A mathematical coincidence is said to occur when two expressions with no direct relationship show a near-equality which has no apparent theoretical explanation.
For example, there is a near-equality close to the round number1000 between powers of 2 and powers of 10:
Some mathematical coincidences are used in engineering when one expression is taken as an approximation of another.
A mathematical coincidence often involves an integer, and the surprising (or "coincidental") 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. Other kinds of mathematical coincidences, such as integers simultaneously satisfying multiple seemingly unrelated criteria or coincidences regarding units of measurement, may also be considered. In the class of those coincidences that are of a purely mathematical sort, some simply result from sometimes very deep mathematical facts, while others appear to come 'out of the blue'.
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. 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). 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. These are explainable in terms of large terms in the 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.
Many other coincidences are combinations of numbers that put them into the form that such rational approximants provide close relationships.
The first convergent of ?, [3; 7] = 22/7 = 3.1428..., was known to Archimedes, and is correct to about 0.04%. The third convergent of ?, [3; 7, 15, 1] = 355/113 = 3.1415929..., found by Zu Chongzhi, is correct to six decimal places; this high accuracy comes about because ? has an unusually large next term in its continued fraction representation: ? = [3; 7, 15, 1, 292, ...].
There is a sequence of six nines in pi that begins at the 762nd decimal place of the decimal representation of pi. For a randomly chosen normal number, the probability of any chosen number sequence of six digits (including 6 of a number, 658 020, or the like) occurring this early in the decimal representation is only 0.08%. Pi is conjectured, but not known, to be a normal number.
; correct to within 0.002%.
; correct to within 0.05%.
The number 1828 repeats twice in a row after the 2nd digit in the decimal expansion of e and immediately after that the next digits are 459045, which show the angles of an isosceles right triangle in degrees. After that, the next 3 digits are the first 3 prime numbers, and after that is 360, the sum of all angles of a square/circle. e = 2.7 1828 1828 45 90 45 2 3 5 360....
Related to the above, . When calculating e in base 60 or 120, it is useful to find 1001 (1+1/1!+1/2! .. +1/13!). These are exact expressions in these bases. Also, 718 + 281 = 999, (i.e. 718 281 718 ... is 719/1001), which makes this one of the convergences to e.
There is a sequence of "99 999 999" within the first 500,000 digits of e.
Concerning base 2
The coincidence , correct to 2.4%, relates to the rational approximation , or to within 0.3%. This relationship is used in engineering, for example to approximate a factor of two in power as 3 dB (actual is 3.0103 dB - see Half-power point), or to relate a kibibyte to a kilobyte; see binary prefix.
This coincidence can also be expressed as (eliminating common factor of , so also correct to 2.4%), which corresponds to the rational approximation , or (also to within 0.3%). This is invoked for instance in shutter speed settings on cameras, as approximations to powers of two (128, 256, 512) in the sequence of speeds 125, 250, 500, etc.
The coincidence leads to the very tiny interval of (about a millicent wide), which is the first 7-limit interval tempered out in 103169-TET.
The coincidence of powers of 2, above, leads to the approximation that three major thirds concatenate to an octave, . This and similar approximations in music are called dieses.
Concerning powers of ?
correct to about 1.3%. This can be understood in terms of the formula for the zeta function This coincidence was used in the design of slide rules, where the "folded" scales are folded on rather than because it is a more useful number and has the effect of folding the scales in about the same place.
The solid angle of the 120-cell of 4 dimensions is 191/600 of the 4-sphere, is very close to 1/pi. This is also the approximation that 1 radian is 57.3 degrees.
This approximation is . The polygon of 377 sides crosses a circle of diameter 120 on every edge. As 3° 8' 30" it is an approximation to pi in sexagesimal numbers.
If the quadrant arc is 10, the chord is 9, is known to the ancient Egyptians. A litre represented as a cylinder of 16 inches circumference and three inches high (48 Hoppus inches) is equal to a cylinder 0.3 feet in circumference and 6 inches high, (0.045 cyl. ft) are equal with this pi. The reflective cell of the symmetry of the [3,3,5] is equal to unity, when the radius of the 4-sphere is taken as 9.[clarification needed]
correct to 0.02%.
correct to 0.004%.
or  accurate to 8 decimal places (due to Ramanujan: Quarterly Journal of Mathematics, XLV, 1914, pp. 350-372). Ramanujan states that this "curious approximation" to was "obtained empirically" and has no connection with the theory developed in the remainder of the paper.
