Mathematical coincidence

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 second convergent of π, [3; 7] = 22/7 = 3.1428..., was known to Archimedes,[3] and is correct to about 0.04%. The fourth convergent of π, [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 π has an unusually large next term in its continued fraction representation: π = [3; 7, 15, 1, 292, ...].[5]
• A coincidence involving π and the golden ratio φ is given by ${\displaystyle \pi \approx 4/{\sqrt {\varphi }}=3.1446\dots }$. 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.
• ${\displaystyle \tan ^{-1}\left({\frac {\pi }{2}}\right)\approx 1,}$ correct to 0.39%.

• The coincidence ${\displaystyle 2^{10}=1024\approx 1000=10^{3}}$, correct to 2.4%, relates to the rational approximation ${\displaystyle \textstyle {\frac {\log 10}{\log 2}}\approx 3.3219\approx {\frac {10}{3}}}$, or ${\displaystyle 2\approx 10^{3/10}}$ 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.[7][8]
• This coincidence can also be expressed as ${\displaystyle 128=2^{7}\approx 5^{3}=125}$ (eliminating common factor of ${\displaystyle 2^{3}}$, so also correct to 2.4%), which corresponds to the rational approximation ${\displaystyle \textstyle {\frac {\log 5}{\log 2}}\approx 2.3219\approx {\frac {7}{3}}}$, or ${\displaystyle 2\approx 5^{3/7}}$ (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 ${\displaystyle 2^{19}\approx 3^{12}}$, from ${\displaystyle {\frac {\log 3}{\log 2}}\approx 1.5849\dots \approx {\frac {19}{12}}}$ 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: ${\displaystyle 2^{7/12}\approx 3/2;}$, correct to about 0.1%. The just fifth is the basis of Pythagorean tuning and most known systems of music. From the consequent approximation ${\displaystyle {(3/2)}^{12}\approx 2^{7},}$ it follows that the circle of fifths terminates seven octaves higher than the origin.[2]
• The coincidence ${\displaystyle {\sqrt[{12}]{2}}{\sqrt[{7}]{5}}=1.33333319\ldots \approx {\frac {4}{3}}}$ leads to the rational version of 12-TET, as noted by Johann Kirnberger.
• The coincidence ${\displaystyle {\sqrt[{8}]{5}}{\sqrt[{3}]{35}}=4.00000559\ldots \approx 4}$ leads to the rational version of quarter-comma meantone temperament.
• The coincidence ${\displaystyle {\sqrt[{9}]{0.6}}{\sqrt[{28}]{4.9}}=0.99999999754\ldots \approx 1}$ leads to the very tiny interval of ${\displaystyle 2^{9}3^{-28}5^{37}7^{-18}}$ (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, ${\displaystyle {(5/4)}^{3}\approx {2/1}}$. This and similar approximations in music are called dieses.

• ${\displaystyle \pi ^{2}\approx 10;}$ correct to about 1.3%.[9] This can be understood in terms of the formula for the zeta function ${\displaystyle \zeta (2)=\pi ^{2}/6.}$[10] This coincidence was used in the design of slide rules, where the "folded" scales are folded on ${\displaystyle \pi }$ rather than ${\displaystyle {\sqrt {10}},}$ because it is a more useful number and has the effect of folding the scales in about the same place.
• ${\displaystyle \pi ^{2}\approx 227/23,}$ correct to 0.0004%.[9]
• ${\displaystyle \pi ^{3}\approx 31}$correct to 0.62%.[11]
• ${\displaystyle \pi ^{4}\approx 2143/22;}$ or ${\displaystyle \pi \approx \left(9^{2}+{\frac {19^{2}}{22}}\right)^{1/4},}$ accurate to 8 decimal places (due to Ramanujan: Quarterly Journal of Mathematics, XLV, 1914, pp. 350–372).[12] Ramanujan states that this "curious approximation" to ${\displaystyle \pi }$ 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

${\displaystyle \int _{0}^{\infty }\cos(2x)\prod _{n=1}^{\infty }\cos \left({\frac {x}{n}}\right)\mathrm {d} x\approx {\frac {\pi }{8}}.}$

The two sides of this expression only differ after the 42nd decimal place.[13]

• ${\displaystyle e^{e}\approx {\frac {500}{33}},}$ correct to 0.018%.
• ${\displaystyle e^{1/e}\approx {\frac {13}{9}},}$ correct to 0.015%.
• ${\displaystyle e^{3}\approx 20,}$ correct to 0.43%.

