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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.

Some examples

$e^\pi\simeq\pi^e$ ; correct to about 3%
$\pi^2\simeq10$ ; correct to about 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.
$\pi\simeq 22/7$ ; correct to about 0.03%; $\pi\simeq 355/113$ , correct to six places or 0.000008%. (The theory of continued fractions gives a systematic treatment of this type of coincidence; and also such coincidences as $2\times 12^2\simeq 17^2$ (ie $\sqrt{2}\simeq 17/12$ ).
$1+1/\log(10)\simeq 1/\log(2)$ ; leading to Donald Knuth's observation that, to within about 5%, $l o g 2 (x) = l o g (x) + l o g 10 (x)$ .
$2^{10}\simeq 10^3$ ; correct to 2.4%; implies that $log 10 2 = 0.3$ ; actual value about 0.30103. Engineers make extensive use of the approximation that 3 dB corresponds to doubling of power level
$e^\pi\simeq\pi+20$ ; correct to about 0.004%
$e^{\pi\sqrt{n}}$ is close to an integer for many values of $n$ , most notably $n = 163$ ; this one has roots in algebraic number theory.
$π$ 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).
$2^{7/12}\simeq 3/2$ ; 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 indeed most known systems of music.