Mathematical Coincidence

A **mathematical coincidence** is said to occur when two expressions show a near-equality which has no known theoretical explanation.

For example, there is a near-equality close to the round number 1000 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^{[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. 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.^{[2]}

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,
^{[3]}and is correct to about 0.04%. The third convergent of ?, [3; 7, 15, 1] = 355/113 = 3.1415929..., found by Zu Chongzhi,^{[4]}is correct to six decimal places;^{[3]}this high accuracy comes about because ? has an unusually large next term in its continued fraction representation: ? = [3; 7, 15, 1, 292, ...].^{[5]} - A coincidence involving ? and the golden ratio ? is given by . This is related to Kepler triangles. Some believe one or the other of these coincidences is to be found in the Great Pyramid of Giza, but it is highly improbable that this was intentional.
^{[6]} - 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%.

- 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. e = 2.7 1828 1828 45 90 45....
- There is a sequence of "99 999 999" within the first 500,000 digits of e.
^{[7]}

- 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.
^{[8]}^{[9]} - 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.
^{[2]}

- The coincidence , from leads to the observation commonly used in music to relate the tuning of 7 semitones of equal temperament to a perfect fifth of just intonation: , correct to about 0.1%. The just fifth is the basis of Pythagorean tuning and most known systems of music. From the consequent approximation it follows that the circle of fifths terminates seven octaves higher than the origin.
^{[2]} - The coincidence leads to the rational version of 12-TET, as noted by Johann Kirnberger.
^{[]} - The coincidence leads to the rational version of quarter-comma meantone temperament.
^{[]} - 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.

- .
- correct to about 1.3%.
^{[10]}This can be understood in terms of the formula for the zeta function^{[11]}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.^{[]} - correct to 0.0004%.
^{[10]} - correct to 0.02%.
- correct to 0.004%.
- or
^{[12]}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.^{[13]}

- , within 0.000 005%
^{[12]} - to better than 14 decimal places.
- is very close to 5, within 0.008%.
- is also very close to 5, approximately 0.000 538% error (Joseph Clarke, 2015)
- is very close to 20 (Conway, Sloane, Plouffe, 1988); this is equivalent to
^{[12]} ^{[12]}

- . In fact, this generalizes to the approximate identity: which can be explained by the Jacobian theta functional identity.
^{[14]}^{[15]}^{[16]}

- , within 0.0005%
^{[12]} - , within 0.000004%
^{[12]} - Ramanujan's constant: , within , discovered in 1859 by Charles Hermite.
^{[17]}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 ).

- .
^{[18]} - and are the only non-trivial (i.e. at least square) consecutive powers of positive integers (Catalan's conjecture).
- is the only positive integer solution of , assuming that
^{[19]}(see Lambert's W function for a formal solution method) - The Fibonacci number
*F*_{296182}is (probably) a semiprime, since*F*_{296182}=*F*_{148091}×*L*_{148091}where*F*_{148091}(30949 digits) and the Lucas number*L*_{148091}(30950 digits) are simultaneously probable primes.^{[20]} - , , , , , , and are all primes, but next such number is not prime: .
- In a discussion of the birthday problem, the number occurs, which is "amusingly" equal to to 4 digits.
^{[21]}

- . This makes 2592 a nice Friedman number.
^{[22]} - . 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.
^{[23]}^{[24]} - . The only such factorions (in base 10) are 1, 2, 145, 40585.
^{[25]} - . In fact, this works so long as the latter part of the number is one less than a power of 10. For instance, and .
- , , , (anomalous cancellation
^{[26]}). Also, the product of these four fractions reduces to exactly 1/100. - , 81 is the only such number other than 0 and 1.
- ; ; and .
^{[27]} - . This can also be written , making 127 the smallest nice Friedman number.
^{[22]} - ; ; ; -- all narcissistic numbers
^{[28]} ^{[29]}- and also when rounded to 8 digits is 0.05882353. Mentioned by Gilbert Labelle in ~1980.
^{[30]}5882353 is also prime. - .
- . The largest such number is 12157692622039623539.
^{[31]} - , where is the golden ratio
^{[32]}(an amusing equality with an angle expressed in degrees) (see Number of the Beast) - , where is Euler's totient function
^{[32]}

The fifth hyperfactorial: 5? × 4? × 3³ × 2² × 1¹ = 86,400,000 is the number of milliseconds in 24 hours.

The number of seconds in six weeks, or 42 days, is exactly 10! (ten factorial) seconds (as and and ). Many have recognized this coincidence in particular because of the importance of 42 in Douglas Adams' *The Hitchhiker's Guide to the Galaxy*.

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.^{[33]} 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%.^{[34]}

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, 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.^{[35]}

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 the square of pi.^{[36]}

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/s^{2} is equal to 1.03 light-year/year^{2}, which numerical value is close to 1. This is related to the fact that *g* is close to 10 in SI units (m/s^{2}), 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.

The Rydberg constant, when multiplied by the speed of light and expressed as a frequency, is close to :^{[33]}

^{[37]}

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.^{[38]}

The multiplier to convert from meters per second to miles per hour is . Accurate to 2 decimal places (within 0.04%)

The fine-structure constant is close to and was once conjectured to be precisely .

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.

- Almost integer
- Anthropic principle
- Birthday problem
- Exceptional isomorphism
- Narcissistic number
- Experimental mathematics
- Kepler triangle#A mathematical coincidence
- Koide formula

**^**Reprinted as Gardner, Martin (2001). "Six Sensational Discoveries".*The Colossal Book of Mathematics*. New York: W. W. Norton & Company. pp. 674-694. ISBN 0-393-02023-1.- ^
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- (in Russian) ?. . -
*? ? ?.*- , ? ? 1970, 256 ?. - Hardy, G. H. -
*A Mathematician's Apology.*- New York: Cambridge University Press, 1993, (ISBN 0-521-42706-1) - Weisstein, Eric W. "Almost Integer".
*MathWorld*. - Various mathematical coincidences in the "Science & Math" section of futilitycloset.com
- Press, W. H., Seemingly Remarkable Mathematical Coincidences Are Easy to Generate

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