1.
Neil Sloane
–
Neil James Alexander Sloane is a British-American mathematician. His major contributions are in the fields of combinatorics, error-correcting codes, Sloane is best known for being the creator and maintainer of the On-Line Encyclopedia of Integer Sequences. Sloane was born in Wales and brought up in Australia and he studied at Cornell University, New York state, under Nick DeClaris, Frank Rosenblatt, Frederick Jelinek and Wolfgang Heinrich Johannes Fuchs, receiving his Ph. D. in 1967. His doctoral dissertation was titled Lengths of Cycle Times in Random Neural Networks, Sloane joined AT&T Bell Labs in 1968 and retired from AT&T Labs in 2012. He became an AT&T Fellow in 1998 and he is also a Fellow of the Learned Society of Wales, an IEEE Fellow, a Fellow of the American Mathematical Society, and a member of the National Academy of Engineering. He is a winner of a Lester R. Ford Award in 1978, in 2005 Sloane received the IEEE Richard W. Hamming Medal. In 2008 he received the Mathematical Association of America David P. Robbins award, in 2014, to celebrate his 75th birthday, Neil Sloane shared some of his favorite integer sequences. Besides mathematics, he loves rock climbing and has authored two rock-climbing guides to New Jersey, N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, NY,1973. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North-Holland, M. Harwit and N. J. A. Sloane, Hadamard Transform Optics, Academic Press, San Diego CA,1979. N. J. A. Sloane and A. D. Wyner, editors, Claude Elwood Shannon, Collected Papers, IEEE Press, N. J. A. Sloane and S. Plouffe, The Encyclopedia of Integer Sequences, Academic Press, San Diego,1995. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, NY, 1st edn. A. S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays, Theory and Applications, Springer-Verlag, NY,1999. G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer-Verlag,2006
2.
Alexa Internet
–
Alexa Internet, Inc. is a California-based company that provides commercial web traffic data and analytics. It is an owned subsidiary of Amazon. com. Founded as an independent company in 1996, Alexa was acquired by Amazon in 1999 and its toolbar collects data on browsing behavior and transmits them to the Alexa website, where they are stored and analyzed. This is the basis for the web traffic reporting. According to its website, Alexa provides traffic data, global rankings, as of 2015, its website has been visited by over 6.5 million people monthly. Alexa Internet was founded in April 1996 by American web entrepreneurs Brewster Kahle, Alexa initially offered a toolbar that gave Internet users suggestions on where to go next, based on the traffic patterns of its user community. The company also offered context for each site visited, to whom it was registered, how many pages it had, how other sites pointed to it. Alexas operations grew to include archiving of web pages as they are crawled and this database served as the basis for the creation of the Internet Archive accessible through the Wayback Machine. In 1998, the company donated a copy of the archive, Alexa continues to supply the Internet Archive with Web crawls. In 1999, as the company moved away from its vision of providing an intelligent search engine. Alexa began a partnership with Google in early 2002, and with the web directory DMOZ in January 2003, in May 2006, replaced Google with Bing as a provider of search results. In December 2006, Amazon released Alexa Image Search, built in-house, it was the first major application built on the companys Web platform. In December 2005, Alexa opened its extensive search index and Web-crawling facilities to third-party programs through a set of Web services. These could be used, for instance, to construct vertical search engines that could run on Alexas own servers or elsewhere. In May 2007, Alexa changed their API to limit comparisons to three websites, reduce the size of embedded graphs in Flash, and add mandatory embedded BritePic advertisements. In April 2007, the company filed a lawsuit, Alexa v. Hornbaker, in the lawsuit, Alexa alleged that Ron Hornbaker was stealing traffic graphs for profit, and that the primary purpose of his site was to display graphs that were generated by Alexas servers. Hornbaker removed the term Alexa from his name on March 19,2007. Thereafter, Alexa became a purely analytics-focused company, on March 31,2009, Alexa launched a major website redesign
3.
AT&T Labs
–
AT&T Labs is the research & development division of AT&T. It employs some 1800 people in locations, including, Bedminster NJ, Middletown, NJ, Manhattan, NY, Warrenville, IL, Austin, TX, Atlanta, GA, San Francisco, CA, San Ramon, CA. AT&T Labs – Research, the 450-person research division of AT&T Labs, is based in the Bedminster, Middletown, San Francisco, AT&T Labs traces its history from AT&T Bell Labs. Much research is in areas associated with networks and systems, ranging from the physics of optical transmission to foundational topics in computing. Other research areas address the challenges of large operational networks. Since its creation in 1996, AT&T Labs has been issued over 2000 US patents, the Online Encyclopedia of Integer Sequences is the creation of AT&T Researcher Neil Sloane. Researchers at AT&T Labs have successfully transmitted 100 Gigabits per second over an optical link. In 2009, AT&T researchers led the team in the Netflix Prize competition. Lucent retained the name Bell Labs and AT&T adopted the name AT&T Laboratories for its R&D organization, AT&T Labs also traces its origin to Southwestern Bell Technology Resources, Inc. which was founded in 1988 as the R&D arm of Southwestern Bell Corporation. It had no connection to Bellcore, the R&D organization owned equally by all of the Baby Bells, in 1995, Southwestern Bell Corporation renamed itself SBC Communications, Inc. resulting in the subsequent name changes of companies such as SWB TRI to SBC Technology Resources, Inc. In 2003, SBC TRI changed its name to SBC Laboratories, SBC Laboratories focused on four core areas, Broadband Internet, Wireless Systems, Network Services, and Network IT. In 2005, SBC Communications and AT&T Corporation merged to form AT&T, AT&T Labs, Inc. became the new name of the combined SBC Laboratories, Inc. and AT&T Laboratories along with its research facilities in New Jersey. In 2006, BellSouth Telecommunications Science and Technology was also merged with AT&T Labs, BellSouth Science and Technology had offices in Birmingham, Alabama and Atlanta, Georgia
4.
Intellectual property
–
Intellectual property refers to creations of the intellect for which a monopoly is assigned to designated owners by law. Intellectual property rights are the protections granted to the creators of IP, and include trademarks, copyright, patents, industrial design rights, and in some jurisdictions trade secrets. Artistic works including music and literature, as well as discoveries, inventions, words, phrases, symbols, the Statute of Monopolies and the British Statute of Anne are seen as the origins of patent law and copyright respectively, firmly establishing the concept of intellectual property. The first known use of the intellectual property dates to 1769. The first clear example of modern usage goes back as early as 1808, the German equivalent was used with the founding of the North German Confederation whose constitution granted legislative power over the protection of intellectual property to the confederation. According to Lemley, it was only at point that the term really began to be used in the United States. The history of patents does not begin with inventions, but rather with royal grants by Queen Elizabeth I for monopoly privileges, the evolution of patents from royal prerogative to common-law doctrine. The term can be used in an October 1845 Massachusetts Circuit Court ruling in the patent case Davoll et al. v. Brown. The statement that discoveries are. property goes back earlier, in Europe, French author A. Nion mentioned propriété intellectuelle in his Droits civils des auteurs, artistes et inventeurs, published in 1846. Until recently, the purpose of property law was to give as little protection as possible in order to encourage innovation. Historically, therefore, they were granted only when they were necessary to encourage invention, limited in time, the concepts origins can potentially be traced back further. In 500 BCE, the government of the Greek state of Sybaris offered one years patent to all who should discover any new refinement in luxury. Intellectual property rights include patents, copyright, industrial design rights, trademarks, plant variety rights, trade dress, geographical indications, a copyright gives the creator of an original work exclusive rights to it, usually for a limited time. Copyright may apply to a range of creative, intellectual, or artistic forms. Copyright does not cover ideas and information themselves, only the form or manner in which they are expressed, an industrial design right protects the visual design of objects that are not purely utilitarian. An industrial design consists of the creation of a shape, configuration or composition of pattern or color, or combination of pattern, an industrial design can be a two- or three-dimensional pattern used to produce a product, industrial commodity or handicraft. Plant breeders rights or plant variety rights are the rights to use a new variety of a plant. The variety must amongst others be novel and distinct and for registration the evaluation of propagating material of the variety is examined, a trademark is a recognizable sign, design or expression which distinguishes products or services of a particular trader from the similar products or services of other traders
5.
