1.
Mechanical calculator
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A mechanical calculator, or calculating machine, was a mechanical device used to perform automatically the basic operations of arithmetic. Most mechanical calculators were comparable in size to small computers and have been rendered obsolete by the advent of the electronic calculator. Surviving notes from Wilhelm Schickard in 1623 reveal that he designed and had built the earliest of the attempts at mechanizing calculation. A study of the surviving notes shows a machine that would have jammed after a few entries on the dial. Schickard abandoned his project in 1624 and never mentioned it again until his death years later in 1635. Two decades after Schickards supposedly failed attempt, in 1642, Blaise Pascal decisively solved these problems with his invention of the mechanical calculator. Co-opted into his fathers labour as tax collector in Rouen, Pascal designed the calculator to help in the amount of tedious arithmetic required. It was called Pascals Calculator or Pascaline, for forty years the arithmometer was the only type of mechanical calculator available for sale. The comptometer, introduced in 1887, was the first machine to use a keyboard which consisted of columns of nine keys for each digit, the Dalton adding machine, manufactured from 1902, was the first to have a 10 key keyboard. Electric motors were used on some mechanical calculators from 1901, the production of mechanical calculators came to a stop in the middle of the 1970s closing an industry that had lasted for 120 years. The first one was a mechanical calculator, his difference engine. In 1855, Georg Scheutz became the first of a handful of designers to succeed at building a smaller and simpler model of his difference engine, a crucial step was the adoption of a punched card system derived from the Jacquard loom making it infinitely programmable. The desire to economize time and mental effort in arithmetical computations and this instrument was probably invented by the Semitic races and later adopted in India, whence it spread westward throughout Europe and eastward to China and Japan. After the development of the abacus, no further advances were made until John Napier devised his numbering rods, or Napiers Bones, in 1617. The 17th century marked the beginning of the history of mechanical calculators, as it saw the invention of its first machines, including Pascals calculator, Blaise Pascal invented a mechanical calculator with a sophisticated carry mechanism in 1642. After three years of effort and 50 prototypes he introduced his calculator to the public and he built twenty of these machines in the following ten years. This machine could add and subtract two numbers directly and multiply and divide by repetition and this suggests that the carry mechanism would have proved itself in practice many times over. Pascals invention of the machine, just three hundred years ago, was made while he was a youth of nineteen
2.
Gottfried Wilhelm Leibniz
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Leibnizs notation has been widely used ever since it was published. It was only in the 20th century that his Law of Continuity and he became one of the most prolific inventors in the field of mechanical calculators. He also refined the number system, which is the foundation of virtually all digital computers. Leibniz, along with René Descartes and Baruch Spinoza, was one of the three great 17th-century advocates of rationalism and he wrote works on philosophy, politics, law, ethics, theology, history, and philology. Leibnizs contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters and he wrote in several languages, but primarily in Latin, French, and German. There is no complete gathering of the writings of Leibniz in English, Gottfried Leibniz was born on July 1,1646, toward the end of the Thirty Years War, in Leipzig, Saxony, to Friedrich Leibniz and Catharina Schmuck. Friedrich noted in his journal,21. Juny am Sontag 1646 Ist mein Sohn Gottfried Wilhelm, post sextam vespertinam 1/4 uff 7 uhr abents zur welt gebohren, in English, On Sunday 21 June 1646, my son Gottfried Wilhelm is born into the world a quarter after six in the evening, in Aquarius. Leibniz was baptized on July 3 of that year at St. Nicholas Church, Leipzig and his father died when he was six and a half years old, and from that point on he was raised by his mother. Her teachings influenced Leibnizs philosophical thoughts in his later life, Leibnizs father had been a Professor of Moral Philosophy at the University of Leipzig, and the boy later inherited his fathers personal library. He was given access to it from the age of seven. Access to his fathers library, largely written in Latin, also led to his proficiency in the Latin language and he also composed 300 hexameters of Latin verse, in a single morning, for a special event at school at the age of 13. In April 1661 he enrolled in his fathers former university at age 15 and he defended his Disputatio Metaphysica de Principio Individui, which addressed the principle of individuation, on June 9,1663. Leibniz earned his masters degree in Philosophy on February 7,1664, after one year of legal studies, he was awarded his bachelors degree in Law on September 28,1665. His dissertation was titled De conditionibus, in early 1666, at age 19, Leibniz wrote his first book, De Arte Combinatoria, the first part of which was also his habilitation thesis in Philosophy, which he defended in March 1666. His next goal was to earn his license and Doctorate in Law, in 1666, the University of Leipzig turned down Leibnizs doctoral application and refused to grant him a Doctorate in Law, most likely due to his relative youth. Leibniz then enrolled in the University of Altdorf and quickly submitted a thesis, the title of his thesis was Disputatio Inauguralis de Casibus Perplexis in Jure. Leibniz earned his license to practice law and his Doctorate in Law in November 1666 and he next declined the offer of an academic appointment at Altdorf, saying that my thoughts were turned in an entirely different direction
3.
