Editing Mathematical coincidence (section)

=== Numerical expressions ===

==== Concerning powers of ''π'' ====
* <math>\pi^2\approx10;</math> correct to about 1.32%.<ref name="Pi">{{cite web |first=Frank |last=Rubin |url=http://www.contestcen.com/pi.htm |title=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>{{cite web |url=http://www.math.harvard.edu/~elkies/Misc/pi10.pdf |title=Why is <math>\pi^2</math> so close to 10? |first=Noam |last=Elkies |authorlink=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}}
* <math>\pi^2+\pi\approx13;</math> correct to about 0.086%.
* <math>\pi^2\approx 227/23,</math> correct to 4 parts per million.<ref name="Pi" />
* <math>\pi^3\approx31,</math> correct to 0.02%.<ref>[http://mathworld.wolfram.com/PiApproximations.html Mathworld, Pi Approximations], Line 47</ref>
* <math>2\pi^3 -\pi^2-\pi \approx7^2,</math> correct to about 0.002% and can be seen as a combination of the above coincidences.
* <math>\pi^4\approx 2143/22;</math> or <math>\pi\approx\left(9^2+\frac{19^2}{22}\right)^{1/4},</math> accurate to 8 decimal places (due to [[Srinivasa Aiyangar Ramanujan|Ramanujan]]: ''Quarterly Journal of Mathematics'', XLV, 1914, pp.&nbsp;350–372).<ref name="wolfram" />  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.
* Some near-equivalences, which hold to a high degree of accuracy, are not actually coincidences. For example,
*: <math>
\int_0^\infty \cos(2x)\prod_{n=1}^\infty \cos\left(\frac{x}{n}\right)\mathrm{d}x \approx \frac{\pi}{8}.
</math>
: The two sides of this expression differ only after the 42nd decimal place; this is [[Borwein integral|not a coincidence]].<ref>{{Cite journal |last1=Bailey |first1=David |last2=Borwein |first2=Jonathan |last3=Kapoor |first3=Vishal |last4=Weisstein |first4=Eric |date=9 March 2006 |title=Ten Problems in Experimental Mathematics |url=http://crd.lbl.gov/~dhbailey/dhbpapers/tenproblems.pdf |url-status=dead |journal=The American Mathematical Monthly |volume=113 |issue=6 |pages=22 |doi=10.1080/00029890.2006.11920330 |s2cid=13560576 |archive-url=https://web.archive.org/web/20070418024214/http://crd.lbl.gov/~dhbailey/dhbpapers/tenproblems.pdf |archive-date=18 April 2007}}</ref><ref>{{cite web |title=Future Prospects for Computer-Assisted Mathematics |first1=David H. |last1=Bailey |first2=Jonathan M. |last2=Borwein |date=December 1, 2005 |url=https://www.davidhbailey.com//dhbpapers/math-future.pdf }}</ref>