Mathematician
–
A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems. Mathematics is concerned with numbers, data, quantity, structure, space, models, one of the earliest known mathematicians was Thales of Miletus, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, the number of known mathematicians grew when Pythagoras of Samos established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was All is number. It was the Pythagoreans who coined the term mathematics, and with whom the study of mathematics for its own sake begins, the first woman mathematician recorded by history was Hypatia of Alexandria. She succeeded her father as Librarian at the Great Library and wrote works on applied mathematics. Because of a dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked. Science and mathematics in the Islamic world during the Middle Ages followed various models and it was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences, an example of a translator and mathematician who benefited from this type of support was al-Khawarizmi. A notable feature of many working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics, maths and astronomy of Ibn al-Haytham, the Renaissance brought an increased emphasis on mathematics and science to Europe. As time passed, many gravitated towards universities. Moving into the 19th century, the objective of universities all across Europe evolved from teaching the “regurgitation of knowledge” to “encourag productive thinking. ”Thus, seminars, overall, science became the focus of universities in the 19th and 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. ”Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education, and then proceed to specialize in topics of their own choice at the graduate level. In some universities, a qualifying exam serves to test both the breadth and depth of an understanding of mathematics, the students, who pass, are permitted to work on a doctoral dissertation. Mathematicians involved with solving problems with applications in life are called applied mathematicians. Applied mathematicians are mathematical scientists who, with their knowledge and professional methodology. With professional focus on a variety of problems, theoretical systems
6.
Graph of a function
–
In mathematics, the graph of a function f is the collection of all ordered pairs. If the function x is a scalar, the graph is a two-dimensional graph. If the function x is an ordered pair of real numbers, the graph is the collection of all ordered triples. Graphing on a Cartesian plane is referred to as curve sketching. The graph of a function on real numbers may be mapped directly to the representation of the function. The concept of the graph of a function is generalized to the graph of a relation, note that although a function is always identified with its graph, they are not the same because it will happen that two functions with different codomain could have the same graph. For example, the cubic polynomial mentioned below is a surjection if its codomain is the real numbers, to test whether a graph of a curve is a function of x, one uses the vertical line test. To test whether a graph of a curve is a function of y, if the function has an inverse, the graph of the inverse can be found by reflecting the graph of the original function over the line y = x. In science, engineering, technology, finance, and other areas, in the simplest case one variable is plotted as a function of another, typically using rectangular axes, see Plot for details. In the modern foundation of mathematics known as set theory, a function, F = { a, if x =1, d, if x =2, c, if x =3, is. The graph of the polynomial on the real line f = x 3 −9 x is. If this set is plotted on a Cartesian plane, the result is a curve, the graph of the trigonometric function f = sin cos is. If this set is plotted on a three dimensional Cartesian coordinate system, the result is a surface, oftentimes it is helpful to show with the graph, the gradient of the function and several level curves. The level curves can be mapped on the surface or can be projected on the bottom plane. The second figure shows such a drawing of the graph of the function, f = −2 Given a function f of n variables, x 1, …, x n, the normal to the graph is. This is seen by considering the graph as a set of the function g = f − z. The graph of a function is contained in a Cartesian product of sets, fibre bundles arent cartesian products, but appear to be up close. There is a notion of a graph on a fibre bundle called a section
7.
Computer music
–
The field of computer music can trace its roots back to the origins of electronic music, and the very first experiments and innovations with electronic instruments at the turn of the 20th century. The worlds first computer to play music was CSIRAC, which was designed and built by Trevor Pearcey, mathematician Geoff Hill programmed the CSIRAC to play popular musical melodies from the very early 1950s. In 1951 it publicly played the Colonel Bogey March of which no known recordings exist, however, CSIRAC played standard repertoire and was not used to extend musical thinking or composition practice which is current computer-music practice. The oldest known recordings of computer generated music were played by the Ferranti Mark 1 computer, the music program was written by Christopher Strachey. During a session recorded by the BBC, the managed to work its way through Baa Baa Black Sheep, God Save the King. Two further major 1950s developments were the origins of digital sound synthesis by computer, max Mathews at Bell Laboratories developed the influential MUSIC I program and its descendents, further popularising computer music through a 1963 article in Science. In Japan, experiments in computer music date back to 1962 and this resulted in a piece entitled TOSBAC Suite, influenced by the Illiac Suite. Later Japanese computer music compositions include a piece by Kenjiro Ezaki presented during Osaka Expo 70, Ezaki also published an article called Contemporary Music and Computers in 1970. Early computer-music programs typically did not run in real time, programs would run for hours or days, on multimillion-dollar computers, to generate a few minutes of music. One way around this was to use a system, most notably the Roland MC-8 Microcomposer. In addition to the Yamaha DX7, the advent of digital chips. By the early 1990s, the performance of microprocessor-based computers reached the point that real-time generation of music using more general programs and algorithms became possible. Interesting sounds must have a fluidity and changeability that allows them to remain fresh to the ear, advances in computing power and software for manipulation of digital media have dramatically affected the way computer music is generated and performed. Current-generation micro-computers are powerful enough to perform very sophisticated audio synthesis using a variety of algorithms. Computer-generated music is composed by, or with the extensive aid of. There is a genre of music that is organized, synthesized. Later, composers such as Gottfried Michael Koenig had computers generate the sounds of the composition as well as the score, Koenig produced algorithmic composition programs which were a generalisation of his own serial composition practice. This is not exactly similar to Xenakis work as he used mathematical abstractions, koenigs software translated the calculation of mathematical equations into codes which represented musical notation
8.
Search engine (computing)
–
A search engine is an information retrieval system designed to help find information stored on a computer system. The search results are presented in a list and are commonly called hits. Search engines help to minimize the required to find information. The most public, visible form of an engine is a Web search engine which searches for information on the World Wide Web. Search engines provide an interface to a group of items that enables users to specify criteria about an item of interest and have the engine find the matching items, the criteria are referred to as a search query. In the case of search engines, the search query is typically expressed as a set of words that identify the desired concept that one or more documents may contain. There are several styles of search query syntax that vary in strictness and it can also switch names within the search engines from previous sites. Some search engines apply improvements to search queries to increase the likelihood of providing a quality set of items through a known as query expansion. Query understanding methods can be used to standardize query language, the list of items that meet the criteria specified by the query is typically sorted, or ranked. Ranking items by relevance reduces the required to find the desired information. Probabilistic search engines rank items based on measures of similarity and sometimes popularity or authority or use relevance feedback, other types of search engines do not store an index. Crawler, or spider type search engines may collect and assess items at the time of the search query, dynamically considering additional items based on the contents of a starting item. Meta search engines store neither an index nor a cache and instead simply reuse the index or results of one or more search engines to provide an aggregated
9.
Combinatorics
–
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general methods were developed. One of the oldest and most accessible parts of combinatorics is graph theory, Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms. A mathematician who studies combinatorics is called a combinatorialist or a combinatorist, basic combinatorial concepts and enumerative results appeared throughout the ancient world. Greek historian Plutarch discusses an argument between Chrysippus and Hipparchus of a rather delicate enumerative problem, which was shown to be related to Schröder–Hipparchus numbers. In the Ostomachion, Archimedes considers a tiling puzzle, in the Middle Ages, combinatorics continued to be studied, largely outside of the European civilization. The Indian mathematician Mahāvīra provided formulae for the number of permutations and combinations, later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. During the Renaissance, together with the rest of mathematics and the sciences, works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging field. In modern times, the works of J. J. Sylvester and Percy MacMahon helped lay the foundation for enumerative, graph theory also enjoyed an explosion of interest at the same time, especially in connection with the four color problem. In the second half of the 20th century, combinatorics enjoyed a rapid growth, in part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc. These connections shed the boundaries between combinatorics and parts of mathematics and theoretical science, but at the same time led to a partial fragmentation of the field. Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of combinatorial objects. Although counting the number of elements in a set is a rather broad mathematical problem, fibonacci numbers is the basic example of a problem in enumerative combinatorics. The twelvefold way provides a framework for counting permutations, combinations and partitions. Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis, in contrast with enumerative combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Partition theory studies various enumeration and asymptotic problems related to integer partitions, originally a part of number theory and analysis, it is now considered a part of combinatorics or an independent field. It incorporates the bijective approach and various tools in analysis and analytic number theory, graphs are basic objects in combinatorics
10.
