Mathematical coincidence

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

For example, there is a near-equality close to the round number 1000 between powers of 2 and powers of 10:

$2^{10} = 1024 \approx 1000 = 10^3$

Some mathematical coincidences are used in engineering when one expression is taken as an approximation of another.

Introduction

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.

Some examples

Rational approximants

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.

Concerning π

• The first convergent of π, [3; 7] = 22/7 = 3.1428..., was known to Archimedes,^[3] and is correct to about 0.04%. The third 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 $\pi \approx 4 / \sqrt{\varphi} = 3.1446\dots$. This is related to Kepler triangles.
• 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.

Concerning e

• The number 1828 repeats twice in a row early in the decimal expansion of e = 2.7 1828 1828....

Concerning base 2

• The coincidence $2^{10} = 1024 \approx 1000 = 10^3$, correct to 2.4%, relates to the rational approximation $\textstyle\frac{\log10}{\log2} \approx 3.3219 \approx \frac{10}{3}$, or $2 \approx 10^{3/10}$ 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 3 dB-point), or to relate a kibibyte to a kilobyte; see binary prefix.^[6][7]
• This coincidence can also be expressed $5^3 = 125 \approx 128 = 2^7$, and 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]

Concerning musical intervals

• The coincidence $2^{19} \approx 3^{12}$, from $\frac{\log3}{\log2} \approx 1.5849\dots \approx \frac{19}{12}$ 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: $2^{7/12}\approx 3/2;$, correct to about 0.1%. The just fifth is the basis of Pythagorean tuning and most known systems of music. From the consequent approximation ${(3/2)}^{12}\approx 2^7,$ it follows that the circle of fifths terminates seven octaves higher than the origin.^[2]
• The coincidence $\sqrt[12]{2}\sqrt[7]{5} = 1.33333319\ldots \approx \frac43$ leads to the rational version of 12-TET, as noted by Johann Kirnberger.
• The coincidence $\sqrt[8]{5}\sqrt[3]{35} = 4.00000559\ldots \approx 4$ leads to the rational version of quarter-comma meantone temperament.
• The coincidence $\sqrt[9]{0.6}\sqrt[28]{4.9} = 0.99999999754\ldots \approx 1$ leads to the very tiny interval of $2^{9}3^{-28}5^{37}7^{-18}$ (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, ${(5/4)}^{3} \approx {2/1}$. This and similar approximations in music are called dieses.

Numerical expressions

Concerning powers of π

• $\pi^2\approx10;$ correct to about 1.3%.^[8] This can be understood in terms of the formula for the zeta function $\zeta(2)=\pi^2/6.$^[9] This coincidence was used in the design of slide rules, where the "folded" scales are folded on $\pi$ rather than $\sqrt{10},$ because it is a more useful number and has the effect of folding the scales in about the same place.
• $\pi^2\approx 227/23,$ correct to 0.0004%.^[8]
• $\pi^3\approx 31,$ correct to 0.02%.
• $\sqrt[5]{\pi^3+1}\approx 2,$ correct to 0.004%.
• $\pi\approx\left(9^2+\frac{19^2}{22}\right)^{1/4},$ or $22\pi^4\approx 2143;$^[10] accurate to 8 decimal places (due to Ramanujan: Quarterly Journal of Mathematics, XLV, 1914, pp. 350–372). Ramanujan states that this "curious approximation" to $\pi$ was "obtained empirically" and has no connection with the theory developed in the remainder of the paper.

$\sqrt[4]{\frac{2143}{22}} = 3.1415926525 \dots$

$\sqrt[5]{306} = 3.14155 \dots$

$\sqrt[6]{\frac{17305}{18}} = 3.1415924 \dots$

Some plausible relations hold to a high degree of accuracy, but are nevertheless coincidental. One example is

$\int_0^\infty \cos(2x)\prod_{n=1}^\infty \cos\left(\frac{x}{n}\right)\mathrm{d}x \approx \frac{\pi}{8}.$

The two sides of this expression only differ after the 42nd decimal place.^[11]

