Editing Mathematical coincidence (section)

=== Numerical coincidences in numbers from the physical world ===

==== Speed of light ====
The [[speed of light]] is (by definition) exactly {{val|299,792,458|u=m/s}}, extremely close to {{val|3.0|e=8|u=m/s}} ({{val|300,000,000|u=m/s}}). This is a pure coincidence, as the metre was originally defined as 1 / {{val|10,000,000}} of the distance between the Earth's pole and equator along the surface at sea level, and the Earth's circumference just happens to be about 2/15 of a light-second.<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|access-date=29 April 2011}}</ref> It is also roughly equal to one foot per [[nanosecond]] (the actual number is 0.9836&nbsp;ft/ns).

==== Angular diameters of the Sun and the Moon ====
As seen from Earth, the [[angular diameter]] of the [[Sun]] varies between 31′27″ and 32′32″, while that of the [[Moon]] is between 29′20″ and 34′6″. The fact that the intervals overlap (the former interval is contained in the latter) is a coincidence, and has implications for the types of [[solar eclipse]]s that can be observed from Earth.

==== Gravitational acceleration ====
{{See also|Seconds pendulum}}
While not constant but varying depending on [[latitude]] and [[altitude]], the numerical value of the [[gravitational acceleration|acceleration caused by Earth's gravity]] on the surface lies between 9.74 and 9.87 [[metre per second squared|m/s<sup>2</sup>]], 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 [[Newton (unit)|newtons]] of force exerted on an object.<ref>{{cite book |title=Cracking the AP Physics B & C Exam, 2004–2005 Edition |page=25 |url=https://books.google.com/books?id=XcX_TvhNjK0C&q=approximation&pg=PA25 |isbn=978-0-375-76387-8 |year=2003 |publisher=Princeton Review Publishing}}</ref>

This is related to the aforementioned coincidence that the square of pi is close to 10. One of the early definitions of the metre was 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 metres per second per second would be exactly equal to ''π''<sup>2</sup>.<ref>{{cite journal|url=https://www.wired.com/2013/03/what-does-pi-have-to-do-with-gravity/|title=What Does Pi Have To Do With Gravity?|journal=[[Wired (website)|Wired]]|date=March 8, 2013|access-date=October 15, 2015}}</ref>
: <math>T \approx 2\pi \sqrt\frac{L}{g}</math>

The upper limit of gravity on Earth's surface (9.87 m/s<sup>2</sup>) is equal to π<sup>2</sup> m/s<sup>2</sup> to four significant figures. It is approximately 0.6% greater than [[standard gravity]] (9.80665 m/s<sup>2</sup>).

==== Rydberg constant ====
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"/>
: <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|access-date=25 July 2011}}</ref>
: <math>\underline{3.2898}68133696\ldots = \frac{\pi^2}{3}</math>

This is also approximately the number of feet in one meter:
: <math>3.28084</math> ft  <math>\approx 1</math> m

==== US customary to metric conversions ====
As discovered by [[Randall Munroe]], a cubic mile is close to <math>\frac{4}{3}\pi</math> cubic kilometres (within 0.5%). This means that a sphere with radius ''n'' kilometres has almost exactly the same volume as a cube with side length ''n'' miles.<ref>{{cite book|title=What If?|author=Randall Munroe|year=2014|page=49|publisher=Hodder & Stoughton |isbn=9781848549562}}</ref><ref>{{Cite web|url=https://what-if.xkcd.com/4/|title=A Mole of Moles|website=what-if.xkcd.com|access-date=2018-09-12}}</ref>

The ratio of a mile to a kilometre is approximately the [[Golden ratio]].  As a consequence, a [[Fibonacci Number|Fibonacci number]] of miles is approximately the next Fibonacci number of kilometres.

The ratio of a mile to a kilometre is also very close to <math>\ln(5)</math> (within 0.006%). That is, <math>5^m \approx e^k</math> where ''m'' is the number of miles, ''k'' is the number of kilometres and ''e'' is [[e (mathematical constant)|Euler's number]].

A density of one ounce per cubic foot is very close to one kilogram per cubic metre: 1&nbsp;oz/ft<sup>3</sup> = 1&nbsp;oz × 0.028349523125&nbsp;kg/oz / (1&nbsp;ft × 0.3048&nbsp;m/ft)<sup>3</sup> ≈ 1.0012&nbsp;kg/m<sup>3</sup>.

The ratio between one troy ounce and one gram is approximately <math> 10\pi-\frac{\pi}{10} = \frac{99}{10}\pi</math>.

==== Fine-structure constant ====
The [[fine-structure constant]] <math>\alpha</math> is close to, and was once conjectured to be precisely equal to {{sfrac|1|137}}.<ref>{{cite journal|last=Whittaker|first=Edmund|date=1945|title=Eddington's Theory of the Constants of Nature|journal=The Mathematical Gazette|volume=29|issue=286|pages=137–144|doi=10.2307/3609461|jstor=3609461|s2cid=125122360 }}</ref>  Its [[CODATA]] recommended value is
: <math>\alpha</math> = {{sfrac|1|{{physconst|alphainv|ref=no}}}}
<math>\alpha</math> is a [[dimensionless physical constant]], so this coincidence is not an artifact of the system of units being used.

==== Earth's Solar Orbit ====
The number of seconds in one year, based on the [[Gregorian calendar]], can be calculated by: <math>365.2425\left(\frac{\text{days}}{\text{year}}\right)\times 24\left(\frac{\text{hours}}{\text{day}}\right)\times 60\left(\frac{\text{minutes}}{\text{hour}}\right) \times 60\left(\frac{\text{seconds}}{\text{minute}}\right)=
31,556,952\left(\frac{\text{seconds}}{\text{year}}\right)</math>

This value can be approximated by <math>\pi\times10^7</math> or 31,415,926.54 with less than one percent of an error: <math>\left[1 - \left(\frac{31,415,926.54}{31,556,952}\right)\right]\times 100 = 0.4489%</math>