Punched card
–
A punched card or punch card is a piece of stiff paper that can be used to contain digital information represented by the presence or absence of holes in predefined positions. The information might be data for data processing applications or, in earlier examples, the terms IBM card, or Hollerith card specifically refer to punched cards used in semiautomatic data processing. Many early digital computers used punched cards, often prepared using keypunch machines, while punched cards are now obsolete as a recording medium, as of 2012, some voting machines still use punched cards to record votes. Basile Bouchon developed the control of a loom by punched holes in paper tape in 1725, in 1801 Joseph Marie Jacquard demonstrated a mechanism to automate loom operation. A number of punched cards were linked into a chain of any length, each card held the instructions for shedding and selecting the shuttle for a single pass. It is considered an important step in the history of computing hardware, semen Korsakov was reputedly the first to use the punched cards in informatics for information store and search. Korsakov announced his new method and machines in September 1832, rather than seeking patents, charles Babbage proposed the use of Number Cards, pierced with certain holes and stand opposite levers connected with a set of figure wheels. Advanced they push in those levers opposite to which there are no holes on the card, Herman Hollerith invented the recording of data on a medium that could then be read by a machine. Prior uses of machine readable media, such as those above, had been for control, after some initial trials with paper tape, he settled on punched cards. Developing punched card data processing technology for the 1890 US census, other companies entering the punched card business included the Powers Accounting Machine Company, Remington Rand, and Groupe Bull. Both IBM and Remington Rand tied punched card purchases to machine leases, in 1932, the US government took both to court on this issue. IBM viewed its business as providing a service and that the cards were part of the machine, IBM fought all the way to the Supreme Court and lost in 1936, the court ruling that IBM could only set card specifications. According to the IBM Archives, By 1937, IBM had 32 presses at work in Endicott, N. Y. printing, cutting and stacking five to 10 million punched cards every day. Punched cards were used as legal documents, such as U. S. Government checks. Punched card technology developed into a tool for business data-processing. By 1950 punched cards had become ubiquitous in industry and government, do not fold, spindle or mutilate, a generalized version of the warning that appeared on some punched cards, became a motto for the post-World War II era. In 1955 IBM signed a consent decree requiring, amongst other things, tom Watson Jr. s decision to sign this decree, where IBM saw the punched card provisions as the most significant point, completed the transfer of power to him from Thomas Watson, Sr. The UNITYPER introduced magnetic tape for data entry in the 1950s, during the 1960s, the punched card was gradually replaced as the primary means for data storage by magnetic tape, as better, more capable computers became available
11.
International Standard Book Number
–
The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
12.
Lexicographical order
–
In mathematics, the lexicographic or lexicographical order is a generalization of the way the alphabetical order of words is based on the alphabetical order of their component letters. This generalization consists primarily in defining a total order over the sequences of elements of a totally ordered set. There are several variants and generalizations of the lexicographical ordering, another generalization defines an order on a Cartesian product of partially ordered sets, this order is a total order if and only if the factors of the Cartesian product are totally ordered. The word lexicographic is derived from lexicon, the set of words that are used in language and appear in dictionaries. The lexicographic order has thus been introduced for sorting the entries of dictionaries and this has been formalized in the following way. Consider a finite set A, often called alphabet, which is totally ordered, in dictionaries, this is the common alphabet, ordered by the alphabetical order. In book indexes, the alphabet is generally extended to all alphanumeric characters, the lexicographic order is a total order on the sequences of elements of A, often called words on A, which is defined as follows. Given two different sequences of the length, a1a2. ak and b1b2. bk, the first one is smaller than the second one for the lexicographical order, if ai<bi, for the first i where ai. To compare sequences of different lengths, the sequence is usually padded at the end with enough blanks. This way of comparing sequences of different lengths is always used in dictionaries, however, in combinatorics, an other convention is frequently used, whereby a shorter sequence is always smaller than a longer sequence. This variant of the order is sometimes called shortlex order. In dictionary order, the word Thomas appears before Thompson because the letter a comes before the letter p in the alphabet, the 5th letter is the first that is different in the two words, the first four letters are Thom in both. Because it is the first difference, the 5th letter is the most significant difference for the alphabetical ordering. An important property of the order on words of a fixed length on a finite alphabet is that it is a well-order. The lexicographical order is used not only in dictionaries, but also commonly for numbers, one of the drawbacks of the Roman numeral system is that it is not always immediately obvious which of two numbers is the smaller. When negative numbers are considered, one has to reverse the order for comparing negative numbers. This is not usually a problem for humans, but it may be for computers and this is one of the reasons for adopting twos complement representation for representing signed integers in computers. Another example of a use of lexicographical ordering appears in the ISO8601 standard for dates
13.
Ishango bone
–
The Ishango bone is a bone tool, dated to the Upper Paleolithic era. It is a dark brown length of bone, the fibula of a baboon, with a piece of quartz affixed to one end. It was first thought to be a stick, as it has a series of what has been interpreted as tally marks carved in three columns running the length of the tool. It has also suggested that the scratches might have been to create a better grip on the handle or for some other non-mathematical reason. The Ishango bone was found in 1960 by Belgian Jean de Heinzelin de Braucourt while exploring what was then the Belgian Congo and it was discovered in the area of Ishango near the Semliki River. Lake Edward empties into the Semliki which forms part of the headwaters of the Nile River, the bone was found among the remains of a small community that fished and gathered in this area of Africa. The settlement had been buried in a volcanic eruption, the artifact was first estimated to have originated between 9,000 BC and 6,500 BC. However, the dating of the site where it was discovered was re-evaluated, the Ishango bone is on permanent exhibition at the Royal Belgian Institute of Natural Sciences, Brussels, Belgium. Some mathematicians, scientists and archaeolgists believe the three columns of asymmetrically grouped notches imply that the implement was used to construct a numeral system, the central column begins with three notches and then doubles to 6 notches. The process is repeated for the number 4, which doubles to 8 notches, and then reversed for the number 10 and these numbers may not be purely random and instead suggest some understanding of the principle of multiplication and division by two. The bone may therefore have been used as a tool for simple mathematical procedures. He also writes that no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10. Claudia Zaslavsky has suggested that this may indicate that the creator of the tool was a woman, during earlier excavations at the Ishango site in 1959, another bone was also found. It is lighter in color and was scraped, thinned, polished, the artifact possibly held a piece of quartz like the more well-known bone or it could have been a tool handle. The 14-cm long bone has 90 notches on six sides, which are categorized as major or minor according to their length, Jean de Heinzelin interpreted the major notches as being units or multiples and the minor notches as fractions or subsidiary. He believed the bone to be an interchange rule between bases 10 and 12, lebombo bone History of mathematics Paleolithic tally sticks Shurkin, J. Engines of the mind, a history of the computer, W. W. Norton & Co.1984. P21 Bogoshi, J. Naidoo, K. and Webb, Africa, The true cradle of mathematical sciences Ishango,22000 and 50 years later, the cradle of mathematics. The On-Line Encyclopedia of Integer Sequences
14.
