Mathematical coincidence

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

In mathematics, a mathematical coincidence can be said to occur when two expressions show a near-equality that lacks direct theoretical explanation.

1 Introduction
2 Some examples
3 See also
4 References
5 External links

[edit] Introduction

A mathematical coincidence often comprises an integer, and the surprising (or "coincidental") feature is the fact that a real number is close to a small integer; or, more generally, to a rational number with a small denominator.

Given the large number of ways of combining mathematical expressions, one might expect a large number of coincidences to occur; this is one aspect of the law of small numbers. Although mathematical coincidences may be useful, they are mainly notable for their curiosity value.

[edit] Some examples

[edit] 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 to the ratios of logs of numbers are often invoked as well, making coincidences between the powers of different numbers.

Many other coincidences are combinations of numbers that put them into the form that such rational approximants provide close relationships.

[edit] Concerning pi

The first convergent of π, [3; 7] = 22/7 = 3.1428..., was known to Archimedes,^[1] and is correct to about 0.04%. The third convergent of π, [3; 7, 15, 1] = 355/113 = 3.1415929..., found by Zu Chongzhi, is correct to six decimal places or about 85 parts in a billion; this extreme accuracy comes about because π has an unusually large next term in its continued fraction representation: π = [3; 7, 15, 1, 292, ...].

[edit] 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), or to relate a kilobyte to a kibibyte; see binary prefix.
- Using 3/10 as the approximate value of log₁₀2, one finds the following approximations for logs of other ratios, starting from approximate integer relationships:
  - $3^4\approx 10\cdot 2^3,$ leading to $\log_{10}3 = (1+3\log_{10}2)/4\approx (1 + 9/10)/4 = 0.475;$ (compare 0.4771, within 0.5%)
  - $7^2\approx 10^2/2,$ leading to $\log_{10}7 \approx 1-\log_{10}2/2 \approx 1 - 3/20 = 0.85$ (compare 0.8451, within 0.6%)

[edit] Concerning musical intervals

The coincidence $2^{11} \approx 3^7$ , from $\frac{\log3}{\log2} \approx 1.5849... \approx \frac{11}{7}$ 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.

[edit] Numeric expressions

[edit] Concerning powers of pi

$\pi^2\approx10;$ correct to about 1.3%. This coincidence was used in the design of slide rules, where the "folded" scales are folded on $π$ 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% (note 2, 227, and 23 are Chen primes).
$\pi\approx \sqrt{2} + \sqrt{3}$ (to about 0.15%).
$\pi\approx\left(9^2+\frac{19^2}{22}\right)^{1/4},$ or $\pi^4\approx 2143/22;$ accurate to 8 decimal places (due to Ramanujan: Quarterly Journal of Mathematics, XLV, 1914, pp350-372). Ramanujan states that this "curious approximation" to $π$ was "obtained empirically" and has no connection with the theory developed in the remainder of the paper. This can be told humorously as: Take the number "1234", transpose the first two digits and the last two digits, so the number becomes "2143". Divide that number by "two-two" (22, so 2143/22 = 97.40909...). Take the two-squaredth root (4th root) of this number. The final outcome is remarkably close to $π$ (within about one part in a billion).
$(1+\frac{1}{81}) \sum_{k=1}^{12} \frac{1}{k} \approx \pi$ , within 0.0023%

[edit] Concerning the constant e

$\sum_{k=1}^8 \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{7} + \frac{1}{8} \approx e$ , within 0.016%.

$(1+\frac{1}{6400}) \sum_{k=1}^8 \frac{1}{k} \approx e$ , within 0.00000076%.

For any integer n, the early decimal digits of $n - e$ are composed of 2,7,1, and 8, the first digits of e
$\exp(-\psi(\sqrt{3}/4+1/2)) \approx 2$ to 0.000015% (one part in ten million), where ψ is Polygamma function and exp is Exponential function.

[edit] Containing both pi and e

$\pi^4+\pi^5\approx e^6$ , within 0.000 005%

$e^\pi - \pi\approx 19.99909998$ is very close to 20 (Conway, Sloane, Plouffe, 1988), and is equivalent to $(\pi+20)^i=-0.999 999 999 2... -i\cdot 0.000 039... \approx -1$ ^[2]

$e^\pi - \pi + \frac{1}{1111}\approx 20.0000001$

${\pi^{3^2}}\div{e^{2^3}}\approx 9.9998$ ^[2]

${\pi\div e} \approx 2/\sqrt3$ , within 0.089%. It is equivalent to: ${e}\approx \pi\cdot \sin (\pi/3)$ ^{[citation needed]}

${\pi - e} \approx 1-1/\sqrt3$ , within 0.15%^{[citation needed]}

${(\pi\div e)/(\pi - e)} \approx \sqrt3 + 1$ , within 0.067%^{[citation needed]}

[edit] Containing pi and e and phi (the Golden ratio)

$\phi\times e\div\pi \approx 7/5$ , within 0.001%.

$\phi+e+\pi \approx 7.5$ , within 0.3%.

[edit] Containing pi or e and number 163

${163}\cdot (\pi - e) \approx 69$ , within 0.0005%^{[citation needed]}

${{163} / \log_{e}{163} } \approx 2^5$ , within 0.000 004%^{[citation needed]}

Ramanujan's constant: $e^{\pi\sqrt{163}} \approx (2^6\cdot 10005)^3+744$ , within $2.9\cdot 10^{-28}%$ ,^[3] it implies $\pi \approx {1\over \sqrt{163}}\log_e{(640320^3+744)}$ , which is correct to about 20 digits.

