A mathematical coincidence can be said to occur when two expressions show a near-equality that lacks direct theoretical explanation. For example, there's a near-equality around the round number

Round number

A round number is mathematically defined as the product of a considerable number of comparatively small factors as compared to its neighbouring numbers, such as 24 = 2*2*2*3 .However, a round number is informally considered to be an integer that ends with one or more zeroes , such...

 1000 between powers of two and powers of ten: . Some of these coincidences are used in engineering

Engineering

Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

 when one expression is taken as an approximation of the other.

Introduction

A mathematical coincidence often involves an integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

, and the surprising (or "coincidental") feature is the fact that a real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

 arising in some context is considered by some ill-defined standard as a "close" approximation to a small integer or to a multiple or power of ten, or more generally, to a rational number

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

 with a small denominator.

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

Strong Law of Small Numbers

"The Strong Law of Small Numbers" is a humorous paper by mathematician Richard K. Guy and also the so-called law that it proclaims: "There aren't enough small numbers to meet the many demands made of them." In other words, any given small number appears in far more contexts than may seem...

 is the sort of thing one has to appeal to with no formal opposing mathematical guidance. Beyond this, some sense of mathematical aesthetics

Mathematical beauty

Many mathematicians derive aesthetic pleasure from their work, and from mathematics in general. They express this pleasure by describing mathematics as beautiful. Sometimes mathematicians describe mathematics as an art form or, at a minimum, as a creative activity...

 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'

April Fools' Day

April Fools' Day is celebrated in different countries around the world on April 1 every year. Sometimes referred to as All Fools' Day, April 1 is not a national holiday, but is widely recognized and celebrated as a day when many people play all kinds of jokes and foolishness...

 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.

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

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 part and another reciprocal, and so on...

 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 pi

• The first convergent
  Convergent (continued fraction)
  A convergent is one of a sequence of values obtained by evaluating successive truncations of a continued fraction The nth convergent is also known as the nth approximant of a continued fraction.-Representation of real numbers:...
  
   of π, [3; 7] = 22/7 = 3.1428..., was known to Archimedes
  Archimedes
  Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
  
  , and is correct to about 0.04%. The third convergent of π, [3; 7, 15, 1] = 355/113 = 3.1415929..., found by Zu Chongzhi
  Zu Chongzhi
  Zu Chongzhi , courtesy name Wenyuan , was a prominent Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties.-Life and works:...
  
  , 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
  Golden ratio
  In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
  
   φ is given by . This is related to Kepler triangles.

• The Feynman point
  Feynman point
  The Feynman point is a sequence of six 9s that begins at the 762nd decimal place of the decimal representation of . It is named after physicist Richard Feynman, who once stated during a lecture he would like to memorize the digits of until that point, so he could recite them and quip "nine nine...
  
   is a sequence of six 9s that begins at the 762nd decimal place of the decimal representation of pi. For a randomly chosen normal number
  Normal number
  In mathematics, a normal number is a real number whose infinite sequence of digits in every base b is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b, also all possible b2 pairs of digits are equally likely with density b−2,...
  
  , the probability of any chosen number sequence of six digits (including 6 of a number, 658020, or the like) occurring this early in the decimal representation is only 0.08%. Pi is conjectured, but not known, to be a normal number.

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 as 3 dB
  Decibel
  The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...
  
   (actual is 3.0103 dB – see 3dB-point), or to relate a kilobyte
  Kilobyte
  The kilobyte is a multiple of the unit byte for digital information. Although the prefix kilo- means 1000, the term kilobyte and symbol KB have historically been used to refer to either 1024 bytes or 1000 bytes, dependent upon context, in the fields of computer science and information...
  
   to a kibibyte
  Kibibyte
  The kibibyte is a multiple of the unit byte for quantities of digital information. The binary prefix kibi means 1024; therefore, 1 kibibyte is . The unit symbol for the kibibyte is KiB. The unit was established by the International Electrotechnical Commission in 1999 and has been accepted for use...
  
  ; see binary prefix
  Binary prefix
  In computing, a binary prefix is a specifier or mnemonic that is prepended to the units of digital information, the bit and the byte, to indicate multiplication by a power of 2...
  
  .

• This coincidence can also be expressed , and is invoked for instance in shutter speed
  Shutter speed
  In photography, shutter speed is a common term used to discuss exposure time, the effective length of time a camera's shutter is open....
  
   settings on cameras, as approximations to powers of two in the sequence of speeds 125, 250, 500, etc.

Concerning musical intervals

• The coincidence , from leads to the observation commonly used in music
  Music
  Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
  
   to relate the tuning of 7 semitone
  Semitone
  A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....
  
  s of equal temperament
  Equal temperament
  An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
  
   to a perfect fifth
  Perfect fifth
  In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
  
   of just intonation
  Just intonation
  In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
  
  : , correct to about 0.1%. The just fifth is the basis of Pythagorean tuning
  Pythagorean tuning
  Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
  
   and most known systems of music. From the consequent approximation it follows that the circle of fifths
  Circle of fifths
  In music theory, the circle of fifths shows the relationships among the 12 tones of the chromatic scale, their corresponding key signatures, and the associated major and minor keys...
  
   terminates seven octave
  Octave
  In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
  
  s higher than the origin.