Some plausible relations hold to a high degree of accuracy, but are nevertheless coincidental. One example is
The two sides of this expression only differ after the 42nd decimal place.
137 ? * [137 * * ? + * + ] to within about 0.0123%, where = alpha fine struct = 0.0072973525664(17) and ? = Golden Ratio = 1.61803398874989484820458683436563811772030917980576286213544862270526046281890.
Ramanujan's constant: , within , discovered in 1859 by Charles Hermite. 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 others here). It is a consequence of the fact that 163 is a Heegner number.
, correct to 0.00024%
, within 0.000028%. Equivalently: (in all bases).
, correct to 0.08%
, correct to 0.6% (for all ).
to 0.000 001 %
Both of these show up nicely in base 18. In the first, the fibonacci numbers are while the multiplication series starting with 2, 3 runs through 18, and descends to 243/256 and 32/27.
shows in the logarithm-approximation log(3) / 13 ~ log(5) / 19 ~ log(7) / 23.
. This makes 3435 a base-10 Münchhausen number. In fact, other than 0 and 1, the number 3435 is the only base-10 Münchhausen number. However, if one is willing to, for these purposes, adopt the convention that , then 438579088 is another Münchhausen number. For a few months, the number 3435 was the favorite number of Matt Parker, a stand-up mathematician, but he soon "got bored" and switched his favorite number to other types of numbers, such as narcissistic numbers and perfect numbers. Parker vehemently disagrees with the notion that 438579088 is another Münchhausen number.
seconds is a nano-century: 100 years × 365.25 days per year (0.25 is for a leap day every fourth year) × 24 hours per day × 3600 seconds per hour = 3,155,760,000 sec ? ? sec × 109 within 0.45%.
Speed of light
The speed of light is (by definition) exactly 299,792,458 m/s, very close to 300,000,000 m/s. This is a pure coincidence, as the meter was originally defined as 1/10,000,000 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. It is also roughly equal to one foot per nanosecond (the actual number is 0.9836 ft/ns).
The polar diameter of the Earth is equal to half a billion inches, to within 0.1%.
This is related to the aforementioned coincidence that the square of pi is close to 10. One of the early definitions of the meter 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 meters per second per second would be exactly equal to .
When it was discovered that the circumference of the earth was very close to 40,000,000 times this value, the meter was redefined to reflect this, as it was a more objective standard (because the gravitational acceleration varies over the surface of the Earth). This had the effect of increasing the length of the meter by less than 1%, which was within the experimental error of the time.
Another coincidence related to the gravitational acceleration g is that its value of approximately 9.8 m/s2 is equal to 1.03 light-year/year2, which numerical value is close to 1. This is related to the fact that g is close to 10 in SI units (m/s2), as mentioned above, combined with the fact that the number of seconds per year happens to be close to the numerical value of c/10, with c the speed of light in m/s. In fact, it has nothing to do with SI as c/g = 354 days, nearly, and 365/354 = 1.03.
The Rydberg constant, when multiplied by the speed of light and expressed as a frequency, is close to :
The pound-mole, being 1 pound / 1 amu, is 273.16*10^24, against the triple-point of water at 273.16 K. This is interesting in terms of the pound-Celcius system.
US customary to metric conversions
inches equals centimeters, which is about 0.0001% less than 20 centimeters. The decimal expansion contains many nines early on.
kilometers equals miles, a difference of 9.4 cm (3.7 in).
As discovered by Randall Munroe, a cubic mile is close to cubic kilometers (within 0.5%). This means that a sphere with radius n kilometers has almost exactly the same volume as a cube with sides length n miles.
The multiplier to convert from meters per second to miles per hour is . Accurate to within 0.04%. As a result, a 1 gram object moving at 100 miles per hour has almost exactly 1 joule of kinetic energy.
One furlong per fortnight is close to one centimetre per minute, to within 0.2%. The first is 201,168 cm/fortnights, the second 201,600 cm/fortnights.
Although this coincidence is not as strong as some of the others in this section, it is notable that is a dimensionless constant, so this coincidence is not an artifact of the system of units being used.
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