696... = \frac{\pi^2}{3}</math>
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| ===Fine-structure constant===
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| The [[Fine-structure constant]] <math>\alpha</math> is close to <math>\frac1{137}</math> and was once conjectured to be precisely <math>\frac1{137}</math>.
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| :<math>\alpha = \frac1{137.035999074\dots}</math>
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| Although this coincidence is not as strong as some of the others in this section, it is notable that <math>\alpha</math> is a dimensionless constant, so this coincidence is not an artifact of the system of units being used.
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| ==See also==
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| * For a list of coincidences in [[physics]], see [[anthropic principle]]
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| * [[Almost integer]]
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| * [[Birthday problem]]
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| * [[Exceptional isomorphism]]
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| * [[Narcissistic number]]
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| * [[Experimental mathematics]]
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| * [[Kepler triangle#A mathematical coincidence]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| * {{ru icon}} В. Левшин. – ''Магистр рассеянных наук.'' – Москва, Детская Литература 1970, 256 с.
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| * [[G. H. Hardy|Hardy, G. H.]] – ''[[A Mathematician's Apology]].'' – New York: Cambridge University Press, 1993, (ISBN 0-521-42706-1)
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| * {{MathWorld|urlname=AlmostInteger|title=Almost Integer}}
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| * [http://www.futilitycloset.com/category/science-math/ Various mathematical coincidences] in the "Science & Math" section of futilitycloset.com
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| * [[William H. Press|Press, W. H.]], [http://www.nr.com/whp/NumericalCoincidences.pdf Seemingly Remarkable Mathematical Coincidences Are Easy to Generate]
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| {{DEFAULTSORT:Mathematical Coincidence}}
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| [[Category:Mathematical terminology]]
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| [[Category:Recreational mathematics]]
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