Fraction (mathematics)
–
A fraction 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-fifths, three-quarters. A common, vulgar, or simple fraction consists of an integer numerator displayed above a line, numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals. The numerator represents a number of parts, and the denominator. For example, in the fraction 3/4, the numerator,3, tells us that the fraction represents 3 equal parts, the picture to the right illustrates 34 or ¾ of a cake. Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, an integer such as the number 7 can be thought of as having an implicit denominator of one,7 equals 7/1. Other uses for fractions are to represent ratios and to represent division, thus the fraction ¾ is also used to represent the ratio 3,4 and the division 3 ÷4. The test for a number being a number is that it can be written in that form. In a fraction, the number of parts being described is the numerator. Informally, they may be distinguished by placement alone but in formal contexts they are separated by a fraction bar. The fraction bar may be horizontal, oblique, or diagonal and these marks are respectively known as the horizontal bar, the slash or stroke, the division slash, and the fraction slash. In typography, horizontal fractions are known as en or nut fractions and diagonal fractions as em fractions. The denominators of English fractions are expressed as ordinal numbers. When the denominator is 1, it may be expressed in terms of wholes but is commonly ignored. When the numerator is one, it may be omitted, a fraction may be expressed as a single composition, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. Fractions should always be hyphenated when used as adjectives, alternatively, a fraction may be described by reading it out as the numerator over the denominator, with the denominator expressed as a cardinal number. The term over is used even in the case of solidus fractions, Fractions with large denominators that are not powers of ten are often rendered in this fashion while those with denominators divisible by ten are typically read in the normal ordinal fashion. A simple fraction is a number written as a/b or a b
15.
Transcendental number
–
In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a non-zero polynomial equation with integer coefficients. The best-known transcendental numbers are π and e, though only a few classes of transcendental numbers are known, transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the numbers are countable while the sets of real. All real transcendental numbers are irrational, since all numbers are algebraic. Another irrational number that is not transcendental is the ratio, φ or ϕ. The name transcendental comes from the root trans meaning across and length of numbers, euler was probably the first person to define transcendental numbers in the modern sense. Johann Heinrich Lambert conjectured that e and π were both transcendental numbers in his 1768 paper proving the number π is irrational, and proposed a tentative sketch of a proof of πs transcendence. In other words, the nth digit of this number is 1 only if n is one of the numbers 1. Liouville showed that number is what we now call a Liouville number. Liouville showed that all Liouville numbers are transcendental, the first number to be proven transcendental without having been specifically constructed for the purpose was e, by Charles Hermite in 1873. In 1874, Georg Cantor proved that the numbers are countable. He also gave a new method for constructing transcendental numbers, in 1878, Cantor published a construction that proves there are as many transcendental numbers as there are real numbers. Cantors work established the ubiquity of transcendental numbers, in 1882, Ferdinand von Lindemann published a proof that the number π is transcendental. He first showed that ea is transcendental when a is algebraic, then, since eiπ = −1 is algebraic, iπ and therefore π must be transcendental. This approach was generalized by Karl Weierstrass to the Lindemann–Weierstrass theorem, the transcendence of π allowed the proof of the impossibility of several ancient geometric constructions involving compass and straightedge, including the most famous one, squaring the circle. The affirmative answer was provided in 1934 by the Gelfond–Schneider theorem and this work was extended by Alan Baker in the 1960s in his work on lower bounds for linear forms in any number of logarithms. The set of numbers is uncountably infinite. Since the polynomials with rational coefficients are countable, and since each such polynomial has a number of zeroes
16.
Complex number
–
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0
17.
Farey sequence
–
Each Farey sequence starts with the value 0, denoted by the fraction 0⁄1, and ends with the value 1, denoted by the fraction 1⁄1. A Farey sequence is called a Farey series, which is not strictly correct. — Beiler Chapter XVI Farey sequences are named after the British geologist John Farey, Farey conjectured, without offering proof, that each new term in a Farey sequence expansion is the mediant of its neighbours. Fareys letter was read by Cauchy, who provided a proof in his Exercices de mathématique, in fact, another mathematician, Charles Haros, had published similar results in 1802 which were not known either to Farey or to Cauchy. Thus it was an accident that linked Fareys name with these sequences. This is an example of Stiglers law of eponymy, the Farey sequence of order n contains all of the members of the Farey sequences of lower orders. In particular Fn contains all of the members of Fn−1 and also contains an additional fraction for each number that is less than n, thus F6 consists of F5 together with the fractions 1/6 and 5/6. The middle term of a Farey sequence Fn is always 1/2, from this, we can relate the lengths of Fn and Fn−1 using Eulers totient function φ, | F n | = | F n −1 | + φ. Using the fact that |F1| =2, we can derive an expression for the length of Fn, the asymptotic behaviour of |Fn| is, | F n | ∼3 n 2 π2. The index I n = k of a fraction a k, n in the Farey sequence F n = is simply the position that a k, n occupies in the sequence. This is of relevance as it is used in an alternative formulation of the Riemann hypothesis. Various useful properties follow, I n =0, I n =1, I n = /2, I n = | F n | −1, I n = | F n | −1 − I n. Fractions which are neighbouring terms in any Farey sequence are known as a Farey pair and have the following properties, if a/b and c/d are neighbours in a Farey sequence, with a/b < c/d, then their difference c/d − a/b is equal to 1/bd. Since c d − a b = b c − a d b d, thus 1/3 and 2/5 are neighbours in F5, and their difference is 1/15. If b c − a d =1 for positive integers a, b, c and d with a < b and c < d then a/b, thus the first term to appear between 1/3 and 2/5 is 3/8, which appears in F8. The Stern-Brocot tree is a data structure showing how the sequence is built up from 0 and 1, Fractions that appear as neighbours in a Farey sequence have closely related continued fraction expansions. Every fraction has two continued fraction expansions — in one the final term is 1, in the other the term is greater than 1. Farey sequences are useful to find rational approximations of irrational numbers
18.
Binary number
–
The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra
19.
ASCII
–
ASCII, abbreviated from American Standard Code for Information Interchange, is a character encoding standard. ASCII codes represent text in computers, telecommunications equipment, and other devices, most modern character-encoding schemes are based on ASCII, although they support many additional characters. ASCII was developed from telegraph code and its first commercial use was as a seven-bit teleprinter code promoted by Bell data services. Work on the ASCII standard began on October 6,1960, the first edition of the standard was published in 1963, underwent a major revision during 1967, and experienced its most recent update during 1986. Compared to earlier telegraph codes, the proposed Bell code and ASCII were both ordered for more convenient sorting of lists, and added features for other than teleprinters. Originally based on the English alphabet, ASCII encodes 128 specified characters into seven-bit integers as shown by the ASCII chart above. The characters encoded are numbers 0 to 9, lowercase letters a to z, uppercase letters A to Z, basic punctuation symbols, control codes that originated with Teletype machines, for example, lowercase j would become binary 1101010 and decimal 106. ASCII includes definitions for 128 characters,33 are non-printing control characters that affect how text and space are processed and 95 printable characters, of these, the IANA encourages use of the name US-ASCII for Internet uses of ASCII. The ASA became the United States of America Standards Institute and ultimately the American National Standards Institute, there was some debate at the time whether there should be more control characters rather than the lowercase alphabet. The X3.2.4 task group voted its approval for the change to ASCII at its May 1963 meeting, the X3 committee made other changes, including other new characters, renaming some control characters and moving or removing others. ASCII was subsequently updated as USAS X3. 4-1967, then USAS X3. 4-1968, ANSI X3. 4-1977 and they proposed a 9-track standard for magnetic tape, and attempted to deal with some punched card formats. The X3.2 subcommittee designed ASCII based on the earlier teleprinter encoding systems, like other character encodings, ASCII specifies a correspondence between digital bit patterns and character symbols. This allows digital devices to communicate each other and to process, store. Before ASCII was developed, the encodings in use included 26 alphabetic characters,10 numerical digits, ITA2 were in turn based on the 5-bit telegraph code Émile Baudot invented in 1870 and patented in 1874. The committee debated the possibility of a function, which would allow more than 64 codes to be represented by a six-bit code. In a shifted code, some character codes determine choices between options for the character codes. It allows compact encoding, but is reliable for data transmission. The standards committee decided against shifting, and so ASCII required at least a seven-bit code, the committee considered an eight-bit code, since eight bits would allow two four-bit patterns to efficiently encode two digits with binary-coded decimal
20.
