Editing Mathematical coincidence (section)

=== Rational approximants ===
Sometimes simple rational approximations are exceptionally close to interesting irrational values. These are explainable in terms of large terms in the [[continued fraction|continued fraction representation]] of the irrational value, but further insight into why such improbably large terms occur is often not available.

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/>

Many other coincidences are combinations of numbers that put them into the form that such rational approximants provide close relationships.

==== Concerning ''π'' ====
* The second [[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 fourth convergent of ''π'', [3; 7, 15, 1] = [[355/113]] = 3.1415929..., found by [[Zu Chongzhi]],<ref>
{{cite book
 | author = Yoshio Mikami
 | title = Development of Mathematics in China and Japan
 | publisher = B. G. Teubner
 | page = 135
 | year = 1913
 | url = https://books.google.com/books?id=4e9LAAAAMAAJ&q=intitle:Development+intitle:%22China+and+Japan%22+355 }}</ref> is correct to six decimal places;<ref name=beckmann>{{Cite book
 | title = A History of Pi
 | author = Petr Beckmann
 | publisher = Macmillan
 | year = 1971
 | isbn = 978-0-312-38185-1
 | pages = 101, 170
 | url = https://books.google.com/books?id=TB6jzz3ZDTEC&q=pi+113+355++digits&pg=PA101
 }}</ref> this high accuracy comes about because π has an unusually large next term in its continued fraction representation: {{pi}} = [3; 7, 15, 1, 292, ...].<ref>{{Cite book
 | title = CRC concise encyclopedia of mathematics
 | author = Eric W. Weisstein
 | publisher = CRC Press
 | year = 2003
 | isbn = 978-1-58488-347-0
 | page = 2232
 | url = https://books.google.com/books?id=_8TyhSqHUiEC&q=pi+113+355++292+convergent&pg=PA2232
 }}</ref>
* A coincidence involving ''π'' and the [[golden ratio]] ''φ'' is given by <math>\pi \approx 4 / \sqrt{\varphi} = 3.1446\dots</math>. Consequently, the square on the middle-sized edge of a [[Kepler triangle]] is similar in perimeter to its circumcircle. Some believe one or the other of these coincidences is to be found in the [[Golden Ratio#Egyptian pyramids|Great Pyramid of Giza]], but it is highly improbable that this was intentional.<ref>
{{cite book
 | title = The Shape of the Great Pyramid
 | author = Roger Herz-Fischler
 | publisher = Wilfrid Laurier University Press
 | year = 2000
 | isbn = 978-0-889-20324-2
 | page = 67
 | url = https://books.google.com/books?id=_8TyhSqHUiEC&q=pi+113+355++292+convergent&pg=PA2232
}}</ref>
* There is a sequence of [[six nines in pi]] beginning at the 762nd decimal place of its decimal representation. For a randomly chosen [[normal number]], the probability of a particular sequence of six consecutive digits—of any type, not just a repeating one—to appear this early is 0.08%.<ref name="ArndtHaenel">{{Citation |last1=Arndt |first1=J. |title=Pi – Unleashed |page=3 |year=2001 |location=Berlin |publisher=Springer |isbn=3-540-66572-2 |name-list-style=amp |last2=Haenel |first2=C.}}.</ref> Pi is conjectured, but not known, to be a normal number.
* The first [[Feigenbaum constant]] is approximately equal to <math>\tfrac{10}{\pi-1}</math>, with an error of 0.0015%.

==== Concerning base 2 ====
* 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 [[Electric power|power]] as 3&nbsp;[[decibel|dB]] (actual is 3.0103&nbsp;dB – see [[Half-power point]]), or to relate a [[kibibyte]] to a [[kilobyte]]; see [[binary prefix]].<ref>
{{cite book
 | title = Matlab und Simulink
 | author = Ottmar Beucher
 | publisher = Pearson Education
 | year = 2008
 | isbn = 978-3-8273-7340-3
 | page = 195
 | url = https://books.google.com/books?id=VgLCb7B3OtYC&q=3.0103+1024+1000&pg=PA195
}}</ref><ref>
{{cite book
 | title = Digital Filters in Hardware: A Practical Guide for Firmware Engineers
 | author = K. Ayob
 | publisher = Trafford Publishing
 | year = 2008
 | isbn = 978-1-4251-4246-9
 | page = 278
 | url = https://books.google.com/books?id=6nmnbIxpY3MC&q=3.0103-db&pg=PA278
}}</ref> The same numerical coincidence is responsible for the near equality between one third of an octave and one tenth of a decade.<ref>Ainslie, M. A., Halvorsen, M. B., & Robinson, S. P. (2021). A terminology standard for underwater acoustics and the benefits of international standardization. IEEE Journal of Oceanic Engineering, 47(1), 179-200.</ref>
* The same coincidence can also be expressed as <math>128 = 2^7 \approx 5^3 = 125</math> (eliminating common factor of <math>2^3</math>, so also correct to 2.4%), which corresponds to the rational approximation <math>\textstyle\frac{\log5}{\log2} \approx 2.3219 \approx \frac{7}{3}</math>, or <math> 2 \approx 5^{3/7}</math> (also to within 0.4%). This is invoked in [[Preferred number|preferred numbers]] in engineering, such as [[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/> and in the original ''[[Who Wants to Be a Millionaire?]]'' game show in the question values ...£16,000, £32,000, £64,000, '''£125,000''', £250,000,...

==== Concerning musical intervals ====
{{See also|Musical temperament}}
In music, the distances between notes (intervals) are measured as ratios of their frequencies, with near-rational ratios often sounding harmonious. In western [[twelve-tone equal temperament]], the ratio between consecutive note frequencies is <math>\sqrt[12]{2}</math>. 
* The coincidence <math>2^{19} \approx 3^{12}</math>, from <math>\frac{\log3}{\log2} = 1.5849\ldots \approx \frac{19}{12}</math>, closely relates the interval of 7 [[semitone]]s in [[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]]; the difference between [[circle of fifths|twelve just fifths]] and seven octaves is the [[Pythagorean comma]].<ref name="schroeder">
{{cite book
 | title = Number theory in science and communication
 | author = Manfred Robert Schroeder
 | publisher = Springer
 | edition = 2nd
 | year = 2008
 | isbn = 978-3-540-85297-1
 | pages = 26–28
 | url = https://books.google.com/books?id=2KV2rfP0yWEC&q=coincidence+circle-of-fifths+1024+7-octaves+%22one+part+in+a+thousand%22&pg=PA27
}}</ref>
* The coincidence <math>{(3/2)}^{4} = (81/16) \approx 5</math> permitted the development of [[meantone temperament]], in which just perfect fifths (ratio <math>3/2</math>) and [[major third]]s (<math>5/4</math>) are "tempered" so that four <math>3/2</math>'s is approximately equal to <math>5/1</math>, or a <math>5/4</math> major third up two octaves. The difference (<math>81/80</math>) between these stacks of intervals is the [[syntonic comma]].{{cn|date=March 2022}}
* 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}}
* 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}}
* The coincidence of powers of 2, above, leads to the approximation that three major thirds concatenate to an octave, <math>{(5/4)}^{3} \approx {2/1}</math>.  This and similar approximations in music are called [[Diesis|dieses]].