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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%

  • e^\pi - \pi\simeq 19.9990999 is very close to 20 in a strange way. (Conway, Sloane, Plouffe, 1988).

  • \sqrt{2 \pi} \simeq 5/2 to about 0.1% (one part in a thousand).

  • \pi\simeq 22/7; correct to about 0.04%; \pi\simeq 355/113, correct to six places or 0.000008%.

  • \pi^2\simeq10; correct to about 1.3%. This coincidence was used in the design of Slide Rules , where the "folded" scales are folded on \pi 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% (note 2 , 227 , and 23 are Chen Prime s).

  • \pi^3\simeq31; correct to about 0.02%.

  • \pi^4\simeq 2143/22, accurate to about one part in 10^{10}; due to .

  • \pi^5\simeq306; correct to about 0.006%.

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

  • 2^{10}\simeq 10^3; correct to 2.4%, see Binary Prefix ; 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. Using this approximate value of \log_{10}2, 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)

  • --- 2^7\simeq 5^3, leading to 5\simeq 2^{7/3}=2^{28/12}, i.e. 5/4\simeq 2^{1/3}=2^{4/12}. The .

  • 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 .

  • \pi Second s is a nanocentury (ie 10^{-7} Year s); 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 \phi 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. Another good approximation is 1 mile = log(5) km, 1 mile = 1.609344 km and log(5) = 1.6094379124341...

  • 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 rac{63}{25}\left( rac{17+15\sqrt{5}}{7+15\sqrt{5}} ight);

    • :accurate to 9 decimal places (due to Ramanujan ).


  • The Speed Of Light is about one Foot per Nanosecond (accurate to 2%) or 3×108 m/s (accurate to about 0.1%)

    • The so-called "strong law of small numbers" {Link without Title} 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 (''n''4−6''n''3+23''n''2−18''n''+24)/24, happens to equal 2''n''−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, ….

  • \lceil e^ rac{n-1}{2} ceil (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 2''n'' 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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