A mathematical coincidence is said to occur when two expressions with no direct relationship show a near-equality which has no apparent theoretical explanation.
A mathematical coincidence often involves an integer, and the surprising feature is the fact that a real number arising in some context is considered by some 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.
Given the countably infinite number of ways of forming mathematical expressions using a finite number of symbols, the number of symbols used and the 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] Beyond this, some sense of 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' joke[1]).
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.
Sometimes simple rational approximations are exceptionally close to interesting irrational values.
[2] Many other coincidences are combinations of numbers that put them into the form that such rational approximants provide close relationships.
In western twelve-tone equal temperament, the ratio between consecutive note frequencies is
This is a pure coincidence, as the metre was originally defined as 1 / 10000000 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.
[39] It is also roughly equal to one foot per nanosecond (the actual number is 0.9836 ft/ns).
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″.
While not constant but varying depending on latitude and altitude, the numerical value of the acceleration caused by Earth's gravity on the surface lies between 9.74 and 9.87 m/s2, which is quite close to 10.
This means that as a result of 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.
[40] 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 π2.
[41] The upper limit of gravity on Earth's surface (9.87 m/s2) is equal to π2 m/s2 to four significant figures.
The Rydberg constant, when multiplied by the speed of light and expressed as a frequency, is close to
:[39] This is also approximately the number of feet in one meter: As discovered by Randall Munroe, a cubic mile is close to
This means that a sphere with radius n kilometres has almost exactly the same volume as a cube with side length n miles.
is a dimensionless physical constant, so this coincidence is not an artifact of the system of units being used.
The number of seconds in one year, based on the Gregorian calendar, can be calculated by: