Mathematical coincidence

This is a page 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. 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.

Some examples

• <math>e^\pi\simeq\pi^e<math>; correct to about 3%
• <math>\pi\simeq 22/7<math>; correct to about 0.03%; <math>\pi\simeq 355/113<math>, correct to six places or 0.000008%.
• <math>\pi^2\simeq10<math>; correct to about 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%.
• <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 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>.
• <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%; 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> 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).
• 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.
• <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).

• N_A ≈ 2⁷⁹, where N is Avogadro's number; correct to about 0.4%. This means that a yottabyte is slightly more than two moles of bytes.

See also

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