Number theory is a branch of pure mathematics devoted to the study of the integers. German mathematician Carl Friedrich Gauss said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of objects made out of integers or defined as generalizations of the integers. Integers can be considered either as solutions to equations. Questions in number theory are best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may study real numbers in relation to rational numbers, for example, as approximated by the latter; the older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory"; the use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is preferred as an adjective to number-theoretic.
The first historical find of an arithmetical nature is a fragment of a table: the broken clay tablet Plimpton 322 contains a list of "Pythagorean triples", that is, integers such that a 2 + b 2 = c 2. The triples are too large to have been obtained by brute force; the heading over the first column reads: "The takiltum of the diagonal, subtracted such that the width..." The table's layout suggests that it was constructed by means of what amounts, in modern language, to the identity 2 + 1 = 2, implicit in routine Old Babylonian exercises. If some other method was used, the triples were first constructed and reordered by c / a for actual use as a "table", for example, with a view to applications, it is not known whether there could have been any. It has been suggested instead. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, Babylonian algebra was exceptionally well developed. Late Neoplatonic sources state.
Much earlier sources state that Pythagoras traveled and studied in Egypt. Euclid IX 21–34 is probably Pythagorean. Pythagorean mystics gave great importance to the even; the discovery that 2 is irrational is credited to the early Pythagoreans. By revealing that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history; this forced a distinction between numbers, on the one hand, lengths and proportions, on the other hand. The Pythagorean tradition spoke of so-called polygonal or figurate numbers. While square numbers, cubic numbers, etc. are seen now as more natural than triangular numbers, pentagonal numbers, etc. the study of the sums of triangular and pentagonal numbers would prove fruitful in the early modern period. We know of no arithmetical material in ancient Egyptian or Vedic sources, though there is some algebra in both; the Chinese remainder theorem appears as an exercise in Sunzi Suanjing There is some numerical mysticism in Chinese mathematics, unlike that of the Pythagoreans, it seems to have led nowhere.
Like the Pythagoreans' perfect numbers, magic squares have passed from superstition into recreation. Aside from a few fragments, the mathematics of Classical Greece is known to us either through the reports of contemporary non-m
Twelfth root of two
The twelfth root of two or 2 12 is an algebraic irrational number. It is most important in Western music theory, where it represents the frequency ratio of a semitone in twelve-tone equal temperament; this number was proposed for the first time in relationship to musical tuning in the sixteenth and seventeenth centuries. It allows measurement and comparison of different intervals as consisting of different numbers of a single interval, the equal tempered semitone. A semitone itself is divided into 100 cents; the twelfth root of two to 20 significant figures is 1.0594630943592952646. Fraction approximations in order of accuracy are 18/17, 196/185, 18904/17843; as of December 2013, its numerical value has been computed to at least twenty billion decimal digits. Since a musical interval is a ratio of frequencies, the equal-tempered chromatic scale divides the octave into twelve equal parts. Applying this value successively to the tones of a chromatic scale, starting from A above middle C with a frequency of 440 Hz, produces the following sequence of pitches: The final A is twice the frequency of the lower A, that is, one octave higher.
The just or Pythagorean perfect fifth is 3/2, the difference between the equal tempered perfect fifth and the just is a grad, the twelfth root of the Pythagorean comma. The equal tempered Bohlen–Pierce scale uses the interval of the thirteenth root of three. Stockhausen's Studie II makes use of the twenty-fifth root of five, a compound major third divided into 5x5 parts; the delta scale is based on ≈50√3/2, the gamma scale is based on ≈20√3/2, the beta scale is based on ≈11√3/2, the alpha scale is based on ≈9√3/2. Since the frequency ratio of a semitone is close to 106%, increasing or decreasing the playback speed of a recording by 6% will shift the pitch up or down by about one semitone, or "half-step". Upscale reel-to-reel magnetic tape recorders have pitch adjustments of up to ±6% used to match the playback or recording pitch to other music sources having different tunings. Modern recording studios utilize digital pitch shifting to achieve similar results, ranging from cents up to several half-steps.
DJ turntables can have an adjustment up to ±20%, but this is more used for beat synchronization between songs than for pitch adjustment, useful only in transitions between beatless and ambient parts. For beatmatching music of high melodic content the DJ would try to look for songs that sound harmonic together when set to equal tempo; this number was proposed for the first time in relationship to musical tuning in 1580 by Simon Stevin. In 1581 Italian musician Vincenzo Galilei may be the first European to suggest twelve-tone equal temperament; the twelfth root of two was first calculated in 1584 by the Chinese mathematician and musician Zhu Zaiyu using an abacus to reach twenty four decimal places calculated circa 1605 by Flemish mathematician Simon Stevin, in 1636 by the French mathematician Marin Mersenne and in 1691 by German musician Andreas Werckmeister. Just intonation § Practical difficulties Music and mathematics Piano key frequencies Scientific pitch notation Twelve-tone technique The Well-Tempered Clavier Barbour, J. M..
