Mathematical coincidence

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In mathematics, a mathematical coincidence can be said to occur when two expressions show a near-equality that lacks direct theoretical explanation. One of the expressions may be an integer and the surprising 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 so-called law of small numbers. Although mathematical coincidences may be useful, they are mainly notable for their curiosity value.

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Some examples

  • <math>e^\pi\simeq\pi^e</math>; correct to about 3%
  • <math>e^\pi - \pi\simeq 19.9990999 </math> is very close to 20 in a strange way. (Conway, Sloane, Plouffe, 1988).
  • <math> \sqrt{2 \pi} \simeq 5/2 </math> to about 0.1% (one part in a thousand).
  • <math>\pi\simeq 22/7</math>; correct to about 0.04%; <math>\pi\simeq 355/113</math>, correct to six places or 0.000008%.
  • <math>\pi^2\simeq10</math>; correct to about 1.3%. This coincidence was used in the design of slide rules, where the "folded" scales are folded on <math>\pi</math> rather than <math>\sqrt{10}</math>, because it is a more useful number and has the effect of folding the scales in about the same place; <math>\pi^2\simeq 227/23</math>, correct to 0.0004% (note 2, 227, and 23 are Chen primes).
  • <math>\pi^3\simeq31</math>; correct to about 0.02%.
  • <math>\pi^4\simeq 2143/22</math>, accurate to about one part in <math>10^{10}</math>; due to Srinivasa Ramanujan, who might have noticed that the continued fraction representation for <math>\pi^4</math> begins <math>[97; 2,2,3,1,16539,1,1,\ldots]</math>. See also: Pi culture.
  • <math>\pi^5\simeq306</math>; correct to about 0.006%.

(The theory of continued fractions gives a systematic treatment of this type of coincidence; and also such coincidences as <math>2\times 12^2\simeq 17^2</math> (ie <math>\sqrt{2}\simeq 17/12</math>). Curiously the continued fractions of the first few powers of <math>\pi</math> have big numbers (>50) quite early, in the case of <math>\pi^3</math> and <math>\pi^5</math> as soon as the first denominator.)

  • <math>1+1/\log(10)\simeq 1/\log(2)</math>; leading to Donald Knuth's observation that, to within about 5%, <math>\log_2(x)=\log(x)+\log_{10}(x)</math>.
  • <math>2^{10}\simeq 10^3</math>; correct to 2.4%, see binary prefix; implies that <math>\log_{10}2=0.3</math>; actual value about 0.30103; engineers make extensive use of the approximation that 3 dB corresponds to doubling of power level. Using this approximate value of <math>\log_{10}2</math>, one can derive the following approximations for logs of other numbers:
    • <math>3^4\simeq 10\cdot 2^3</math>, leading to <math>\log_{10}3=(1+3\log_{10})/4\simeq 0.475</math>; compare the true value of about 0.4771
    • <math>7^2\simeq 10^2/2</math>, leading to <math>\log_{10}7\simeq 1-\log_{10}2/2</math>, or about 0.85 (compare 0.8451)
    • <math>2^7\simeq 5^3</math>, leading to <math>5\simeq 2^{7/3}=2^{28/12}</math>, i.e. <math>5/4\simeq 2^{1/3}=2^{4/12}</math>. The major third in equal temperament (four semitones) thus approximates the ratio 5:4 corresponding to the major third in just intonation.
  • <math> e^\pi\simeq\pi+20</math>; correct to about 0.004%
  • <math>e^{\pi\sqrt{n}}</math> is close to an integer for many values of <math>n</math>, most notably <math>n=163</math>; this one has roots in algebraic number theory.
  • <math>\pi</math> seconds is a nanocentury (ie <math>10^{-7}</math> 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 <math>\phi</math> kilometers (correct to about 0.5%), where <math>\phi={1+\sqrt 5\over 2}</math> 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 <math>F_n</math> mi = <math>F_{n+1}</math> km, e.g. 5 mi = 8 km, 8 mi = 13 km. Another good approximation is 1 mile = log(5) km, 1 mile = 1.609344 km and log(5) = 1.6094379124341...
  • <math>2^{7/12}\simeq 3/2</math>; 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 indeed most known systems of music.
  • <math>\pi\simeq\frac{63}{25}\left(\frac{17+15\sqrt{5}}{7+15\sqrt{5}}\right)</math>;
accurate to 9 decimal places (due to Ramanujan).

The so-called "strong law of small numbers" [1] states that functions which look equal if we just look at small values can reveal different if higher values are taken in 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, ….
  • <math>\lceil e^\frac{n-1}{2} \rceil </math> (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.)
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See also

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