math lessons - 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%
$\pi\simeq 22/7$ ; correct to about 0.03%; $\pi\simeq 355/113$ , correct to six places or 0.000008%.
$\pi^2\simeq10$ ; 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; $\pi^2\simeq 227/23$ , correct to 0.0004%.
$\pi^3\simeq31$ ; correct to about 0.02%.
$\pi^4\simeq 2143/22$ , accurate to about one part in $1010$ ; due to Ramanujan, who might have noticed that the continued fraction representation for $π 4$ begins $[97; 2,2,3,1,16539,1,1,\ldots]$ .
$\pi^5\simeq306$ ; correct to about 0.006%.

(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 $\sqrt{2}\simeq 17/12$ ). Curiously the continued fractions of the first few powers of $π$ have big numbers (>50) quite early, in the case of $π 3$ and $π 5$ as soon as the first denominator.)

$1+1/\log(10)\simeq 1/\log(2)$ ; leading to Donald Knuth's observation that, to within about 5%, $log 2 (x) = log(x) + log 10 (x)$ .
; 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. Using this approximate value of log₁₀2, one can derive the following approximations for logs of other numbers:
- $3^4\simeq 10\cdot 2^3$ , leading to $\log_{10}3=(1+3\log_{10})/4\simeq 0.475$ ; compare the true value of about 0.4771
- $7^2\simeq 10^2/2$ , leading to $\log_{10}7\simeq 1-\log_{10}2/2$ , or about 0.85 (compare 0.8451)
$e^\pi\simeq\pi+20$ ; 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).
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.
$2^{7/12}\simeq 3/2$ ; 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.
$\pi\simeq\frac{63}{25}\left(\frac{17+15\sqrt{5}}{7+15\sqrt{5}}\right)$ ;