"A Sixteenth Century Chinese Approximation for π". American Mathematical Monthly. 40: 69–73. Doi:10.2307/2300937. JSTOR 2300937. Ellis, Alexander. On the Sensations of Tone. Dover Publications. ISBN 0-486-60753-4. Partch, Harry. Genesis of a Music. Da Capo Press. ISBN 0-306-80106-X
Square root of 2
The square root of 2, or the th power of 2, written in mathematics as √2 or 21⁄2, is the positive algebraic number that, when multiplied by itself, gives the number 2. Technically, it is called the principal square root of 2, to distinguish it from the negative number with the same property. Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length, it was the first number known to be irrational. As a good rational approximation for the square root of two, with a reasonable small denominator, the fraction 99/70 is sometimes used; the sequence A002193 in the OEIS gives the numerical value for the square root of two, truncated to 65 decimal places: 1.41421356237309504880168872420969807856967187537694807317667973799... The Babylonian clay tablet YBC 7289 gives an approximation of √2 in four sexagesimal figures, 1 24 51 10, accurate to about six decimal digits, is the closest possible three-place sexagesimal representation of √2: 1 + 24 60 + 51 60 2 + 10 60 3 = 305470 216000 = 1.41421 296 ¯.
Another early close approximation is given in ancient Indian mathematical texts, the Sulbasutras as follows: Increase the length by its third and this third by its own fourth less the thirty-fourth part of that fourth. That is, 1 + 1 3 + 1 3 × 4 − 1 3 × 4 × 34 = 577 408 = 1.41421 56862745098039 ¯. This approximation is the seventh in a sequence of accurate approximations based on the sequence of Pell numbers, which can be derived from the continued fraction expansion of √2. Despite having a smaller denominator, it is only less accurate than the Babylonian approximation. Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, that the square root of two is irrational. Little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is mentioned. For a while, the Pythagoreans treated as an official secret the discovery that the square root of two is irrational, according to legend, Hippasus was murdered for divulging it.
The square root of two is called Pythagoras' number or Pythagoras' constant, for example by Conway & Guy. In ancient Roman architecture, Vitruvius describes the use of the square root of 2 progression or ad quadratum technique, it consists in a geometric, rather than arithmetic, method to double a square, in which the diagonal of the original square is equal to the side of the resulting square. Vitruvius attributes the idea to Plato; the system was employed to build pavements by creating a square tangent to the corners of the original square at 45 degrees of it. The proportion was used to design atria by giving them a length equal to a diagonal taken from a square which sides are equivalent to the intended atrium's width. There are a number of algorithms for approximating √2, which in expressions as a ratio of integers or as a decimal can only be approximated; the most common algorithm for this, one used as a basis in many computers and calculators, is the Babylonian method of computing square roots, one of many methods of computing square roots.
It goes as follows: First, pick a guess, a0 > 0. Using that guess, iterate through the following recursive computation: a n + 1 = a n + 2 a n 2 = a n 2 + 1 a n; the more iterations through the algorithm, the better approximation of the square root of 2 is achieved. Each iteration doubles the number of correct digits. Starting with a0 = 1 the next approximations are 3/2 = 1.5 17/12 = 1.416... 577/408 = 1.414215... 665857/470832 = 1.4142135623746... The value of √2 was calculated to 137,438,953,444 decimal places by Yasumasa Kanada's team in 1997. In February 2006 the record for the calculation of √2 was eclipsed with the use of a home computer. Shigeru Kondo calculated 1 trillion decimal places in 2010. For a development of this record, see the table below. Among mathematical constants with computationally challenging decimal expansions, only π has been calculated more precisely; such computations aim to check empirically. A simple rational approximation 99/70 is sometimes used. Despite having a denominator of only 70, it differs from the correct value by less than 1/10,000.
