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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Mathematical coincidence | 1/5 | https://en.wikipedia.org/wiki/Mathematical_coincidence | reference | science, encyclopedia | 2026-05-05T07:23:33.671306+00:00 | kb-cron |
A mathematical coincidence is said to occur when two expressions with no direct relationship show a near-equality which has no apparent theoretical explanation. For example, there is a near-equality close to the round number 1000 between powers of 2 and powers of 10:
2
10
=
1024
≈
1000
=
10
3
.
{\displaystyle 2^{10}=1024\approx 1000=10^{3}.}
Some mathematical coincidences are used in engineering when one expression is taken as an approximation of another.
== Introduction == A mathematical coincidence often involves an integer, and the surprising feature is the fact that a real number arising in some context is considered by some standard as a "close" approximation to a small integer or to a multiple or power of ten, or more generally, to a rational number with a small denominator. Other kinds of mathematical coincidences, such as integers simultaneously satisfying multiple seemingly unrelated criteria or coincidences regarding units of measurement, may also be considered. In the class of those coincidences that are of a purely mathematical sort, some simply result from sometimes very deep mathematical facts, while others appear to come 'out of the blue'. Given the countably infinite number of ways of forming mathematical expressions using a finite number of symbols, the number of symbols used and the precision of approximate equality might be the most obvious way to assess mathematical coincidences; but there is no standard, and the strong law of small numbers is the sort of thing one has to appeal to with no formal opposing mathematical guidance. Beyond this, some sense of mathematical aesthetics could be invoked to adjudicate the value of a mathematical coincidence, and there are in fact exceptional cases of true mathematical significance (see Ramanujan's constant below, which made it into print some years ago as a scientific April Fools' joke). All in all, though, they are generally to be considered for their curiosity value, or perhaps to encourage new mathematical learners at an elementary level.
== Some examples ==
=== Rational approximants === Sometimes simple rational approximations are exceptionally close to interesting irrational values. These are explainable in terms of large terms in the continued fraction representation of the irrational value, but further insight into why such improbably large terms occur is often not available. Rational approximants (convergents of continued fractions) to ratios of logs of different numbers are often invoked as well, making coincidences between the powers of those numbers. Many other coincidences are combinations of numbers that put them into the form that such rational approximants provide close relationships.
==== Concerning π ==== The second convergent of π, [3; 7] = 22/7 = 3.1428..., was known to Archimedes, and is correct to about 0.04%. The fourth convergent of π, [3; 7, 15, 1] = 355/113 = 3.1415929..., found by Zu Chongzhi, is correct to six decimal places; this high accuracy comes about because π has an unusually large next term in its continued fraction representation: π = [3; 7, 15, 1, 292, ...]. A coincidence involving π and the golden ratio φ is given by
π
≈
4
/
φ
=
3.1446
…
{\displaystyle \pi \approx 4/{\sqrt {\varphi }}=3.1446\dots }
. Consequently, the square on the middle-sized edge of a Kepler triangle is similar in perimeter to its circumcircle. Some believe one or the other of these coincidences is to be found in the Great Pyramid of Giza, but it is highly improbable that this was intentional. There is a sequence of six nines in pi beginning at the 762nd decimal place of its decimal representation. For a randomly chosen normal number, the probability of a particular sequence of six consecutive digits—of any type, not just a repeating one—to appear this early is 0.08%. Pi is conjectured, but not known, to be a normal number. The first Feigenbaum constant is approximately equal to
10
π
−
1
{\displaystyle {\tfrac {10}{\pi -1}}}
, with an error of 0.0047%.
==== Concerning base 2 ==== The coincidence
2
10
=
1024
≈
1000
=
10
3
{\displaystyle 2^{10}=1024\approx 1000=10^{3}}
, correct to 2.4%, relates to the rational approximation
log
2
(
10
)
≈
3.3219
≈
10
3
{\displaystyle \textstyle \log _{2}(10)\approx 3.3219\approx {\frac {10}{3}}}
, or
2
≈
10
3
/
10
{\displaystyle 2\approx 10^{3/10}}
to within 0.3%. This relationship is used in engineering, for example to approximate a factor of two in power as 3 dB (actual is 3.0103 dB – see Half-power point), or to relate a kibibyte to a kilobyte; see binary prefix. The same numerical coincidence is responsible for the near equality between one third of an octave and one tenth of a decade. The same coincidence can also be expressed as
128
=
2
7
≈
5
3
=
125
{\displaystyle 128=2^{7}\approx 5^{3}=125}
(eliminating common factor of
2
3
{\displaystyle 2^{3}}
, so also correct to 2.4%), which corresponds to the rational approximation
log
2
(
5
)
≈
2.3219
≈
7
3
{\displaystyle \textstyle \log _{2}(5)\approx 2.3219\approx {\frac {7}{3}}}
, or
2
≈
5
3
/
7
{\displaystyle 2\approx 5^{3/7}}
(also to within 0.4%). This is invoked in preferred numbers in engineering, such as shutter speed settings on cameras, as approximations to powers of two (128, 256, 512) in the sequence of speeds 125, 250, 500, etc., and in the original Who Wants to Be a Millionaire? game show in the question values ...£16,000, £32,000, £64,000, £125,000, £250,000,...
==== Concerning musical intervals ====
In music, the distances between notes (intervals) are measured as ratios of their frequencies, with near-rational ratios often sounding harmonious. In western twelve-tone equal temperament, the ratio between consecutive note frequencies is
2
12
{\displaystyle {\sqrt[{12}]{2}}}
.