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Farey Fraction
In mathematics, the Farey sequence of order ''n'' is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which when in lowest terms have denominators less than or equal to ''n'', arranged in order of increasing size. With the restricted definition, each Farey sequence starts with the value 0, denoted by the fraction , and ends with the value 1, denoted by the fraction (although some authors omit these terms). A ''Farey sequence'' is sometimes called a Farey ''series'', which is not strictly correct, because the terms are not summed. Examples The Farey sequences of orders 1 to 8 are : :''F''1 = :''F''2 = :''F''3 = :''F''4 = :''F''5 = :''F''6 = :''F''7 = :''F''8 = Plotting the numerators versus the denominators of a Farey sequence gives a shape like the one to the right, shown for 6. Reflecting this shape around the diagonal and main axes generates the ''Farey sunburst'', shown below. The Farey sunburst of orde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Farey Diagram Horizontal Arc 9 , a mathematical construct named after John Farey Sr.
{{Surname, Farey ...
Farey is the surname of: * Cyril Farey (1888–1954), British architect and architectural illustrator * John Farey Sr. (1766-1826), English geologist * John Farey Jr. (1791-1851), English mechanical engineer, son of John Farey Sr. * Joseph Farey (1796–1829), English mechanical engineer and draughtsman, son of John Farey Sr. * Lizzie Farey (born 1962), Scottish artist See also *Farey sequence In mathematics, the Farey sequence of order ''n'' is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which when in lowest terms have denominators less than or equal to ''n'', arranged in ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Farey, Sr
John is a common English name and surname: * John (given name) * John (surname) John may also refer to: New Testament Works * Gospel of John, a title often shortened to John * First Epistle of John, often shortened to 1 John * Second Epistle of John, often shortened to 2 John * Third Epistle of John, often shortened to 3 John People * John the Baptist (died c. AD 30), regarded as a prophet and the forerunner of Jesus Christ * John the Apostle (lived c. AD 30), one of the twelve apostles of Jesus * John the Evangelist, assigned author of the Fourth Gospel, once identified with the Apostle * John of Patmos, also known as John the Divine or John the Revelator, the author of the Book of Revelation, once identified with the Apostle * John the Presbyter, a figure either identified with or distinguished from the Apostle, the Evangelist and John of Patmos Other people with the given name Religious figures * John, father of Andrew the Apostle and Saint Peter * Pope Joh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by , after whom it is named. The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of twenty-three unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems, which offers a million dollars to anyone who solves any of them. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann zeta function ζ(''s'') is a function whose argument ''s'' may be any complex numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floor And Ceiling Functions
In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or equal to , denoted or . For example, , , , and . Historically, the floor of has been–and still is–called the integral part or integer part of , often denoted (as well as a variety of other notations). Some authors may define the integral part as if is nonnegative, and otherwise: for example, and . The operation of truncation generalizes this to a specified number of digits: truncation to zero significant digits is the same as the integer part. For an integer, . Notation The ''integral part'' or ''integer part'' of a number ( in the original) was first defined in 1798 by Adrien-Marie Legendre in his proof of the Legendre's formula. Carl Friedrich Gauss introduced the square bracket notation in hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Function
The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula. Following work of Gian-Carlo Rota in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted . Definition For any positive integer , define as the sum of the primitive th roots of unity. It has values in depending on the factorization of into prime factors: * if is a square-free positive integer with an even number of prime factors. * if is a square-free positive integer with an odd number of prime factors. * if has a squared prime factor. The Möbius function can alternatively be represented as : \mu(n) = \delta_ \lambda(n), where is the Kronecker delta, is the Liouville function, is the number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Inversion Formula
In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius. A large generalization of this formula applies to summation over an arbitrary locally finite partially ordered set, with Möbius' classical formula applying to the set of the natural numbers ordered by divisibility: see incidence algebra. Statement of the formula The classic version states that if and are arithmetic functions satisfying : g(n)=\sum_f(d)\quad\textn\ge 1 then :f(n)=\sum_\mu(d)g\left(\frac\right)\quad\textn\ge 1 where is the Möbius function and the sums extend over all positive divisors of (indicated by d \mid n in the above formulae). In effect, the original can be determined given by using the inversion formula. The two sequences are said to be Möbius transforms of each other. The formula is also correct if and are fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Totient Summatory Function