Containing both π and e

• $\pi^4+\pi^5\approx e^6$, within 0.000 005%^[10]
• $\sqrt[4]{3^3 e^\pi}$ is very close to 5, within 0.008%.
• ${ 3 }^{ \frac { \pi +e }{ 4 } }$ is also very close to 5, approximately 0.000 538% error (Joseph Clarke, 2015)
• $e^\pi - \pi\approx 19.99909998$ is very close to 20 (Conway, Sloane, Plouffe, 1988); this is equivalent to $(\pi+20)^i=-0.999 999 999 2\ldots -i\cdot 0.000 039\ldots \approx -1$^[10]
• $\pi^{3^2}/e^{2^3}=9.9998\ldots\approx 10$^[10]

Containing π or e and 163

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

Concerning logarithms

$\ln 2\approx \left(\frac{2}{5}\right)^{\frac{2}{5}}$ (correct to 0.00024%).

Other numerical curiosities

• $10! = 6! \cdot 7! = 1! \cdot 3! \cdot 5! \cdot 7!$.^[13]
• $\, 2^3=8$ and $3^2=9\,$ are the only non-trivial (i.e. at least square) consecutive powers of positive integers (Catalan's conjecture).
• $\,4^2 = 2^4$ is the only positive integer solution of $a^b = b^a, a\neq b$^[14] (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.^[15]
• In a discussion of the birthday problem, the number $\lambda=\frac{1}{365}{23\choose 2}=\frac{253}{365}$ occurs, which is "amusingly" equal to $\ln(2)$ to 4 digits.^[16]

Decimal coincidences

• $2^5 \cdot 9^2 = 2592$. This makes 2592 a nice Friedman number.^[17]
• $3^3+4^4+3^3+5^5=3435$.
• $\,1! + 4! + 5! = 145$. The only such factorions (in base 10) are 1, 2, 145, 40585.^[18]
• $\frac {16} {64} = \frac {1\!\!\!\not6} {\not6 4} = \frac {1} {4}$,    $\frac {26} {65} = \frac {2\!\!\!\not6} {\not6 5} = \frac {2} {5}$,    $\frac {19} {95} = \frac {1\!\!\!\not9} {\not9 5} = \frac {1} {5}$,    $\frac{49}{98}=\frac{4\!\!\!\not9}{\not98}=\frac{4}{8}$ (anomalous cancellation^[19]). Also, the product of these four fractions reduces to exactly 1/100.
• $\,(4 + 9 + 1 + 3)^3 = 4{,}913$; $\,(5 + 8 + 3 + 2)^3 = 5{,}832$; and $\,(1 + 9 + 6 + 8 + 3)^3=19{,}683$.^[20]
• $\,2^7 - 1 = 127$. This can also be written $\,127 = -1 + 2^7$, making 127 the smallest nice Friedman number.^[17]
• $\,1^3 + 5^3 + 3^3 = 153$ ; $\,3^3 + 7^3 + 0^3 = 370$ ; $\,3^3 + 7^3 +1^3 = 371$ ; $\,4^3 + 0^3 +7^3 = 407$ — all narcissistic numbers^[21]
• $\,(3 + 4)^3 = 343$^[22]
• $\,588^2+2353^2 = 5882353$ and also $\, 1/17 = 0.0588235294117647\ldots$ when rounded to 8 digits is 0.05882353. Mentioned by Gilbert Labelle in ~1980. 5882353 also happens to be prime.
• $\,2646798 = 2^1+6^2+4^3+6^4+7^5+9^6+8^7$. The largest such number is 12157692622039623539.^[23]
• $\sin(666^\circ) = \cos(6\cdot6\cdot6^\circ) = - \varphi/2$, where $\varphi$ is the golden ratio^[24] (an amusing equality with an angle expressed in degrees) (see Number of the Beast)
• $\,\phi(666)=6\cdot6\cdot6$, where $\phi$ is Euler's totient function^[24]

Numerical coincidences in numbers from the physical world

Length of six weeks

The number of seconds in six weeks, or 42 days, is exactly 10! (ten factorial) seconds (as $24 = 4!$ and $42 = 6 \cdot 7$ and $60^2 = 5 \cdot 8 \cdot 9 \cdot 10$). Many have recognized this coincidence in particular because of the importance of 42 in Douglas Adams' The Hitchhiker's Guide to the Galaxy.