• ${\displaystyle 2\pi +e\approx 9.001467}$ is close to 9, with an error of only 0.01%.
• ${\displaystyle \pi ^{4}+\pi ^{5}\approx e^{6}}$, within 0.000 005%[12]
• ${\displaystyle \left({\frac {\pi }{2}}-\ln \left({\frac {3\pi }{2}}\right)\right)42\pi \approx e}$, within 0.000 000 03% [14]
• ${\displaystyle {\sqrt {\frac {e}{(\pi -e)^{3/2}}}}\approx \pi }$, within 0.000 06% [15]
• ${\displaystyle \exp \left(\left({\frac {\pi }{2}}\right)^{\frac {\pi -e}{\sqrt {2}}}\right)\approx \pi }$, within 0.000 02%
• ${\displaystyle \left({\frac {\varphi }{e^{\pi }-\pi ^{e}}}\right)^{\frac {\pi }{e}}\approx e}$
• ${\displaystyle {3}^{\frac {\pi +e}{4}}\approx {5}}$, approximately 0.000 538% error (Joseph Clarke, 2015)
• ${\displaystyle e^{\pi }-\pi \approx 19.99909998}$ is very close to 20 (Conway, Sloane, Plouffe, 1988); this is equivalent to ${\displaystyle (\pi +20)^{i}=-0.9999999992\ldots -i\cdot 0.000039\ldots \approx -1}$[12]
• ${\displaystyle \pi ^{3^{2}}/e^{2^{3}}=9.9998\ldots \approx 10}$[12]
• ${\textstyle e^{-{\frac {\pi }{9}}}+e^{-4{\frac {\pi }{9}}}+e^{-9{\frac {\pi }{9}}}+e^{-16{\frac {\pi }{9}}}+e^{-25{\frac {\pi }{9}}}+e^{-36{\frac {\pi }{9}}}+e^{-49{\frac {\pi }{9}}}+e^{-64{\frac {\pi }{9}}}=1.00000000000105...\approx 1}$. In fact, this generalizes to the approximate identity:${\displaystyle \sum _{k=0}^{n-1}{e^{-{\frac {k^{2}\pi }{n}}}}\approx {\frac {-1+{\sqrt {n}}}{2}}}$ which can be explained by the Jacobian theta functional identity.[16][17][18]
• ${\displaystyle e/\pi +\pi /e\approx 2.02098}$

• ${\displaystyle {163}\cdot (\pi -e)\approx 69}$, within 0.0005%[12]
• ${\displaystyle {\frac {163}{\ln 163}}\approx 2^{5}}$, within 0.000004%[12]
• Ramanujan's constant: ${\displaystyle e^{\pi {\sqrt {163}}}\approx (2^{6}\cdot 10005)^{3}+744}$, within ${\displaystyle 2.9\cdot 10^{-28}\%}$, discovered in 1859 by Charles Hermite.[19] 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.

• ${\displaystyle 10!=6!\cdot 7!=1!\cdot 3!\cdot 5!\cdot 7!}$.[20]
• ${\displaystyle {\sqrt {7!+1}}=71}$
• ${\displaystyle \,2^{3}=8}$ and ${\displaystyle 3^{2}=9}$ are the only non-trivial (i.e. at least square) consecutive powers of positive integers (Catalan's conjecture).
• ${\displaystyle \,4^{2}=2^{4}}$ is the only positive integer solution of ${\displaystyle a^{b}=b^{a}}$, assuming that ${\displaystyle a\neq b}$[21] (see Lambert's W function for a formal solution method)
• The Fibonacci number F₂₉₆₁₈₂ is (probably) a semiprime, since F₂₉₆₁₈₂ = F₁₄₈₀₉₁ × L₁₄₈₀₉₁ where F₁₄₈₀₉₁ (30949 digits) and the Lucas number L₁₄₈₀₉₁ (30950 digits) are simultaneously probable primes.[22]
• In a discussion of the birthday problem, the number ${\displaystyle \lambda ={\frac {1}{365}}{23 \choose 2}={\frac {253}{365}}}$ occurs, which is "amusingly" equal to ${\displaystyle \ln(2)}$ to 4 digits.[23]
• Integers 1 through 10 divide the number that is at the same distance from them as the distance between their factorial and the next smallest square number. There is no rigorous reason for this to hold, and it holds true chaotically for number 11 and integers above it. [24]