In number theory, the totient summatory function \Phi(n) is a summatory function of Euler's totient function defined by: :\Phi(n) := \sum_^n \varphi(k), \quad n\in \mathbf It is the number of coprime integer pairs . Properties Using Möbius inversion to the totient function, we obtain :\Phi(n) = \sum_^n k\sum _ \frac = \frac \sum _^n \mu(k) \left\lfloor \frac \right\rfloor \left(1 + \left\lfloor \frac \right\rfloor \right) has the asymptotic expansion :\Phi(n) \sim \fracn^+O\left( n\log n \right ), where is the Riemann zeta function for the value 2. is the number of coprime integer pairs . The summatory of reciprocal totient function The summatory of reciprocal totient function is defined as :S(n) := \sum _^ Edmund Landau showed in 1900 that this function has the asymptotic behavior :S(n) \sim A (\gamma+\log n)+ B +O\left(\frac n\right) where is the Euler–Mascheroni constant, :A = \sum_^\infty \frac = \frac = \prod_p \left(1+\frac 1 \right) and :B = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Totient Function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In other words, it is the number of integers in the range for which the greatest common divisor is equal to 1. The integers of this form are sometimes referred to as totatives of . For example, the totatives of are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since and . Therefore, . As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring \Z/n\Z). It is also used for defining the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also '' is prime to '' or '' is coprime with ''. The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing Standard notations for relatively prime integers and are: and . In their 1989 textbook '' Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed that the notation a\perp b be used to indicate that and are relatively prime and that the term "prime" be used instead of coprime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Farey Diagram Circle Packing 5
Farey is the surname of: * Cyril Farey (1888–1954), British architect and architectural illustrator * John Farey Sr. (1766-1826), English geologist * John Farey Jr. (1791-1851), English mechanical engineer, son of John Farey Sr. * Joseph Farey (1796–1829), English mechanical engineer and draughtsman, son of John Farey Sr. * Lizzie Farey Lizzie Farey (born 1962) is a contemporary willow sculptor and artist based in the Stewartry in Galloway, Scotland. She designs and creates wall sculptures and 3D forms using her own grown and harvested willow, as well as other local woods such ... (born 1962), Scottish artist See also * Farey sequence, a mathematical construct named after John Farey Sr. {{Surname, Farey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stigler's Law Of Eponymy
Stigler's law of eponymy, proposed by University of Chicago statistics professor Stephen Stigler in his 1980 publication ''Stigler’s law of eponymy'', states that no scientific discovery is named after its original discoverer. Examples include Hubble's law, which was derived by Georges Lemaître two years before Edwin Hubble, the Pythagorean theorem, which was known to Babylonian mathematicians before Pythagoras, and Halley's Comet, which was observed by astronomers since at least 240 BC (although its official designation is due to the first ever mathematical prediction of such astronomical phenomenon in the sky, not to its discovery). Stigler himself named the sociologist Robert K. Merton as the discoverer of "Stigler's law" to show that it follows its own decree, though the phenomenon had previously been noted by others. Derivation Historical acclaim for discoveries is often assigned to persons of note who bring attention to an idea that is not yet widely known, whether o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Haros
Charles Haros was a geometer (mathematician) in the French Bureau du Cadastre at the end of the eighteenth century and the beginning of the nineteenth century. Haros' conversion table One of the primary tasks of the Bureau du Cadastre was the accurate mapping of France for the purpose of taxation but from time to time the bureau also provided computational services to other parts of the government. One of the changes instituted by the French revolution was to convert France to the metric system and this necessitated changing from a fractional to a decimal representation of rational numbers. While Haros was involved many computation projects at the Bureau du Cadastre including the computation of de Prony’s tables of logarithms and the preparation of the French ephemeris, Connaissance des Temps, he is best known for a small table he prepared to convert fractions to their decimal equivalents. Haros’ conversion table appeared in a tract, ''Instruction Abrégée sur les nouve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