Since it is a convergent of the continued fraction representation of the square root of two
ArXiv is a repository of electronic preprints approved for posting after moderation, but not full peer review. It consists of scientific papers in the fields of mathematics, astronomy, electrical engineering, computer science, quantitative biology, mathematical finance and economics, which can be accessed online. In many fields of mathematics and physics all scientific papers are self-archived on the arXiv repository. Begun on August 14, 1991, arXiv.org passed the half-million-article milestone on October 3, 2008, had hit a million by the end of 2014. By October 2016 the submission rate had grown to more than 10,000 per month. ArXiv was made possible by the compact TeX file format, which allowed scientific papers to be transmitted over the Internet and rendered client-side. Around 1990, Joanne Cohn began emailing physics preprints to colleagues as TeX files, but the number of papers being sent soon filled mailboxes to capacity. Paul Ginsparg recognized the need for central storage, in August 1991 he created a central repository mailbox stored at the Los Alamos National Laboratory which could be accessed from any computer.
Additional modes of access were soon added: FTP in 1991, Gopher in 1992, the World Wide Web in 1993. The term e-print was adopted to describe the articles, it began as a physics archive, called the LANL preprint archive, but soon expanded to include astronomy, computer science, quantitative biology and, most statistics. Its original domain name was xxx.lanl.gov. Due to LANL's lack of interest in the expanding technology, in 2001 Ginsparg changed institutions to Cornell University and changed the name of the repository to arXiv.org. It is now hosted principally with eight mirrors around the world, its existence was one of the precipitating factors that led to the current movement in scientific publishing known as open access. Mathematicians and scientists upload their papers to arXiv.org for worldwide access and sometimes for reviews before they are published in peer-reviewed journals. Ginsparg was awarded a MacArthur Fellowship in 2002 for his establishment of arXiv; the annual budget for arXiv is $826,000 for 2013 to 2017, funded jointly by Cornell University Library, the Simons Foundation and annual fee income from member institutions.
This model arose in 2010, when Cornell sought to broaden the financial funding of the project by asking institutions to make annual voluntary contributions based on the amount of download usage by each institution. Each member institution pledges a five-year funding commitment to support arXiv. Based on institutional usage ranking, the annual fees are set in four tiers from $1,000 to $4,400. Cornell's goal is to raise at least $504,000 per year through membership fees generated by 220 institutions. In September 2011, Cornell University Library took overall administrative and financial responsibility for arXiv's operation and development. Ginsparg was quoted in the Chronicle of Higher Education as saying it "was supposed to be a three-hour tour, not a life sentence". However, Ginsparg remains on the arXiv Scientific Advisory Board and on the arXiv Physics Advisory Committee. Although arXiv is not peer reviewed, a collection of moderators for each area review the submissions; the lists of moderators for many sections of arXiv are publicly available, but moderators for most of the physics sections remain unlisted.
Additionally, an "endorsement" system was introduced in 2004 as part of an effort to ensure content is relevant and of interest to current research in the specified disciplines. Under the system, for categories that use it, an author must be endorsed by an established arXiv author before being allowed to submit papers to those categories. Endorsers are not asked to review the paper for errors, but to check whether the paper is appropriate for the intended subject area. New authors from recognized academic institutions receive automatic endorsement, which in practice means that they do not need to deal with the endorsement system at all. However, the endorsement system has attracted criticism for restricting scientific inquiry. A majority of the e-prints are submitted to journals for publication, but some work, including some influential papers, remain purely as e-prints and are never published in a peer-reviewed journal. A well-known example of the latter is an outline of a proof of Thurston's geometrization conjecture, including the Poincaré conjecture as a particular case, uploaded by Grigori Perelman in November 2002.
Perelman appears content to forgo the traditional peer-reviewed journal process, stating: "If anybody is interested in my way of solving the problem, it's all there – let them go and read about it". Despite this non-traditional method of publication, other mathematicians recognized this work by offering the Fields Medal and Clay Mathematics Millennium Prizes to Perelman, both of which he refused. Papers can be submitted in any of several formats, including LaTeX, PDF printed from a word processor other than TeX or LaTeX; the submission is rejected by the arXiv software if generating the final PDF file fails, if any image file is too large, or if the total size of the submission is too large. ArXiv now allows one to store and modify an incomplete submission, only finalize the submission when ready; the time stamp on the article is set. The standard access route is through one of several mirrors. Sev
The number π is a mathematical constant. Defined as the ratio of a circle's circumference to its diameter, it now has various equivalent definitions and appears in many formulas in all areas of mathematics and physics, it is equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, though it is sometimes spelled out as "pi", it is called Archimedes' constant. Being an irrational number, π cannot be expressed as a common fraction. Still, fractions such as 22/7 and other rational numbers are used to approximate π; the digits appear to be randomly distributed. In particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date, no proof of this has been discovered. Π is a transcendental number. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. Ancient civilizations required accurate computed values to approximate π for practical reasons, including the Egyptians and Babylonians.
