--- title: "Mathematical coincidence" chunk: 2/5 source: "https://en.wikipedia.org/wiki/Mathematical_coincidence" category: "reference" tags: "science, encyclopedia" date_saved: "2026-05-05T07:23:33.671306+00:00" instance: "kb-cron" --- The coincidence 2 19 ≈ 3 12 {\displaystyle 2^{19}\approx 3^{12}} , from log 2 ⁡ ( 3 ) = 1.5849 … ≈ 19 12 {\displaystyle \log _{2}(3)=1.5849\ldots \approx {\frac {19}{12}}} , closely relates the interval of 7 semitones in equal temperament to a perfect fifth of just intonation: 2 7 / 12 ≈ 3 / 2 {\displaystyle 2^{7/12}\approx 3/2} , correct to about 0.1%. The just fifth is the basis of Pythagorean tuning; the difference between twelve just fifths and seven octaves is the Pythagorean comma. The coincidence ( 3 / 2 ) 4 = ( 81 / 16 ) ≈ 5 {\displaystyle {(3/2)}^{4}=(81/16)\approx 5} permitted the development of meantone temperament, in which just perfect fifths (ratio 3 / 2 {\displaystyle 3/2} ) and major thirds ( 5 / 4 {\displaystyle 5/4} ) are "tempered" so that four 3 / 2 {\displaystyle 3/2} 's is approximately equal to 5 / 1 {\displaystyle 5/1} , or a 5 / 4 {\displaystyle 5/4} major third up two octaves. The difference ( 81 / 80 {\displaystyle 81/80} ) between these stacks of intervals is the syntonic comma. The coincidence 2 12 5 7 = 1.33333319 … ≈ 4 3 {\displaystyle {\sqrt[{12}]{2}}{\sqrt[{7}]{5}}=1.33333319\ldots \approx {\frac {4}{3}}} leads to the rational version of 12-TET, as noted by Johann Kirnberger. The coincidence 5 8 35 3 = 4.00000559 … ≈ 4 {\displaystyle {\sqrt[{8}]{5}}{\sqrt[{3}]{35}}=4.00000559\ldots \approx 4} leads to the rational version of quarter-comma meantone temperament. The coincidence of powers of 2, above, leads to the approximation that three major thirds concatenate to an octave, ( 5 / 4 ) 3 ≈ 2 / 1 {\displaystyle {(5/4)}^{3}\approx {2/1}} . This and similar approximations in music are called dieses. === Numerical expressions === ==== Concerning powers of π ==== π 2 ≈ 10 ; {\displaystyle \pi ^{2}\approx 10;} correct to about 1.32%. This can be understood in terms of the formula for the zeta function ζ ( 2 ) = π 2 / 6. {\displaystyle \zeta (2)=\pi ^{2}/6.} This coincidence was used in the design of slide rules, where the "folded" scales are folded on π {\displaystyle \pi } rather than 10 , {\displaystyle {\sqrt {10}},} because it is a more useful number and has the effect of folding the scales in about the same place. π 2 + π ≈ 13 ; {\displaystyle \pi ^{2}+\pi \approx 13;} correct to about 0.086%. π 2 ≈ 227 / 23 , {\displaystyle \pi ^{2}\approx 227/23,} correct to 4 parts per million. π 3 ≈ 31 , {\displaystyle \pi ^{3}\approx 31,} correct to 0.02%. 2 π 3 − π 2 − π ≈ 7 2 , {\displaystyle 2\pi ^{3}-\pi ^{2}-\pi \approx 7^{2},} correct to about 0.002% and can be seen as a combination of the above coincidences. π 4 ≈ 2143 / 22 ; {\displaystyle \pi ^{4}\approx 2143/22;} or π ≈ ( 9 2 + 19 2 22 ) 1 / 4 , {\displaystyle \pi \approx \left(9^{2}+{\frac {19^{2}}{22}}\right)^{1/4},} accurate to 9 decimal places (due to Ramanujan: Quarterly Journal of Mathematics, XLV, 1914, pp. 350–372). Ramanujan states that this "curious approximation" to π {\displaystyle \pi } was "obtained empirically" and has no connection with the theory developed in the remainder of the paper. Some near-equivalences, which hold to a high degree of accuracy, are not actually coincidences. For example, ∫ 0 ∞ cos ⁡ ( 2 x ) ∏ n = 1 ∞ cos ⁡ ( x n ) d x ≈ π 8 . {\displaystyle \int _{0}^{\infty }\cos(2x)\prod _{n=1}^{\infty }\cos \left({\frac {x}{n}}\right)\mathrm {d} x\approx {\frac {\pi }{8}}.} The two sides of this expression differ only after the 42nd decimal place; this is not a coincidence. ==== Containing both π and e ==== π ≈ 1 + e − γ {\displaystyle \pi \approx 1+e-\gamma } to 4 digits, where γ is the Euler–Mascheroni constant. π 4 + π 5 ≈ e 6 {\displaystyle \pi ^{4}+\pi ^{5}\approx e^{6}} , to about 7 decimal places. Equivalently, 4 ⋅ ln ⁡ ( π ) + ln ⁡ ( π + 1 ) ≈ 6 {\displaystyle 4\cdot \ln(\pi )+\ln(\pi +1)\approx 6} . ( e − 1 ) π ≈ 5 + 10 {\displaystyle (e-1)\pi \approx {\sqrt {5}}+{\sqrt {10}}} , to about 4 decimal places. ( π 2 − ln ⁡ ( 3 π 2 ) ) 42 π ≈ e {\displaystyle \left({\frac {\pi }{2}}-\ln \left({\frac {3\pi }{2}}\right)\right)42\pi \approx e} , to about 9 decimal places. e π − π ≈ 20 {\displaystyle e^{\pi }-\pi \approx 20} to about 4 decimal places. (Conway, Sloane, Plouffe, 1988); this is equivalent to ( π + 20 ) i = − 0.9999999992 … − i ⋅ 0.000039 … ≈ − 1. {\displaystyle (\pi +20)^{i}=-0.9999999992\ldots -i\cdot 0.000039\ldots \approx -1.} Once considered a textbook example of a mathematical coincidence, the fact that e π − π {\displaystyle e^{\pi }-\pi } is close to 20 is itself not a coincidence, although the approximation is an order of magnitude closer than would be expected. It is a consequence of the infinite sum ∑ k = 1 ∞ ( 8 π k 2 − 2 ) e ( − π k 2 ) = 1 , {\displaystyle \textstyle \sum _{k=1}^{\infty }\left(8\pi k^{2}-2\right)e^{\left(-\pi k^{2}\right)}=1,} resulting from the Jacobian theta functional identity. The first term of the sum is by far the largest, which gives the approximation ( 8 π − 2 ) e − π ≈ 1 , {\displaystyle \left(8\pi -2\right)e^{-\pi }\approx 1,} or e π ≈ 8 π − 2. {\displaystyle e^{\pi }\approx 8\pi -2.} Using the estimate π ≈ 22 / 7 {\displaystyle \pi \approx 22/7} then gives e π ≈ π + ( 7 ⋅ 22 7 − 2 ) = π + 20. {\displaystyle e^{\pi }\approx \pi +(7\cdot {\frac {22}{7}}-2)=\pi +20.} Although not widely known, an explanation for it has been circulating for more than a decade, at least. π e + e π ≈ 45 3 5 {\displaystyle \pi ^{e}+e^{\pi }\approx 45{\frac {3}{5}}} , within 4 parts per million.