Greek alphabet
–
It is the ancestor of the Latin and Cyrillic scripts. In its classical and modern forms, the alphabet has 24 letters, Modern and Ancient Greek use different diacritics. In standard Modern Greek spelling, orthography has been simplified to the monotonic system, examples In both Ancient and Modern Greek, the letters of the Greek alphabet have fairly stable and consistent symbol-to-sound mappings, making pronunciation of words largely predictable. Ancient Greek spelling was generally near-phonemic, among consonant letters, all letters that denoted voiced plosive consonants and aspirated plosives in Ancient Greek stand for corresponding fricative sounds in Modern Greek. This leads to groups of vowel letters denoting identical sounds today. Modern Greek orthography remains true to the spellings in most of these cases. The following vowel letters and digraphs are involved in the mergers, Modern Greek speakers typically use the same, modern, in other countries, students of Ancient Greek may use a variety of conventional approximations of the historical sound system in pronouncing Ancient Greek. Several letter combinations have special conventional sound values different from those of their single components, among them are several digraphs of vowel letters that formerly represented diphthongs but are now monophthongized. In addition to the three mentioned above, there is also ⟨ου⟩, pronounced /u/, the Ancient Greek diphthongs ⟨αυ⟩, ⟨ευ⟩ and ⟨ηυ⟩ are pronounced, and respectively in voicing environments in Modern Greek. The Modern Greek consonant combinations ⟨μπ⟩ and ⟨ντ⟩ stand for and respectively, ⟨τζ⟩ stands for, in addition, both in Ancient and Modern Greek, the letter ⟨γ⟩, before another velar consonant, stands for the velar nasal, thus ⟨γγ⟩ and ⟨γκ⟩ are pronounced like English ⟨ng⟩. There are also the combinations ⟨γχ⟩ and ⟨γξ⟩ and these signs were originally designed to mark different forms of the phonological pitch accent in Ancient Greek. The letter rho, although not a vowel, also carries a rough breathing in word-initial position, if a rho was geminated within a word, the first ρ always had the smooth breathing and the second the rough breathing leading to the transiliteration rrh. The vowel letters ⟨α, η, ω⟩ carry an additional diacritic in certain words, the iota subscript. This iota represents the former offglide of what were originally long diphthongs, ⟨ᾱι, ηι, ωι⟩, another diacritic used in Greek is the diaeresis, indicating a hiatus. In 1982, a new, simplified orthography, known as monotonic, was adopted for use in Modern Greek by the Greek state. Although it is not a diacritic, the comma has a function as a silent letter in a handful of Greek words, principally distinguishing ό. There are many different methods of rendering Greek text or Greek names in the Latin script, the form in which classical Greek names are conventionally rendered in English goes back to the way Greek loanwords were incorporated into Latin in antiquity. In this system, ⟨κ⟩ is replaced with ⟨c⟩, the diphthongs ⟨αι⟩ and ⟨οι⟩ are rendered as ⟨ae⟩ and ⟨oe⟩ respectively, and ⟨ει⟩ and ⟨ου⟩ are simplified to ⟨i⟩ and ⟨u⟩ respectively
21.
Magic square
–
In recreational mathematics, a magic square is a n × n square grid filled with distinct positive integers in the range 1,2. N2 such that each contains a different integer and the sum of the integers in each row, column. The sum is called the constant or magic sum of the magic square. A square grid with n cells on each side is said to have order n. In regard to magic sum, the problem of magic squares only requires the sum of row, column and diagonal to be equal. Thus, although magic squares may contain negative integers, they are just variations by adding or multiplying a number to every positive integer in the original square. Magic squares are called normal magic squares, in the sense that there are non-normal magic squares which integers are not restricted in 1,2. However, in places, magic squares is used as a general term to cover both the normal and non-normal ones, especially when non-normal ones are under discussion. Moreover, the term magic squares is also used to refer to various types of word squares. Magic squares have a history, dating back to at least 650 BC in China. At various times they have acquired magical or mythical significance, and have appeared as symbols in works of art, the constant that is the sum of every row, column and diagonal is called the magic constant or magic sum, M. Every normal magic square has a constant dependent on the n, calculated by the formula M = n /2. N2 is n 2 /2 which when divided by the n is the magic constant. For normal magic squares of orders n =3,4,5,6,7, and 8, the constants are, respectively,15,34,65,111,175. Normal magic squares of all sizes can be constructed except 2×2, any magic square can be rotated and reflected to produce 8 trivially distinct squares. In magic square theory, all of these are deemed equivalent. Excluding rotations and reflections, there is exactly one 3×3 magic square, exactly 880 4×4 magic squares, for the 6×6 case, there are estimated to be approximately 1.8 ×1019 squares. Then all magic squares of an order have the same moment of inertia as each other
22.
Euler's totient function
–
In number theory, Eulers totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as φ or ϕ and it can be defined more formally as the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd is equal to 1. The integers k of this form are referred to as totatives of n. For example, the totatives of n =9 are the six numbers 1,2,4,5,7 and 8. They are all relatively prime to 9, but the three numbers in this range,3,6, and 9 are not, because gcd = gcd =3. As another example, φ =1 since for n =1 the only integer in the range from 1 to n is 1 itself, Eulers totient function is a multiplicative function, meaning that if two numbers m and n are relatively prime, then φ = φφ. This function gives the order of the group of integers modulo n. It also plays a key role in the definition of the RSA encryption system, leonhard Euler introduced the function in 1763. However, he did not at that time choose any specific symbol to denote it. In a 1784 publication, Euler studied the function further, choosing the Greek letter π to denote it, he wrote πD for the multitude of less than D. This definition varies from the current definition for the totient function at D =1 but is otherwise the same, the now-standard notation φ comes from Gausss 1801 treatise Disquisitiones Arithmeticae. Although Gauss didnt use parentheses around the argument and wrote φA, thus, it is often called Eulers phi function or simply the phi function. In 1879, J. J. Sylvester coined the term totient for this function, so it is referred to as Eulers totient function. Jordans totient is a generalization of Eulers, the cototient of n is defined as n − φ. It counts the number of positive integers less than or equal to n that have at least one factor in common with n. There are several formulas for computing φ and it states φ = n ∏ p ∣ n, where the product is over the distinct prime numbers dividing n. The proof of Eulers product formula depends on two important facts and this means that if gcd =1, then φ = φ φ. If p is prime and k ≥1, then φ = p k − p k −1 = p k −1 = p k, proof, since p is a prime number the only possible values of gcd are 1, p, p2
23.
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 number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
24.
Fibonacci number
–
The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, the sequence described in Liber Abaci began with F1 =1. Fibonacci numbers are related to Lucas numbers L n in that they form a complementary pair of Lucas sequences U n = F n and V n = L n. They are intimately connected with the ratio, for example. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is a journal dedicated to their study. The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody, in the Sanskrit tradition of prosody, there was interest in enumerating all patterns of long syllables that are 2 units of duration, and short syllables that are 1 unit of duration. Counting the different patterns of L and S of a given duration results in the Fibonacci numbers, susantha Goonatilake writes that the development of the Fibonacci sequence is attributed in part to Pingala, later being associated with Virahanka, Gopāla, and Hemachandra. He dates Pingala before 450 BC, however, the clearest exposition of the sequence arises in the work of Virahanka, whose own work is lost, but is available in a quotation by Gopala, Variations of two earlier meters. For example, for four, variations of meters of two three being mixed, five happens, in this way, the process should be followed in all mātrā-vṛttas. The sequence is also discussed by Gopala and by the Jain scholar Hemachandra, outside India, the Fibonacci sequence first appears in the book Liber Abaci by Fibonacci. The puzzle that Fibonacci posed was, how many pairs will there be in one year, at the end of the first month, they mate, but there is still only 1 pair. At the end of the month the female produces a new pair. At the end of the month, the original female produces a second pair. At the end of the month, the original female has produced yet another new pair. At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs plus the number of pairs alive last month and this is the nth Fibonacci number. The name Fibonacci sequence was first used by the 19th-century number theorist Édouard Lucas, the most common such problem is that of counting the number of compositions of 1s and 2s that sum to a given total n, there are Fn+1 ways to do this. For example, if n =5, then Fn+1 = F6 =8 counts the eight compositions, 1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 = 1+2+2, all of which sum to 5. The Fibonacci numbers can be found in different ways among the set of strings, or equivalently
25.
