Mathematical coincidence

This article is about numerical curiosities. For the technical mathematical concept of coincidence, see Coincidence point.

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

Concerning e

Concerning base 2

Concerning musical intervals

Numerical expressions

Concerning powers of π

 \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

Containing π or e and 163

Concerning logarithms

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

Other numerical curiosities

Decimal coincidences

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

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. 1 2 3 Manfred Robert Schroeder (2008). Number theory in science and communication (2nd ed.). Springer. pp. 26–28. ISBN 978-3-540-85297-1.
  3. 1 2 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. 1 2 Frank Rubin, The Contest Center – Pi.
  9. \pi^2, Noam Elkies
  10. 1 2 3 4 5 6 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. 1 2 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. 1 2 Weisstein, Eric W., "Beast Number", MathWorld.
  25. 1 2 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

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