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# Mathematical coincidence

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%, log2(x) = log(x) + log10(x).
• $2^{10}\simeq 10^3$; correct to 2.4%; implies that log102 = 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.
• $\pi\simeq\frac{63}{25}\left(\frac{17+15\sqrt{5}}{7+15\sqrt{5}}\right)$;

accurate to 9 decimal places (due to Ramanujan).