Lazy caterer's sequence
–
The lazy caterers sequence, more formally known as the central polygonal numbers, describes the maximum number of pieces of a circle that can be made with a given number of straight cuts. For example, three cuts across a pancake will produce six pieces if the cuts all meet at a point inside the circle. This problem can be formalized mathematically as one of counting the cells in an arrangement of lines, for generalizations to higher dimensions, the analogue of this sequence in 3 dimensions is the cake number. The maximum number p of pieces that can be created with a number of cuts n. Using binomial coefficients, the formula can be expressed as p =1 + = + +. This sequence, starting with n =0, results in 1,2,4,7,11,16,22,29,37,46,56,67,79,92,106,121,137,154,172,191,211. Each number equals 1 plus a triangular number, to obtain the maximum number of pieces, the nth cut line should cross all the other previous cut lines inside the circle, but not cross any intersection of previous cut lines. Thus, the nth line itself is cut in n −1 places, each segment divides one piece of the -cut pancake into 2 parts, adding exactly n to the number of pieces. The new line cant have any more segments since it can cross each previous line once. Thus, the number of pieces after n cuts is f = n + f. This recurrence relation can be solved, if ƒ is expanded one term the relation becomes f = n + + f. Expansion of the term ƒ can continue until the last term is reduced to ƒ, thus, since f =1, because there is one piece before any cuts are made, this can be rewritten as f =1 +. This can be simplified, using the formula for the sum of an arithmetic progression, using Eulers formula to solve plane separation problems, The College Mathematics Journal, Mathematical Association of America,22, 125–130, doi,10. 2307/2686448, JSTOR2686448. Steiner, J. Einige Gesetze über die Theilung der Ebene und des Raumes, on the division of the plane by lines, American Mathematical Monthly, Mathematical Association of America,85, 647–656, doi,10. 2307/2320333, JSTOR2320333. Weisstein, Eric W. Circle Division by Lines
26.
Russell's paradox
–
According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. Symbolically, Let R =, then R ∈ R ⟺ R ∉ R In 1908, two ways of avoiding the paradox were proposed, Russells type theory and the Zermelo set theory, the first constructed axiomatic set theory. Zermelos axioms went well beyond Gottlob Freges axioms of extensionality and unlimited set abstraction, Let us call a set abnormal if it is a member of itself, and normal otherwise. For example, take the set of all squares in the plane and that set is not itself a square in the plane, and therefore is not a member of the set of all squares in the plane. On the other hand, if we take the set that contains all non-. Now we consider the set of all sets, R. This leads to the conclusion that R is neither normal nor abnormal, then by existential instantiation and universal instantiation we have y ∈ y ⟺ y ∉ y a contradiction. Modifications to this axiomatic theory proposed in the 1920s by Abraham Fraenkel, Thoralf Skolem and this theory became widely accepted once Zermelos axiom of choice ceased to be controversial, and ZFC has remained the canonical axiomatic set theory down to the present day. ZFC does not assume that, for property, there is a set of all things satisfying that property. Rather, it asserts that any set X, any subset of X definable using first-order logic exists. The object R discussed above cannot be constructed in this fashion, in some extensions of ZFC, objects like R are called proper classes. ZFC is silent about types, although the hierarchy has a notion of layers that resemble types. Zermelo himself never accepted Skolems formulation of ZFC using the language of first-order logic and this 2nd order ZFC preferred by Zermelo, including axiom of foundation, allowed a rich cumulative hierarchy. Ferreirós writes that Zermelos layers are essentially the same as the types in the versions of simple TT offered by Gödel. One can describe the cumulative hierarchy into which Zermelo developed his models as the universe of a cumulative TT in which types are allowed. Thus, simple TT and ZFC could now be regarded as systems that talk essentially about the same intended objects, the main difference is that TT relies on a strong higher-order logic, while Zermelo employed second-order logic, and ZFC can also be given a first-order formulation. The first-order description of the hierarchy is much weaker, as is shown by the existence of denumerable models
27.
Serial code
–
A serial code is a unique identifier assigned incrementally or sequentially to an item. It is also called a number, although it may be a character string that includes letters and other typographical symbols. Serial numbers identify otherwise identical individual units with many, obvious uses, Serial numbers are a deterrent against theft and counterfeit products, as they can be recorded, and stolen or otherwise irregular goods can be identified. Banknotes and other documents of value bear serial numbers to assist in preventing counterfeiting and tracing stolen ones. They are valuable in quality control, as once a defect is found in the production of a batch of product. Serial numbers may be used to identify individual physical or intangible objects, the purpose and application is different. A software serial number, otherwise called product key, is not embedded in the software. The software will function if a potential user enters a valid product code. The vast majority of codes are rejected by the software. If an unauthorised user is found to be using the software, the term serial number is sometimes used for codes which do not identify a single instance of something. It takes its name from the library use of the word serial to mean a periodical. Certificates and certificate authorities are necessary for use of cryptography. These depend on applying mathematically rigorous serial numbers and serial number arithmetic, the term serial number is also used in military formations as an alternative to the expression service number. Because of this, the number is sometimes called a tail number. LZ548/G—the prototype de Havilland Vampire jet fighter, or ML926/G—a de Havilland Mosquito XVI experimentally fitted with H2S radar, the serial number follows the aircraft throughout its period of service. In 2009 the U. S. FDA published draft guidance for the industry to use serial numbers on prescription drug packages. This measure will enhance the traceability of drugs and help to prevent counterfeiting, Serial numbers are often used in network protocols. However, most sequence numbers in computer protocols are limited to a number of bits
28.
Ceva's theorem
–
Cevas theorem is a theorem about triangles in Euclidean plane geometry. Given a triangle ABC, let the lines AO, BO and CO be drawn from the vertices to a common point O, then, using signed lengths of segments, A F F B ⋅ B D D C ⋅ C E E A =1. In other words, the length AB is taken to be positive or negative according to whether A is to the left or right of B in some fixed orientation of the line. For example, AF/FB is defined as having positive value when F is between A and B and negative otherwise, the converse is often included as part of the theorem. The theorem is attributed to Giovanni Ceva, who published it in his 1678 work De lineis rectis. But it was much earlier by Yusuf Al-Mutaman ibn Hűd. Associated with the figures are several terms derived from Cevas name, cevian, cevian triangle, cevian nest, anticevian triangle, the theorem is very similar to Menelaus theorem in that their equations differ only in sign. A standard proof is as follows, Posamentier and Salkind give four proofs, to check the magnitude, note that the area of a triangle of a given height is proportional to its base. So | △ B O D | | △ C O D | = B D D C = | △ B A D | | △ C A D |. Similarly, C E E A = | △ B C O | | △ A B O |, multiplying these three equations gives | A F F B ⋅ B D D C ⋅ C E E A | =1, as required. The theorem can also be proven easily using Menelaus theorem, the theorem follows by dividing these two equations. The converse follows as a corollary, let D, E and F be given on the lines BC, AC and AB so that the equation holds. Let AD and BE meet at O and let F′ be the point where CO crosses AB, the theorem can be generalized to higher-dimensional simplexes using barycentric coordinates. Define a cevian of an n-simplex as a ray from each vertex to a point on the opposite -face, then the cevians are concurrent if and only if a mass distribution can be assigned to the vertices such that each cevian intersects the opposite facet at its center of mass. Moreover, the point of the cevians is the center of mass of the simplex. Rouths theorem gives the area of the triangle formed by three cevians in the case that they are not concurrent, Cevas theorem can be obtained from it by setting the area equal to zero and solving. The analogue of the theorem for general polygons in the plane has been known since the nineteenth century. The theorem has also been generalized to triangles on other surfaces of constant curvature, projective geometry Median – an application Hogendijk, J. B
29.