Note: $e^{\pi\sqrt{n}}$ is close to an integer for many values of $n$ , most notably $n = 163$ is explained by algebraic number theory; see Heegner number.

[edit] Coincidences of units

$π$ seconds is a nanocentury (ie $10 - 7$ years); correct to within about 0.5%
one attoparsec per microfortnight approximately equals 1 inch per second (the actual figure is about 1.0043 inch per second).
one furlong per fortnight is approximately equal to one centimetre per minute
a cubic attoparsec (a cube where each edge is one attoparsec) is within 1% of a U.S. fluid ounce.
one mile is about $φ$ kilometers (correct to about 0.5%), where $\phi={1+\sqrt 5\over 2}$ is the golden ratio. Since this is the limit of the ratio of successive terms of the Fibonacci sequence, this gives a sequence of approximations $F n$ mi = $F n + 1$ km, e.g. 5 mi = 8 km, 8 mi = 13 km. Another good approximation is 1 mile = ln(5) km, 1 mile = 1.609344 km and ln(5) = 1.6094379124341...
N_A ≈ 2⁷⁹, where N_A is Avogadro's number; correct to about 0.4%. This means that a yobibyte is slightly more than two moles of bytes. It also means that 1 mole of any material (e.g. 12 g carbon, or 25 l of gas at room temperature and normal pressure) cannot be halved more than 79 times.
The speed of light in a vacuum is about one foot per nanosecond (accurate to 2%) or 3×10⁸ m/s (accurate to about 0.07%) or 1 billion km/h (accurate to 7.93%)

[edit] Other numeric curiosities

$10! = 6! \cdot 7! = 3! \cdot 5! \cdot 7!$ .
$\sin(666^\circ) = \cos(6\cdot6\cdot6^\circ) = - \phi/2$ , where φ is Golden ratio (an amusing equality with an angle expressed in degrees) (see Number of the Beast)
$\,5^2=25$ and $3^3=27\,$ are the only square and cube of positive integers that differ by 2.
$\,3 + 2 = \log_2{32}$ .
$\,4^2 = 2^4$ is the only integer solution of $a^b = b^a, a\neq b$ (see Lambert's W function for a formal solution method)
$\,3^2 - 2^3 = 3 - 2$
Not only $\,3^2 + 4^2 = 5^2$ , but also $\,3^3 + 4^3 + 5^3 = 6^3$ .
$\!\ \sin 9 - \cos 9$ is equal to the plastic constant to within 0.111%
31, 331, 3331 etc. up to 33333331 are all prime numbers, but then 333333331 is not. See also Formula for primes.
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.^[4]
In a discussion of the birthday problem, the number $\lambda=\frac{1}{365}{23\choose 2}$ occurs which Arriata states is "amusingly" equal to $ln(2)$ to 4 digits.^[5]

[edit] Decimal coincidences

$2^5 \cdot 9^2 = 2592$ .
$\,1! + 4! + 5! = 145$ .
$\frac {16} {64} = \frac {1\!\!\!\not6} {\not6 4} = \frac {1} {4}$ , $\frac {26} {65} = \frac {2\!\!\!\not6} {\not6 5} = \frac {2} {5}$ , $\frac {19} {95} = \frac {1\!\!\!\not9} {\not9 5} = \frac {1} {5}$ (Anomalous cancellation)
$\,(4 + 9 + 1 + 3)^3 = 4,913$ and $\,(1 + 9 + 6 + 8 + 3)^3=19,683$ .
$\,2^7 - 1 = 127$
$\,1^3 + 5^3 + 3^3 = 153$ ; $\,3^3 + 7^3 + 0^3 = 370$ ; $\,3^3 + 7^3 +1^3 = 371$ ; $\,4^3 + 0^3 +7^3 = 407$
$3^2 + 7^2 - 3 \cdot 7 = (3^3 + 7^3)/(3 + 7) = 37$ .
$\,(3 + 4)^3 = 343$ (important in the numerological symbolism of St. Stephen's Cathedral, Vienna)
$\,35 - 3^2 - 5^2 = 75 - 7^2 - 5^2$ .
$\,588^2+2353^2 = 5882353$ and also $\, 1/17 = 0.0588235294117647...$ when rounded to 8 digits is 0.05882353

mentioned by Gilbert Labelle in ~1980.

The number which equals the sum of its digits in consecutive powers: $\,2646798 = 2^1+6^2+4^3+6^4+7^5+9^6+8^7.$

[edit] See also

For a list of coincidences in physics, see anthropic principle.
Almost integer
Birthday problem
Narcissistic number

[edit] References

^ Petr Beckmann (1993). A History of Pi. Barnes & Noble. ISBN 0880294183.
^ ^a ^b Almost Integer. Wolfram MathWorld.
^ Ramanujan, S. "Modular Equations and Approximations to pi." Quart. J. Pure Appl. Math. 45, 350-372, 1913-1914.
^ David Broadhurst, "Prime Curios!: 10660...49391 (61899-digits)" at The Prime Pages.
^ Arratia, Richard & and others (1990), “Poisson approximation and the Chen-Stein method”, Statistical Science 5, (4,): 403-434

[edit] 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)
Archimedes' Laboratory
Online Encyclopedia of Integer Sequences
Almost Integer from Wolfram MathWorld
Various mathematical coincidences in the "Science & Math" section of futilitycloset.com
e to the pi Minus pi, xkcd comic

Categories: Mathematical terminology

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