• The coincidence leads to the rational version
  Schisma
  In music, the schisma is the ratio between a Pythagorean comma and a syntonic comma and equals 32805:32768, which is 1.9537 cents...
  
   of 12-TET, as noted by Johann Kirnberger
  Johann Kirnberger
  Johann Philipp Kirnberger was a musician, composer , and music theorist. A pupil of Johann Sebastian Bach, he became a violinist at the court of Frederick II of Prussia in 1751. He was the music director to the Prussian Princess Anna Amalia from 1758 until his death. Kirnberger greatly admired J.S...
  
  .

• The coincidence leads to the rational version of quarter-comma meantone
  Quarter-comma meantone
  Quarter-comma meantone, or 1/4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. This method is a variant of Pythagorean tuning...
  
   temperament.

• The coincidence leads to the very tiny interval of (about a millicent
  Cent (music)
  The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
  
   wide), which is the first 7-limit
  Limit (music)
  In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...
  
   interval tempered out in 103169-TET.

Concerning powers of pi

• correct to about 1.3%. This can be understood in terms of the formula for the zeta function  This coincidence was used in the design of slide rule
  Slide rule
  The slide rule, also known colloquially as a slipstick, is a mechanical analog computer. The slide rule is used primarily for multiplication and division, and also for functions such as roots, logarithms and trigonometry, but is not normally used for addition or subtraction.Slide rules come in a...
  
  s, where the "folded" scales are folded on rather than because it is a more useful number and has the effect of folding the scales in about the same place.
• correct to 0.0004%.
• or accurate to 8 decimal places (due to Ramanujan: Quarterly Journal of Mathematics, XLV, 1914, pp350–372). Ramanujan states that this "curious approximation" to was "obtained empirically" and has no connection with the theory developed in the remainder of the paper.

Some plausible relations hold to a high degree of accuracy, but are nevertheless coincidental. One example is:



The two sides of this expression only differ after the 42nd decimal place.

Containing both pi and e

• , within 0.000 005%

• is very close to 20 (Conway, Sloane, Plouffe, 1988); this is equivalent to

• 

Containing pi or e and numbers 163 and 22

• , within 0.0005%

• , within 0.0000003%

• , within 0.0000008%

• Ramanujan's constant: , within . This fact is not a mathematical coincidence; it is a deep consequence of the fact that 163 is a Heegner number
  Heegner number
  In number theory, a Heegner number is a square-free positive integer d such that the imaginary quadratic field Q has class number 1...
  
  .

Other numerical curiosities

• .
• , where is the golden ratio
  Golden ratio
  In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
  
   (an amusing equality with an angle expressed in degrees) (see Number of the Beast
  Number of the Beast
  The Number of the Beast is a term in the Book of Revelation, of the New Testament, that is associated with the first Beast of Revelation chapter 13, the Beast of the sea. In most manuscripts of the New Testament and in English translations of the Bible, the number of the Beast is...
  
  )
• , where is Euler's totient function
  Euler's totient function
  In number theory, the totient \varphi of a positive integer n is defined to be the number of positive integers less than or equal to n that are coprime to n In number theory, the totient \varphi(n) of a positive integer n is defined to be the number of positive integers less than or equal to n that...
  
  
• and are the only non-trivial (i.e. at least square) consecutive powers of positive integers (Catalan's conjecture).
• = 0.57735, very close to the Euler-Mascheroni constant
  Euler-Mascheroni constant
  The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....
  
  .
• is the only positive integer solution of (see Lambert's W function for a formal solution method)
• 31, 331, 3331 etc. up to 33333331 are all prime number
  Prime number
  A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
  
  s, but then 333333331 is not. See also Formula for primes
  Formula for primes
  In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. No such formula which is easily computable is presently known...
  
  .
• The Fibonacci number
  Fibonacci number
  In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....
  
   F₂₉₆₁₈₂ is (probably) a semiprime
  Semiprime
  In mathematics, a semiprime is a natural number that is the product of two prime numbers. The first few semiprimes are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ... ....
  
  , since F₂₉₆₁₈₂ = F₁₄₈₀₉₁ × L₁₄₈₀₉₁ where F₁₄₈₀₉₁ (30949 digits) and the Lucas number
  Lucas number
  The Lucas numbers are an integer sequence named after the mathematician François Édouard Anatole Lucas , who studied both that sequence and the closely related Fibonacci numbers...
  
