14 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Mathematical coincidence | 2/5 | https://en.wikipedia.org/wiki/Mathematical_coincidence | reference | science, encyclopedia | 2026-05-05T07:23:33.671306+00:00 | 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.