==== Containing both ''π'' and ''e'' ====
* <math>\pi \approx 1 + e - \gamma</math> to 4 digits, where γ is the Euler–Mascheroni constant.
* <math>\pi^4+\pi^5\approx e^6</math>, to about 7 decimal places.<ref name="wolfram" /> Equivalently, <math>4 \cdot \ln(\pi) + \ln(\pi+1) \approx 6</math>.
* <math>(e-1)\pi\approx\sqrt{5}+\sqrt{10}</math>, to about 4 decimal places.
* <math>\left(\frac{\pi}{2} - \ln\left( \frac{3\pi}{2}\right) \right)42\pi \approx e</math>, to about 9 decimal places.<ref>{{Cite web|url=http://rtomas.web.cern.ch/rtomas/Coincidences/|title=Rogelio Tomas' web page}}</ref>
* <math>e^\pi - \pi\approx 20</math> to about 4 decimal places. (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> Once considered a textbook example of a mathematical coincidence,<ref>{{Citation|last1=Maze|first1=G.|last2=Minder|first2=L.|title=A New Family of Almost Identities|date=28 June 2005|arxiv=math/0409014|pages=1}}</ref><ref>{{Cite web|url=https://mathworld.wolfram.com/AlmostInteger.html|title=Almost Integer|archive-url=https://web.archive.org/web/20231127023501/https://mathworld.wolfram.com/AlmostInteger.html|archive-date=27 November 2023|date=10 November 2023}}</ref> the fact that <math>e^\pi - \pi</math> is close to 20 is itself not a coincidence, although the approximation is an order of magnitude closer than would be expected. No explanation for the near-identity was known until 2023. It is a consequence of the infinite sum <math>\textstyle\sum_{k=1}^{\infty }\left ( 8\pi k^{2}-2 \right )e^{\left (-\pi k^{2}  \right )}=1,</math> resulting from the [[Theta function|Jacobian theta functional identity]]. The first term of the sum is by far the largest, which gives the approximation <math>\left (8\pi-2  \right )e^{-\pi}\approx 1,</math> or <math>e^{\pi}\approx 8\pi-2.</math> Using the estimate <math>\pi \approx 22/7</math> then gives <math>e^{\pi}\approx \pi+(7\cdot\frac{22}{7}-2) = \pi+20.</math><ref>{{Cite web|url=https://mathworld.wolfram.com/AlmostInteger.html|title=Almost Integer|archive-url=https://web.archive.org/web/20231203115913/https://mathworld.wolfram.com/AlmostInteger.html|archive-date=3 December 2023|date=1 December 2023|quote=(A. Doman, Sep. 18, 2023, communicated by D. Bamberger, Nov. 26, 2023). Amusingly, the choice of π≈22/7 (which is not mathematically significant compared to other choices except that it makes the final form very simple) in the last step makes the formula an order of magnitude more precise than it would otherwise be.}}</ref>
* <math>\pi^e+e^\pi \approx 45\frac{3}{5}</math>, within 4 parts per million.
* <math>\pi^9/e^8\approx 10</math>, to about 5 decimal places.<ref name="wolfram">{{MathWorld|urlname=AlmostInteger|title=Almost Integer}}</ref> That is, <math>\ln(\pi) \approx {\ln(10)+8 \over 9}</math>, within 0.0002%.
* <math>2\pi + e \approx 9</math>, within 0.02%.<ref>{{Citation|last=Irkhin|first=V. Yu.|title=Relations between <math>e</math> and <math>\pi</math>: Nilakantha's series and Stirling's formula|date=13 June 2022|arxiv=2206.07174|pages=1}}</ref>
* <math display="inline">e^{-\frac{\pi}{9}} + e^{-4\frac{\pi}{9}} + e^{-9\frac{\pi}{9}} + e^{-16\frac{\pi}{9}} + e^{-25\frac{\pi}{9}} + e^{-36\frac{\pi}{9}} + e^{-49\frac{\pi}{9}} + e^{-64\frac{\pi}{9}} = 1.00000000000105\ldots \approx 1</math>. In fact, this generalizes to the approximate identity <math>\textstyle\sum_{k=1}^{n-1}{e^{-\frac{k^2\pi}{n}}}\approx\frac{-1+\sqrt{n}}{2},</math> which can be explained by the Jacobian theta functional identity.<ref>{{Cite web|url=https://math.stackexchange.com/q/2072897 |title=Curious relation between <math>e</math> and <math>\pi</math> that produces almost integers|work=Math Stack Exchange |date=December 26, 2016 |access-date=2017-12-04 }}</ref><ref>{{Cite journal |url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN600494829_0015%7CLOG_0020 |via=Göttinger Digitalisierungszentrum |title=An Approximate Numerical Theorem Involving <var>e</var> and <var>π</var> |first= J. W. L. |last=Glaisher |journal=The Quarterly Journal of Pure and Applied Mathematics }}</ref><ref>{{Cite web|url=https://math.stackexchange.com/q/30804 |title=Proving the identity <math>\textstyle\sum_{n=-\infty}^\infty e^{-\pi n^2x}=x^{-1/2}\sum_{n=-\infty}^\infty e^{-\pi n^2/x}</math>|work=Stack Exchange |date=December 5, 2013 |access-date=2017-12-04}}</ref>
* [[Ramanujan's constant]]: <math>e^{\pi\sqrt{163}} \approx 262537412640768744 = 12^3(231^2-1)^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 = 978-0-224-06135-3 }}</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).  It is a consequence of the fact that 163 is a [[Heegner number]].
* There are several integers <math> k= 2198, 422151, 614552, 2508952, 6635624, 199148648,\dots</math> ({{oeis|A019297}}) such that <math>\pi \approx \frac{\ln(k)}{\sqrt{n}}</math> for some integer ''n'', or equivalently <math>k \approx e^{\pi\sqrt{n}}</math> for the same <math>n = 6, 17, 18, 22, 25, 37,\dots</math> These are not strictly coincidental because they are related to both [[Ramanujan's constant]] above and the [[Heegner number]]s. For example, <math>k=199148648 = 14112^2+104,</math> so these integers ''k'' are near-squares or near-cubes and note the consistent forms for ''n'' = 18, 22, 37,
:<math>\pi \approx \frac{\ln (784^2-104)}{\sqrt{18}}</math>
:<math>\pi \approx \frac{\ln (1584^2-104)}{\sqrt{22}}</math>
:<math>\pi \approx \frac{\ln (14112^2+104)}{\sqrt{37}}</math> 
with the last accurate to 14 or 15 decimal places.
* <math>(e^e)^e \approx 1000\varphi</math>
* <math>\frac{10(e^\pi-\ln3)}{\ln2} = 318.000000033\ldots</math> is [[Almost integer|almost an integer]], to about 8th decimal place.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Almost Integer |url=https://mathworld.wolfram.com/ |access-date=2022-07-15 |website=mathworld.wolfram.com |language=en}}</ref>