Maple (software)
–
Maple is a symbolic and numeric computing environment, and is also a multi-paradigm programming language. Developed by Maplesoft, Maple also covers aspects of technical computing, including visualization, data analysis, matrix computation. A toolbox, MapleSim, adds functionality for physical modeling. Users can enter mathematics in traditional mathematical notation, custom user interfaces can also be created. There is support for numeric computations, to arbitrary precision, as well as symbolic computation and visualization, examples of symbolic computations are given below. Maple incorporates a dynamically typed programming language which resembles Pascal. The language permits variables of lexical scope, there are also interfaces to other languages. There is also an interface to Excel, Maple supports MathML2.0, a W3C format for representing and interpreting mathematical expressions, including their display in Web pages. Maple is based on a kernel, written in C. Most functionality is provided by libraries, which come from a variety of sources, most of the libraries are written in the Maple language, these have viewable source code. Many numerical computations are performed by the NAG Numerical Libraries, ATLAS libraries, different functionality in Maple requires numerical data in different formats. Symbolic expressions are stored in memory as directed acyclic graphs, the standard interface and calculator interface are written in Java. The first concept of Maple arose from a meeting in November 1980 at the University of Waterloo, researchers at the university wished to purchase a computer powerful enough to run Macsyma. Instead, it was decided that they would develop their own computer system that would be able to run on lower cost computers. The first limited version appearing in December 1980 with Maple demonstrated first at conferences beginning in 1982, the name is a reference to Maples Canadian heritage. By the end of 1983, over 50 universities had copies of Maple installed on their machines, in 1984, the research group arranged with Watcom Products Inc to license and distribute the first commercially available version, Maple 3.3. In 1988 Waterloo Maple Inc. was founded, the company’s original goal was to manage the distribution of the software. In 1989, the first graphical user interface for Maple was developed and included with version 4.3 for the Macintosh, x11 and Windows versions of the new interface followed in 1990 with Maple V
30.
Wolfram Mathematica
–
Wolfram Mathematica is a mathematical symbolic computation program, sometimes termed a computer algebra system or program, used in many scientific, engineering, mathematical, and computing fields. It was conceived by Stephen Wolfram and is developed by Wolfram Research of Champaign, the Wolfram Language is the programming language used in Mathematica. The kernel interprets expressions and returns result expressions, all content and formatting can be generated algorithmically or edited interactively. Standard word processing capabilities are supported, including real-time multi-lingual spell-checking, documents can be structured using a hierarchy of cells, which allow for outlining and sectioning of a document and support automatic numbering index creation. Documents can be presented in an environment for presentations. Notebooks and their contents are represented as Mathematica expressions that can be created, modified or analyzed by Mathematica programs or converted to other formats, the front end includes development tools such as a debugger, input completion, and automatic syntax highlighting. Among the alternative front ends is the Wolfram Workbench, an Eclipse based integrated development environment and it provides project-based code development tools for Mathematica, including revision management, debugging, profiling, and testing. There is a plugin for IntelliJ IDEA based IDEs to work with Wolfram Language code which in addition to syntax highlighting can analyse and auto-complete local variables, the Mathematica Kernel also includes a command line front end. Other interfaces include JMath, based on GNU readline and MASH which runs self-contained Mathematica programs from the UNIX command line, version 5.2 added automatic multi-threading when computations are performed on multi-core computers. This release included CPU specific optimized libraries, in addition Mathematica is supported by third party specialist acceleration hardware such as ClearSpeed. Support for CUDA and OpenCL GPU hardware was added in 2010, also, since version 8 it can generate C code, which is automatically compiled by a system C compiler, such as GCC or Microsoft Visual Studio. A free-of-charge version, Wolfram CDF Player, is provided for running Mathematica programs that have saved in the Computable Document Format. It can also view standard Mathematica files, but not run them and it includes plugins for common web browsers on Windows and Macintosh. WebMathematica allows a web browser to act as a front end to a remote Mathematica server and it is designed to allow a user written application to be remotely accessed via a browser on any platform. It may not be used to full access to Mathematica. Due to bandwidth limitations interactive 3D graphics is not fully supported within a web browser, Wolfram Language code can be converted to C code or to an automatically generated DLL. Wolfram Language code can be run on a Wolfram cloud service as a web-app or as an API either on Wolfram-hosted servers or in an installation of the Wolfram Enterprise Private Cloud. Communication with other applications occurs through a protocol called Wolfram Symbolic Transfer Protocol and it allows communication between the Wolfram Mathematica kernel and front-end, and also provides a general interface between the kernel and other applications
31.
PARI/GP
–
PARI/GP is a computer algebra system with the main aim of facilitating number theory computations. Versions 2.1.0 and higher are distributed under the GNU General Public License and it runs on most common operating systems. The PARI/GP system is a package that is capable of doing formal computations on recursive types at high speed and its three main strengths are its speed, the possibility of directly using data types that are familiar to mathematicians, and its extensive algebraic number theory module. The PARI/GP system consists of the standard components, PARI is a C library, allowing for fast computations. Gp is an interactive command line interface giving access to the PARI functions. It functions as a programmable calculator which contains most of the control instructions of a standard language like C. GP is the name of gps scripting language which can be used to program gp, also available is gp2c, the GP-to-C compiler, which compiles GP scripts into the C language and transparently loads the resulting functions into gp. The advantage of this is that gp2c-compiled scripts will typically run three to four times faster, gp2c understands almost all of GP, with the exception of Lists. It can compute factorizations, perform elliptic curve computations and perform algebraic number theory calculations and it also allows computations with matrices, polynomials, power series, algebraic numbers and implements many special functions. PARI/GP comes with its own built-in graphical plotting capability, PARI/GP has some symbolic manipulation capability, e. g. multivariate polynomial and rational function handling. It also has some formal integration and differentiation capabilities, PARI/GP can be compiled with GMP providing faster computations than PARI/GPs native arbitrary precision kernel. PARI/GPs progenitor was a program named Isabelle, an interpreter for higher arithmetic, written in 1979 by Henri Cohen and François Dress at the Université Bordeaux 1. The name PARI is a pun about the early stages when the authors started to implement a library for Pascal ARIthmetic in the Pascal programming language. The first version of the gp calculator was originally called GPC, the trailing C was eventually dropped. Below are some samples of the gp calculator usage, \p 212 realprecision =221 significant digits. %3 = x - 1/6*x^3 + 1/120*x^5 - 1/5040*x^7 + 1/362880*x^9 - 1/39916800*x^11 + 1/6227020800*x^13 - 1/1307674368000*x^15 + O. for time =5 ms, K = bnfinit, K. cyc time = 1ms. %4 = /* This number field has class number 3
32.