   L₁₄₈₀₉₁ (30950 digits) are simultaneously probable prime
  Probable prime
  In number theory, a probable prime is an integer that satisfies a specific condition also satisfied by all prime numbers. Different types of probable primes have different specific conditions...
  
  s.
• The solution to F_x=x+1, when multiplied by 200 is, to 4 decimal places is 1119.0000, almost exactly 1119.
• In a discussion of the birthday problem, the number occurs, which is "amusingly" equal to to 4 digits.

Decimal coincidences

• . This makes 2592 a nice Friedman number
  Friedman number
  A Friedman number is an integer which, in a given base, is the result of an expression using all its own digits in combination with any of the four basic arithmetic operators and sometimes exponentiation. For example, 347 is a Friedman number since 347 = 73 + 4...
  
  .
• . The only such factorion
  Factorion
  A factorion is a natural number that equals the sum of the factorials of its decimal digits. For example, 145 is a factorion because 1! + 4! + 5! = 1 + 24 + 120 = 145.There are just four factorions and they are 1, 2, 145 and 40585 .-Upper bound:...
  
  s (in base 10) are 1, 2, 145, 40585.
• ,    ,    ,     (anomalous cancellation
  Anomalous cancellation
  An anomalous cancellation or accidental cancellation is a particular kind of arithmetic procedural error that gives a numerically correct answer. An attempt is made to reduce a fraction by canceling individual digits in the numerator and denominator...
  
  )
• and .
• . This can also be written , making 127 the smallest nice Friedman number.
• ; ; ; — all narcissistic numbers
• 
• and also when rounded to 8 digits is 0.05882353. Mentioned by Gilbert Labelle in ~1980. 5882353 also happens to be prime.
• . The largest such number is 12157692622039623539.
• 73 is the 21st prime and 37 is the 12th prime. Both are reverse numbers. It is the only known combination.

Speed of light

The speed of light

Speed of light

The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

 is exactly 299,792,458 m/s, very close to 300,000 km/s. This is a pure coincidence.

Earth's diameter

The diameter of the Earth is almost exactly half a billion inches. The polar diameter of Earth is 500531678 inches and the equatorial diameter is 502215511 inches.

Gravitational acceleration

While not constant but varying depending on latitude

Latitude

In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

, the acceleration caused by Earth's gravity

Gravitational acceleration

In physics, gravitational acceleration is the acceleration on an object caused by gravity. Neglecting friction such as air resistance, all small bodies accelerate in a gravitational field at the same rate relative to the center of mass....

 on the surface lies between 9.78 and 9.82 m/s², which is quite close to 10. This means that as a result of Newton's second law

Newton's laws of motion

Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...

, the weight of a kilogram of mass on Earth's surface corresponds roughly to 10 Newtons of force exerted on an object.

Rydberg constant

The Rydberg constant

Rydberg constant

The Rydberg constant, symbol R∞, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to atomic spectra in the science of spectroscopy. Rydberg initially determined its value empirically from spectroscopy, but Niels Bohr later showed that its value could be calculated...

, when multiplied by the speed of light and expressed as a frequency, is close to :

See also

• For a list of coincidences in physics
  Physics
  Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
  
  , see anthropic principle
  Anthropic principle
  In astrophysics and cosmology, the anthropic principle is the philosophical argument that observations of the physical Universe must be compatible with the conscious life that observes it. Some proponents of the argument reason that it explains why the Universe has the age and the fundamental...
  
  
• Almost integer
• Birthday problem
• Exceptional isomorphism
  Exceptional isomorphism
  In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ai and bj of two families of mathematical objects, that is not an example of a pattern of such isomorphisms.Because these series of objects are presented differently, they are not...
  
  
• Narcissistic number
  Narcissistic number
  In recreational number theory, a narcissistic number is a number that is the sum of its own digits each raised to the power of the number of digits. This definition depends on the base b of the number system used, e.g...
  
  
• Experimental mathematics
  Experimental mathematics
  Experimental mathematics is an approach to mathematics in which numerical computation is used to investigate mathematical objects and identify properties and patterns...
  
  
• Kepler triangle#A mathematical coincidence

External links

В. Левшин. - Магистр рассеянных наук. - Москва, Детская Литература 1970, 256 с.

• Hardy, G. H. - A Mathematician's Apology
  A Mathematician's Apology
  A Mathematician's Apology is a 1940 essay by British mathematician G. H. Hardy. It concerns the aesthetics of mathematics with some personal content, and gives the layman an insight into the mind of a working mathematician.-Summary:...
  
  . - New York: Cambridge University Press, 1993, (ISBN 0-521-42706-1)
• Sequence in the On-Line Encyclopedia of Integer Sequences
  On-Line Encyclopedia of Integer Sequences
  The On-Line Encyclopedia of Integer Sequences , also cited simply as Sloane's, is an online database of integer sequences, created and maintained by N. J. A. Sloane, a researcher at AT&T Labs...
  
  
• Various mathematical coincidences in the "Science & Math" section of futilitycloset.com
• Press, W. H., Seemingly Remarkable Mathematical Coincidences Are Easy to Generate