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.^[25] It is also roughly equal to one foot per nanosecond (the actual number is 0.9836 ft/ns).

Earth's diameter

The polar diameter of the Earth is equal to half a billion inches, to within 0.1%.^[26]

Gravitational acceleration

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.^[27]

This is actually 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.^[28]

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

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.

Rydberg constant

The Rydberg constant, when multiplied by the speed of light and expressed as a frequency, is close to $\frac{\pi^2}{3}\times 10^{15}\ \text{Hz}$:^[25]

$\underline{3.2898}41960364(17) \times 10^{15}\ \text{Hz} = R_\infty c$^[29]

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

Cubic miles and kilometers

A cubic mile is close to $\frac{4}{3}\pi$ cubic kilometers (within 0.5%). This means that a sphere with radius n kilometers has almost the exact same volume as a cube with sides length n miles.^[30]

Fine-structure constant

The fine-structure constant $\alpha$ is close to $\frac1{137}$ and was once conjectured to be precisely $\frac1{137}$.

$\alpha = \frac1{137.035999074\dots}$

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

See also

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

References

1. 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.
2. Manfred Robert Schroeder (2008). Number theory in science and communication (2nd ed.). Springer. pp. 26–28. ISBN 978-3-540-85297-1.
3. Petr Beckmann (1971). A History of Pi. Macmillan. pp. 101, 170. ISBN 978-0-312-38185-1.
4. Yoshio Mikami (1913). Development of Mathematics in China and Japan. B. G. Teubner. p. 135.
5. Eric W. Weisstein (2003). CRC concise encyclopedia of mathematics. CRC Press. p. 2232. ISBN 978-1-58488-347-0.
6. Ottmar Beucher (2008). Matlab und Simulink. Pearson Education. p. 195. ISBN 978-3-8273-7340-3.
7. K. Ayob (2008). Digital Filters in Hardware: A Practical Guide for Firmware Engineers. Trafford Publishing. p. 278. ISBN 978-1-4251-4246-9.
8. Frank Rubin, The Contest Center – Pi.
9. $\pi^2$, Noam Elkies
10. Weisstein, Eric W., "Almost Integer", MathWorld.
11. http://crd.lbl.gov/~dhbailey/dhbpapers/math-future.pdf
12. Barrow, John D (2002). The Constants of Nature. London: Jonathan Cape. ISBN 0-224-06135-6.
13. Harvey Heinz, Narcissistic Numbers.
14. Ask Dr. Math, "Solving the Equation x^y = y^x".
15. David Broadhurst, "Prime Curios!: 10660...49391 (61899-digits)".
16. Arratia, Richard; Goldstein, Larry; Gordon, Louis (1990). "Poisson approximation and the Chen-Stein method". Statistical Science 5 (4): 403–434. doi:10.1214/ss/1177012015. JSTOR 2245366. MR 1092983.
17. Erich Friedman, Problem of the Month (August 2000).
18. (sequence A014080 in OEIS)
19. Weisstein, Eric W., "Anomalous Cancellation", MathWorld.
20. (sequence A061209 in OEIS)
21. (sequence A005188 in OEIS)
22. Prime Curios!: 343.
23. (sequence A032799 in OEIS)
24. Weisstein, Eric W., "Beast Number", MathWorld.
25. Michon, Gérard P. "Numerical Coincidences in Man-Made Numbers". Mathematical Miracles. Retrieved 29 April 2011.
26. Smythe, Charles (2004). Our Inheritance in the Great Pyramid. Kessinger Publishing. p. 39. ISBN 1-4179-7429-X.
27. Cracking the AP Physics B & C Exam, 2004–2005 Edition. Princeton Review Publishing. 2003. p. 25. ISBN 0-375-76387-2.
28. "What Does Pi Have To Do With Gravity?". Wired. March 8, 2013. Retrieved October 15, 2015.
29. "Rydberg constant times c in Hz". Fundamental physical constants. NIST. Retrieved 25 July 2011.
30. Randall Munroe (2014). What If?. p. 49. ISBN 9781848549562.

External links

• (Russian) В. Левшин. – Магистр рассеянных наук. – Москва, Детская Литература 1970, 256 с.
• 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