Around 250 BC the Greek mathematician Archimedes created an algorithm for calculating it. In the 5th century AD Chinese mathematics approximated π to seven digits, while Indian mathematics made a five-digit approximation, both using geometrical techniques; the first exact formula for π, based on infinite series, was not available until a millennium when in the 14th century the Madhava–Leibniz series was discovered in Indian mathematics. In the 20th and 21st centuries and computer scientists discovered new approaches that, when combined with increasing computational power, extended the decimal representation of π to many trillions of digits after the decimal point. All scientific applications require no more than a few hundred digits of π, many fewer, so the primary motivation for these computations is the quest to find more efficient algorithms for calculating lengthy numeric series, as well as the desire to break records; the extensive calculations involved have been used to test supercomputers and high-precision multiplication algorithms.
Because its most elementary definition relates to the circle, π is found in many formulae in trigonometry and geometry those concerning circles and spheres. In more modern mathematical analysis, the number is instead defined using the spectral properties of the real number system, as an eigenvalue or a period, without any reference to geometry, it appears therefore in areas of mathematics and the sciences having little to do with the geometry of circles, such as number theory and statistics, as well as in all areas of physics. The ubiquity of π makes it one of the most known mathematical constants both inside and outside the scientific community. Several books devoted to π have been published, record-setting calculations of the digits of π result in news headlines. Attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits; the symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the lowercase Greek letter π, sometimes spelled out as pi, derived from the first letter of the Greek word perimetros, meaning circumference.
In English, π is pronounced as "pie". In mathematical use, the lowercase letter π is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation; the choice of the symbol π is discussed in the section Adoption of the symbol π. Π is defined as the ratio of a circle's circumference C to its diameter d: π = C d The ratio C/d is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle it will have twice the circumference, preserving the ratio C/d; this definition of π implicitly makes use of flat geometry. Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be formally defined independently of geometry using limits, a concept in calculus. For example, one may directly compute the arc length of the top half of the unit circle, given in Cartesian coordinates by the equation x2 + y2 = 1, as the integral: π = ∫ − 1 1 d x 1 − x 2.
An integral such as this was adopted as the definition of π by Karl Weierstrass, who defined it directly as an integral in 1841. Definitions of π such as these that rely on a notion of circumference, hence implicitly on concepts of the integral calculus, are no longer common in the literature. Remmert explains that this is because in many modern treatments of calculus, differential calculus precedes integral calculus in the university curriculum, so it is desirable to have a definition of π that does not rely on the latter. One such definition, due to Richard Baltzer, popularized by Edmund Landau, is the following: π is twice the smallest positive number at which the cosine function equals 0; the cosine can be defined independently of geometry as a power series, or as the solution of a differen
Cambridge University Press
Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the world's oldest publishing house and the second-largest university press in the world, it holds letters patent as the Queen's Printer. The press mission is "to further the University's mission by disseminating knowledge in the pursuit of education and research at the highest international levels of excellence". Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. With a global sales presence, publishing hubs, offices in more than 40 countries, it publishes over 50,000 titles by authors from over 100 countries, its publishing includes academic journals, reference works and English language teaching and learning publications. Cambridge University Press is a charitable enterprise that transfers part of its annual surplus back to the university. Cambridge University Press is both the oldest publishing house in the world and the oldest university press.
It originated from letters patent granted to the University of Cambridge by Henry VIII in 1534, has been producing books continuously since the first University Press book was printed. Cambridge is one of the two privileged presses. Authors published by Cambridge have included John Milton, William Harvey, Isaac Newton, Bertrand Russell, Stephen Hawking. University printing began in Cambridge when the first practising University Printer, Thomas Thomas, set up a printing house on the site of what became the Senate House lawn – a few yards from where the press's bookshop now stands. In those days, the Stationers' Company in London jealously guarded its monopoly of printing, which explains the delay between the date of the university's letters patent and the printing of the first book. In 1591, Thomas's successor, John Legate, printed the first Cambridge Bible, an octavo edition of the popular Geneva Bible; the London Stationers objected strenuously. The university's response was to point out the provision in its charter to print "all manner of books".
Thus began the press's tradition of publishing the Bible, a tradition that has endured for over four centuries, beginning with the Geneva Bible, continuing with the Authorized Version, the Revised Version, the New English Bible and the Revised English Bible. The restrictions and compromises forced upon Cambridge by the dispute with the London Stationers did not come to an end until the scholar Richard Bentley was given the power to set up a'new-style press' in 1696. In July 1697 the Duke of Somerset made a loan of £200 to the university "towards the printing house and presse" and James Halman, Registrary of the University, lent £100 for the same purpose, it was in Bentley's time, in 1698, that a body of senior scholars was appointed to be responsible to the university for the press's affairs. The Press Syndicate's publishing committee still meets and its role still includes the review and approval of the press's planned output. John Baskerville became University Printer in the mid-eighteenth century.
