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
A round number is an integer that ends with one or more " 0"s (zero-digit) in a given base. So, 590 is rounder than 592, but 590 is less round than 600. In both technical and informal language, a round number is often interpreted to stand for a ...
1000 between powers of 2 and powers of 10:
:
Some mathematical coincidences are used in
engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
when one expression is taken as an approximation of another.
Introduction
A mathematical coincidence often involves an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
, and the surprising feature is the fact that a
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...
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
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
with a small
denominator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
. 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
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
number of ways of forming mathematical expressions using a finite number of symbols, the number of symbols used and the
precision
Precision, precise or precisely may refer to:
Science, and technology, and mathematics Mathematics and computing (general)
* Accuracy and precision, measurement deviation from true value and its scatter
* Significant figures, the number of digit ...
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
In mathematics, the "strong law of small numbers" is the humorous law that proclaims, in the words of Richard K. Guy (1988):
In other words, any given small number appears in far more contexts than may seem reasonable, leading to many apparentl ...
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 In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer ''d'' such that the imaginary quadratic field \Q\left sqrt\right/math> has class number 1. Equivalently, its ring of integers has unique factorizat ...
below, which made it into print some years ago as a scientific
April Fools' joke
[Reprinted as ]). 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
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
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, ; 7= 22/7 = 3.1428..., was known to Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists i ...
,[ and is correct to about 0.04%. The fourth convergent of π, ; 7, 15, 1= 355/113 = 3.1415929..., found by ]Zu Chongzhi
Zu Chongzhi (; 429–500 AD), courtesy name Wenyuan (), was a Chinese astronomer, mathematician, politician, inventor, and writer during the Liu Song and Southern Qi dynasties. He was most notable for calculating pi as between 3.1415926 and 3 ...
, is correct to six decimal places; this high accuracy comes about because π has an unusually large next term in its continued fraction representation: = ; 7, 15, 1, 292, ...
* A coincidence involving π and the golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
φ is given by . Consequently, the square on the middle-sized edge of a Kepler triangle
A Kepler triangle is a special right triangle with edge lengths in geometric progression. The ratio of the progression is \sqrt\varphi where \varphi=(1+\sqrt)/2 is the golden ratio, and the progression can be written: or approximately . Square ...
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
The Great Pyramid of Giza is the biggest Egyptian pyramids, Egyptian pyramid and the tomb of Fourth Dynasty of Egypt, Fourth Dynasty pharaoh Khufu. Built in the early 26th century BC during a period of around 27 years, the pyramid is the oldes ...
, but it is highly improbable that this was intentional.
* There is a sequence of six nines in pi
A sequence of six consecutive nines occurs in the decimal representation of the number pi (), starting at the 762nd decimal place.. It has become famous because of the mathematical coincidence and because of the idea that one could memorize the ...
, popularly known as the Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superf ...
point, beginning at the 762nd decimal place of its decimal representation. For a randomly chosen normal number
In mathematics, a real number is said to be simply normal in an integer base b if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b. A number is said to ...
, 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
In mathematics, specifically bifurcation theory, the Feigenbaum constants are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the physicist Mitchell J. Feigenbaum.
Histor ...
is approximately equal to , with an error of 0.0015%.
Concerning base 2
* The coincidence , correct to 2.4%, relates to the rational approximation , or to within 0.3%. This relationship is used in engineering, for example to approximate a factor of two in power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may ...
as 3 dB (actual is 3.0103 dB – see Half-power point), or to relate a kibibyte
The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable uni ...
to a kilobyte
The kilobyte is a multiple of the unit byte for digital information.
The International System of Units (SI) defines the prefix ''kilo'' as 1000 (103); per this definition, one kilobyte is 1000 bytes.International Standard IEC 80000-13 Quantitie ...
; see binary prefix
A binary prefix is a unit prefix for multiples of units. It is most often used in data processing, data transmission, and digital information, principally in association with the bit and the byte, to indicate multiplication by a power of ...
.
* This coincidence can also be expressed as (eliminating common factor of , so also correct to 2.4%), which corresponds to the rational approximation , or (also to within 0.3%). This is invoked for instance in shutter speed
In photography, shutter speed or exposure time is the length of time that the film or digital sensor inside the camera is exposed to light (that is, when the camera's shutter is open) when taking a photograph.
The amount of light that rea ...
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?
''Who Wants to Be a Millionaire?'' (often informally called ''Millionaire'') is an international television game show franchise of British origin, created by David Briggs, Mike Whitehill and Steven Knight. In its format, currently owned and l ...
'' 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
Twelve-tone equal temperament (12-TET) is the musical system that divides the octave into 12 parts, all of which are equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the 12th root of 2 ( ≈ 1.05946). That result ...
, the ratio between consecutive note frequencies is