Editing Mathematical coincidence (section)

==== 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>