Baskerville's concern was the production of the finest possible books using his own type-design and printing techniques. Baskerville wrote, "The importance of the work demands all my attention. Caxton would have found nothing to surprise him if he had walked into the press's printing house in the eighteenth century: all the type was still being set by hand. A technological breakthrough was badly needed, it came when Lord Stanhope perfected the making of stereotype plates; this involved making a mould of the whole surface of a page of type and casting plates from that mould. The press was the first to use this technique, in 1805 produced the technically successful and much-reprinted Cambridge Stereotype Bible. By the 1850s the press was using steam-powered machine presses, employing two to three hundred people, occupying several buildings in the Silver Street and Mill Lane area, including the one that the press still occupies, the Pitt Building, built for the press and in honour of William Pitt the Younger.
Under the stewardship of C. J. Clay, University Printer from 1854 to 1882, the press increased the size and scale of its academic and educational publishing operation. An important factor in this increase was the inauguration of its list of schoolbooks. During Clay's administration, the press undertook a sizeable co-publishing venture with Oxford: the Revised Version of the Bible, begun in 1870 and completed in 1885, it was in this period as well that the Syndics of the press turned down what became the Oxford English Dictionary—a proposal for, brought to Cambridge by James Murray before he turned to Oxford. The appointment of R. T. Wright as Secretary of the Press Syndicate in 1892 marked the beginning of the press's development as a modern publishing business with a defined editorial policy and administrative structure, it was Wright who devised the plan for one of the most distinctive Cambridge contributions to publishing—the Cambridge Histories. The Cambridge Modern History was published
Joseph Liouville FRS FRSE FAS · was a French mathematician. He was born in Saint-Omer in France on 24 March 1809. Liouville graduated from the École Polytechnique in 1827. After some years as an assistant at various institutions including the École Centrale Paris, he was appointed as professor at the École Polytechnique in 1838, he obtained a chair in mathematics at the Collège de France in 1850 and a chair in mechanics at the Faculté des Sciences in 1857. Besides his academic achievements, he was talented in organisational matters. Liouville founded the Journal de Mathématiques Pures et Appliquées which retains its high reputation up to today, in order to promote other mathematicians' work, he was the first to read, to recognize the importance of, the unpublished work of Évariste Galois which appeared in his journal in 1846. Liouville was involved in politics for some time, he became a member of the Constituting Assembly in 1848. However, after his defeat in the legislative elections in 1849, he turned away from politics.
Liouville worked in a number of different fields in mathematics, including number theory, complex analysis, differential geometry and topology, but mathematical physics and astronomy. He is remembered for Liouville's theorem. In number theory, he was the first to prove the existence of transcendental numbers by a construction using continued fractions. In mathematical physics, Liouville made two fundamental contributions: the Sturm–Liouville theory, joint work with Charles François Sturm, is now a standard procedure to solve certain types of integral equations by developing into eigenfunctions, the fact that time evolution is measure preserving for a Hamiltonian system. In Hamiltonian dynamics, Liouville introduced the notion of action-angle variables as a description of integrable systems; the modern formulation of this is sometimes called the Liouville–Arnold theorem, the underlying concept of integrability is referred to as Liouville integrability. In 1851, he was elected a foreign member of the Royal Swedish Academy of Sciences.
The crater Liouville on the Moon is named after him. So is the Liouville function, an important function in number theory. List of things named after Joseph Liouville Liouville's theorem O'Connor, John J.. Lützen, Joseph Liouville 1809–1882: Master of Pure and Applied Mathematics, Studies in the History of Mathematics and Physical Sciences, 15, Springer-Verlag, ISBN 3-540-97180-7 Lutzen J. "Liouville's differential calculus of arbitrary order and its electrodynamical origin",in. 1985. Icelandic Mathematical Society, Reykjavik, pp. 149–160. Williams, Kenneth S. Number theory in the spirit of Liouville, London Mathematical Society Student Texts, 76, Cambridge: Cambridge University Press, ISBN 978-0-521-17562-3, Zbl 1227.11002 Works by Joseph Liouville at Project Gutenberg Works by or about Joseph Liouville at Internet Archive Joseph Liouville at the Mathematics Genealogy Project