Riemann zeta function
–
More general representations of ζ for all s are given below. The Riemann zeta function plays a role in analytic number theory and has applications in physics, probability theory. As a function of a variable, Leonhard Euler first introduced and studied it in the first half of the eighteenth century without using complex analysis. The values of the Riemann zeta function at even positive integers were computed by Euler, the first of them, ζ, provides a solution to the Basel problem. In 1979 Apéry proved the irrationality of ζ, the values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms. Many generalizations of the Riemann zeta function, such as Dirichlet series, the Riemann zeta function ζ is a function of a complex variable s = σ + it. It can also be defined by the integral ζ =1 Γ ∫0 ∞ x s −1 e x −1 d x where Γ is the gamma function. The Riemann zeta function is defined as the continuation of the function defined for σ >1 by the sum of the preceding series. Leonhard Euler considered the series in 1740 for positive integer values of s. The above series is a prototypical Dirichlet series that converges absolutely to a function for s such that σ >1. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values s ≠1, for s =1 the series is the harmonic series which diverges to +∞, and lim s →1 ζ =1. Thus the Riemann zeta function is a function on the whole complex s-plane. For any positive even integer 2n, ζ = n +1 B2 n 2 n 2, where B2n is the 2nth Bernoulli number. For odd positive integers, no simple expression is known, although these values are thought to be related to the algebraic K-theory of the integers. For nonpositive integers, one has ζ = B n +1 n +1 for n ≥0 In particular, ζ = −12, Similarly to the above, this assigns a finite result to the series 1 +1 +1 +1 + ⋯. ζ ≈ −1.4603545 This is employed in calculating of kinetic boundary layer problems of linear kinetic equations, ζ =1 +12 +13 + ⋯ = ∞, if we approach from numbers larger than 1. Then this is the harmonic series, but its Cauchy principal value lim ε →0 ζ + ζ2 exists which is the Euler–Mascheroni constant γ =0. 5772…. ζ ≈2.612, This is employed in calculating the critical temperature for a Bose–Einstein condensate in a box with periodic boundary conditions, and for spin wave physics in magnetic systems
33.
Mersenne prime
–
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a number that can be written in the form Mn = 2n −1 for some integer n. They are named after Marin Mersenne, a French Minim friar, the first four Mersenne primes are 3,7,31, and 127. If n is a number then so is 2n −1. The definition is therefore unchanged when written Mp = 2p −1 where p is assumed prime, more generally, numbers of the form Mn = 2n −1 without the primality requirement are called Mersenne numbers. The smallest composite pernicious Mersenne number is 211 −1 =2047 =23 ×89, Mersenne primes Mp are also noteworthy due to their connection to perfect numbers. As of January 2016,49 Mersenne primes are known, the largest known prime number 274,207,281 −1 is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the “Great Internet Mersenne Prime Search”, many fundamental questions about Mersenne primes remain unresolved. It is not even whether the set of Mersenne primes is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture asserts that there are infinitely many Mersenne primes,23 | M11,47 | M23,167 | M83,263 | M131,359 | M179,383 | M191,479 | M239, and 503 | M251. Since for these primes p, 2p +1 is congruent to 7 mod 8, so 2 is a quadratic residue mod 2p +1, since p is a prime, it must be p or 1. The first four Mersenne primes are M2 =3, M3 =7, M5 =31, a basic theorem about Mersenne numbers states that if Mp is prime, then the exponent p must also be prime. This follows from the identity 2 a b −1 = ⋅ = ⋅ and this rules out primality for Mersenne numbers with composite exponent, such as M4 =24 −1 =15 =3 ×5 = ×. Though the above examples might suggest that Mp is prime for all p, this is not the case. The evidence at hand does suggest that a randomly selected Mersenne number is more likely to be prime than an arbitrary randomly selected odd integer of similar size. Nonetheless, prime Mp appear to grow increasingly sparse as p increases, in fact, of the 2,270,720 prime numbers p up to 37,156,667, Mp is prime for only 45 of them. The lack of any simple test to determine whether a given Mersenne number is prime makes the search for Mersenne primes a difficult task, the Lucas–Lehmer primality test is an efficient primality test that greatly aids this task. The search for the largest known prime has somewhat of a cult following, consequently, a lot of computer power has been expended searching for new Mersenne primes, much of which is now done using distributed computing
34.
Eigenvalues and eigenvectors
–
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector whose direction does not change when that linear transformation is applied to it. This condition can be written as the equation T = λ v, there is a correspondence between n by n square matrices and linear transformations from an n-dimensional vector space to itself. For this reason, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices or the language of linear transformations. Geometrically an eigenvector, corresponding to a real eigenvalue, points in a direction that is stretched by the transformation. If the eigenvalue is negative, the direction is reversed, Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix eigen- is adopted from the German word eigen for proper, inherent, own, individual, special, specific, peculiar, or characteristic. In essence, an eigenvector v of a linear transformation T is a vector that. Applying T to the eigenvector only scales the eigenvector by the scalar value λ and this condition can be written as the equation T = λ v, referred to as the eigenvalue equation or eigenequation. In general, λ may be any scalar, for example, λ may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero or complex. The Mona Lisa example pictured at right provides a simple illustration, each point on the painting can be represented as a vector pointing from the center of the painting to that point. The linear transformation in this example is called a shear mapping, the vectors pointing to each point in the original image are therefore tilted right or left and made longer or shorter by the transformation. Notice that points along the horizontal axis do not move at all when this transformation is applied, therefore, any vector that points directly to the right or left with no vertical component is an eigenvector of this transformation because the mapping does not change its direction. Moreover, these all have an eigenvalue equal to one because the mapping does not change their length. Linear transformations can take different forms, mapping vectors in a variety of vector spaces. Alternatively, the transformation could take the form of an n by n matrix. For a matrix, eigenvalues and eigenvectors can be used to decompose the matrix, the set of all eigenvectors of T corresponding to the same eigenvalue, together with the zero vector, is called an eigenspace or characteristic space of T. If the set of eigenvectors of T form a basis of the domain of T, Eigenvalues are often introduced in the context of linear algebra or matrix theory. Historically, however, they arose in the study of quadratic forms, in the 18th century Euler studied the rotational motion of a rigid body and discovered the importance of the principal axes
35.
Egyptian fraction
–
An Egyptian fraction is a finite sum of distinct unit fractions, such as 12 +13 +116. That is, each fraction in the expression has an equal to 1 and a denominator that is a positive integer. The value of an expression of type is a positive rational number a/b. Every positive rational number can be represented by an Egyptian fraction, in modern mathematical notation, Egyptian fractions have been superseded by vulgar fractions and decimal notation. However, Egyptian fractions continue to be an object of study in modern theory and recreational mathematics. Beyond their historical use, Egyptian fractions have some advantages over other representations of fractional numbers. For instance, Egyptian fractions can help in dividing a number of objects into equal shares, for more information on this subject, see Egyptian numerals, Eye of Horus, and Egyptian mathematics. Egyptian fraction notation was developed in the Middle Kingdom of Egypt, five early texts in which Egyptian fractions appear were the Egyptian Mathematical Leather Roll, the Moscow Mathematical Papyrus, the Reisner Papyrus, the Kahun Papyrus and the Akhmim Wooden Tablet. A later text, the Rhind Mathematical Papyrus, introduced improved ways of writing Egyptian fractions, the Rhind papyrus was written by Ahmes and dates from the Second Intermediate Period, it includes a table of Egyptian fraction expansions for rational numbers 2/n, as well as 84 word problems. Solutions to each problem were written out in scribal shorthand, with the answers of all 84 problems being expressed in Egyptian fraction notation. 2/n tables similar to the one on the Rhind papyrus also appear on some of the other texts, however, as the Kahun Papyrus shows, vulgar fractions were also used by scribes within their calculations. To write the unit used in their Egyptian fraction notation, in hieroglyph script. Similarly in hieratic script they drew a line over the letter representing the number. For example, The Egyptians had special symbols for 1/2, 2/3, the remaining number after subtracting one of these special fractions was written using as a sum of distinct unit fractions according to the usual Egyptian fraction notation. These have been called Horus-Eye fractions after a theory that they were based on the parts of the Eye of Horus symbol, the unit fraction 1/n is expressed as n, and the fraction 2/n is expressed as n, and the plus sign “＋” is omitted. For example, 2/3 = 1/2 + 1/6 is expressed as 3 =26, modern historians of mathematics have studied the Rhind papyrus and other ancient sources in an attempt to discover the methods the Egyptians used in calculating with Egyptian fractions. In particular, study in this area has concentrated on understanding the tables of expansions for numbers of the form 2/n in the Rhind papyrus, although these expansions can generally be described as algebraic identities, the methods used by the Egyptians may not correspond directly to these identities. This method is available for not only odd prime denominators but also all odd denominators, for larger prime denominators, an expansion of the form 2/p = 1/A + 2A − p/Ap was used, where A is a number with many divisors between p/2 and p