• ${\displaystyle 3^{3}+4^{4}+3^{3}+5^{5}=3435}$, making 3435 the only non-trivial Münchhausen number in base 10 (excluding 0 and 1). If one adopts the convention that ${\displaystyle 0^{0}=0}$, however, then 438579088 is another Münchhausen number.[25]
• ${\displaystyle \,1!+4!+5!=145}$ and ${\displaystyle \,4!+0!+5!+8!+5!=40585}$ are the only non-trivial factorions in base 10 (excluding 1 and 2).[26]
• ${\displaystyle {\frac {16}{64}}={\frac {1\!\!\!\not 6}{\not 64}}={\frac {1}{4}}}$,    ${\displaystyle {\frac {26}{65}}={\frac {2\!\!\!\not 6}{\not 65}}={\frac {2}{5}}}$,    ${\displaystyle {\frac {19}{95}}={\frac {1\!\!\!\not 9}{\not 95}}={\frac {1}{5}}}$,  and  ${\displaystyle {\frac {49}{98}}={\frac {4\!\!\!\not 9}{\not 98}}={\frac {4}{8}}}$. If the end result of these four anomalous cancellations[27] are multiplied, their product reduces to exactly 1/100.
• ${\displaystyle \,(4+9+1+3)^{3}=4913}$, ${\displaystyle \,(5+8+3+2)^{3}=5832}$, and ${\displaystyle \,(1+9+6+8+3)^{3}=19683}$.[28] (Along a similar vein, ${\displaystyle \,(3+4)^{3}=343}$.)[29]
• ${\displaystyle \,-1+2^{7}=127}$, making 127 the smallest nice Friedman number. A similar example is ${\displaystyle 2^{5}\cdot 9^{2}=2592}$.[30]
• ${\displaystyle \,1^{3}+5^{3}+3^{3}=153}$, ${\displaystyle \,3^{3}+7^{3}+0^{3}=370}$, ${\displaystyle \,3^{3}+7^{3}+1^{3}=371}$, and ${\displaystyle \,4^{3}+0^{3}+7^{3}=407}$ are all narcissistic numbers.[31]
• ${\displaystyle \,588^{2}+2353^{2}=5882353}$,[32] a prime number. The fraction 1/17 also produces 0.05882353 when rounded to 8 digits.
• ${\displaystyle \,2^{1}+6^{2}+4^{3}+6^{4}+7^{5}+9^{6}+8^{7}=2646798}$. The largest known number with this pattern is ${\displaystyle \,1^{1}+2^{2}+1^{3}+5^{4}+7^{5}+6^{6}+9^{7}+2^{8}+6^{9}+2^{10}+2^{11}+0^{12}+3^{13}+9^{14}+6^{15}+2^{16}+3^{17}+5^{18}+3^{19}+9^{20}=12157692622039623539}$.[33]
• ${\displaystyle \sin(666^{\circ })=\cos(6\cdot 6\cdot 6^{\circ })=-\varphi /2}$ (where ${\displaystyle \varphi }$ is the golden ratio), and ${\displaystyle \,\phi (666)=6\cdot 6\cdot 6}$ (where ${\displaystyle \phi }$ is Euler's totient function).[34]

The speed of light is (by definition) exactly 299,792,458 m/s, extremely 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.[35] 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%.[36]

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 eclipses that can be observed from Earth.

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.[37]

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 ${\displaystyle \pi ^{2}}$.[38]

${\displaystyle T\approx 2\pi {\sqrt {\frac {L}{g}}}}$

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 ${\displaystyle {\frac {\pi ^{2}}{3}}\times 10^{15}\ {\text{Hz}}}$:[35]

${\displaystyle {\underline {3.2898}}41960364(17)\times 10^{15}\ {\text{Hz}}=R_{\infty }c}$[39]

${\displaystyle {\underline {3.2898}}68133696\ldots ={\frac {\pi ^{2}}{3}}}$

As discovered by Randall Munroe, a cubic mile is close to ${\displaystyle {\frac {4}{3}}\pi }$ 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.[40][41]

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

The fine-structure constant ${\displaystyle \alpha }$ is close to ${\displaystyle {\frac {1}{137}}}$ and was once conjectured to be precisely ${\displaystyle {\frac {1}{137}}}$.[42]

${\displaystyle \alpha ={\frac {1}{137.035999074\dots }}}$

Although this coincidence is not as strong as some of the others in this section, it is notable that ${\displaystyle \alpha }$ is a Physical constant, so this coincidence is not an artifact of the system of units being used.