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 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] Numeric expressions

[edit] Concerning the constant "e"

$e^\pi - \pi\approx 19.9990999$ is very close to 20. (Conway, Sloane, Plouffe, 1988).
$e^{\pi\sqrt{n}}$ is close to an integer for many values of $n$ , most notably $n = 163$ $(e^{\pi\sqrt{163}} \approx 640,320^3+744-7.5\times10^{-13});$ this one is explained by algebraic number theory; see Heegner number.
${{2e} \over \pi} \approx \sqrt3$ , within 0.089%
${\pi \over {e(\pi - e)}} \approx \sqrt3 + 1$ , within 0.067%
$\sum_{k=1}^8 \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{7} + \frac{1}{8} \approx e$ , within 0.016%.
For any integer n, the early decimal digits of $n - e$ are composed of 2,7,1, and 8, the first digits of e
$\log_e (x) \approx \log_2 (x) - \log_{10} (x)$ to within 1%
$\exp(-\Psi(\sqrt{3}/4+1/2)))= 2$ to one part in ten million.
$e + π + Φ = 7.5$ , within 0.29%.

[edit] Concerning pi

$\pi\approx 22/7,$ correct to about 0.04%; $\pi\approx 355/113,$ correct to six decimal places or about 85 parts in a billion. ( $π$ has unusually large terms in its continued fraction representation very early: $π$ = [3, 7, 15, 1, 292, ...]).
$\pi^2\approx10;$ 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).
$\pi^3\approx 31$ (actually 31.0062...); $\pi^5\approx 306$ (actually 306.0196...).
$\pi\approx\left(9^2+\frac{19^2}{22}\right)^{1/4},$ 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).
$\pi^4+\pi^5\approx e^6;$ correct to about 44 parts in a billion.

[edit] Concerning base 2

correct to 2.4%, see binary prefix; implies that actual value about 0.30103; engineers make extensive use of the approximation that 3 dB corresponds to doubling of power level.
- This is also very useful for estimating the size of a binary number, for example a 512 bit number is $2^{512}= 2^2*(2^{10})^{51}\approx 4*(10^3)^{51}=4*10^{153}$
- Using this approximate value of $log 10 2,$ one can derive the following approximations for logs of other numbers:
- $3^4\approx 10\cdot 2^3,$ leading to $\log_{10}3=(1+3\log_{10}2)/4\approx 0.475;$ compare the true value of about 0.4771
- $7^2\approx 10^2/2,$ leading to $\log_{10}7\approx 1-\log_{10}2/2,$ or about 0.85 (compare 0.8451)
- $2^7\approx 5^3,$ leading to $5\approx 2^{7/3}=2^{28/12},$ i.e. $5/4\approx 2^{1/3}=2^{4/12}.$ The major third in equal temperament (four semitones) thus approximates the ratio 5:4 corresponding to the major third in just intonation.
$2^{7/12}\approx 3/2;$ correct to about 0.1%. In music, this coincidence means that the chromatic scale of twelve pitches includes, for each note (in a system of equal temperament, which depends on this coincidence), a note related by the 3/2 ratio. This 3/2 ratio of frequencies is the musical interval of a fifth and lies at the basis of Pythagorean tuning, just intonation, and most known systems of music.

[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).
a cubic attoparsec (a cube where each edge is one attoparsec) is within 1% of a 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.
The speed of light in a vacuum is about one foot per nanosecond (accurate to 2%) or 3×10⁸ m/s (accurate to about 0.1%)
If you are planning an interstellar cruise, and you want to accelerate your spaceship to the speed of light using a power source that gives you 1 g, how long will it take? Ignoring relativity, it will take about one year: 1 light-year per year squared is an approximation of 1 g (accurate to about 3%).
Units of pressure: One millimetre of water (mmH₂O) is equal to 9.807 pascals, , or 0.9807 decapascals (daPa). The mmH₂O unit was commonly used by audiologists, but is now considered obsolete because it is not an SI unit. Consequently, the decapascal has taken its place. This is a rare instance of the very obscure "deca" SI prefix.

[edit] Functions

The so-called "strong law of small numbers" [1] states that functions which look equal if we just look at small values can reveal differences if higher values are taken into account. For example:

The maximum number of areas into which a circle can be divided by choosing n points on its circumference and joining them with straight lines, given by the polynomial (n⁴−6n³+23n²−18n+24)/24, happens to equal 2ⁿ⁻¹ for any n = 1, 2, 3, 4 and 5, i.e. 1, 2, 4, 8, and 16, but for n = 6, 7, … it gives 31, 57, ….
$\lceil e^\frac{n-1}{2} \rceil$ (see ceiling function) happens to equal the n-th Fibonacci number for n = 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, i.e. 1, 1, 2, 3, 5, 8, 13, 21, 34, and 55, but for n = 10, 11, … it gives 91, 149, ….
The number of letters needed to spell out the word for 2n in Italian language happens to equal n for n = 3, 4, 5, and 6 (as 6, 8, 10, 12 is sei, otto, dieci, dodici in Italian), but for n = 7, 8, … it gives 11, 6, …. (Notice that a non-standard variant of diciotto = 18, i.e. dieciotto, is indeed spelt with 9 letters.)
The so-called Euler's formula $n 2 - n + 41$ generates prime numbers for all integer n from 0 to 40, but fails when n equals 41.

[edit] Other numeric curiosities

To put it the way Hardy did: "These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals to the mathematician."[2]

$10! = 6! \cdot 7!$ .
Golden ratio $\phi = -2 \cdot \sin(666^\circ)$ (an amusing equality with an angle expressed in degrees)
$\,5^2=25$ and $3^3=27\,$ are the only square and cube 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]
$\,(27/8)^{9/4} = (9/4)^{27/8} = 15.438887358552...$ is the smallest irrational example found by S. Ramanujan, cited by G.H. Hardy.
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%

[edit] Decimal coincidences

$2^5 \cdot 9^2 = 2592$ .
$\frac {16} {64} = \frac {1\!\!\!\not6} {\not6 4} = \frac {1} {4}$ , $\frac {26} {65} = \frac {2\!\!\!\not6} {\not6 5} = \frac {2} {5}$ , $\frac {19} {95} = \frac {1\!\!\!\not9} {\not9 5} = \frac {1} {5}$
$\,(4 + 9 + 1 + 3)^3 = 4,913$ and $\,(1 + 9 + 6 + 8 + 3)^3=19,683$ .
$\,1^3 + 5^3 + 3^3 = 153$ ; $\,3^3 + 7^3 + 0^3 = 370$ ; $\,3^3 + 7^3 +1^3 = 371$ ; $\,4^3 + 0^3 +7^3 = 407$
$3^2 + 7^2 - 3 \cdot 7 = (3^3 + 7^3)/(3 + 7) = 370$ .
$\,35 - 3^2 - 5^2 = 75 - 7^2 - 5^2$ .
$\,588^2+2353^2 = 5882353$ 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: $\,2646798 = 2^1+6^2+4^3+6^4+7^5+9^6+8^7.$
$\,111111111^2 = 12345678987654321$

[edit] See also

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

[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

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