Arithmetic operations
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Arithmetic is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations between them—addition, subtraction, multiplication and division. Arithmetic is an part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are still used to refer to a wider part of number theory. The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC and these artifacts do not always reveal the specific process used for solving problems, but the characteristics of the particular numeral system strongly influence the complexity of the methods. The hieroglyphic system for Egyptian numerals, like the later Roman numerals, in both cases, this origin resulted in values that used a decimal base but did not include positional notation. Complex calculations with Roman numerals required the assistance of a board or the Roman abacus to obtain the results. Early number systems that included positional notation were not decimal, including the system for Babylonian numerals. Because of this concept, the ability to reuse the same digits for different values contributed to simpler. The continuous historical development of modern arithmetic starts with the Hellenistic civilization of ancient Greece, prior to the works of Euclid around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs. For example, Nicomachus summarized the viewpoint of the earlier Pythagorean approach to numbers, Greek numerals were used by Archimedes, Diophantus and others in a positional notation not very different from ours. Because the ancient Greeks lacked a symbol for zero, they used three separate sets of symbols, one set for the units place, one for the tens place, and one for the hundreds. Then for the place they would reuse the symbols for the units place. Their addition algorithm was identical to ours, and their multiplication algorithm was very slightly different. Their long division algorithm was the same, and the square root algorithm that was taught in school was known to Archimedes. He preferred it to Heros method of successive approximation because, once computed, a digit doesnt change, and the square roots of perfect squares, such as 7485696, terminate immediately as 2736. For numbers with a part, such as 546.934. The ancient Chinese used a positional notation. Because they also lacked a symbol for zero, they had one set of symbols for the place
4.
Lower Saxony
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Lower Saxony is a German state situated in northwestern Germany and is second in area, with 47,624 square kilometres, and fourth in population among the sixteen Länder of Germany. In rural areas Northern Low Saxon, a dialect of Low German, and Saterland Frisian, a variety of Frisian, are still spoken, but the number of speakers is declining. Lower Saxony borders on the North Sea, the states of Schleswig-Holstein, Hamburg, Mecklenburg-Vorpommern, Brandenburg, Saxony-Anhalt, Thuringia, Hesse and North Rhine-Westphalia, and the Netherlands. Furthermore, the state of Bremen forms two enclaves within Lower Saxony, one being the city of Bremen, the other, its city of Bremerhaven. In fact, Lower Saxony borders more neighbours than any other single Bundesland, the states principal cities include the state capital Hanover, Braunschweig, Lüneburg, Osnabrück, Oldenburg, Hildesheim, Wolfenbüttel, Wolfsburg and Göttingen. The northwestern area of Lower Saxony, which lies on the coast of the North Sea, is called East Frisia, in the extreme west of Lower Saxony is the Emsland, a traditionally poor and sparsely populated area, once dominated by inaccessible swamps. The northern half of Lower Saxony, also known as the North German Plains, is almost invariably flat except for the hills around the Bremen geestland. Towards the south and southwest lie the northern parts of the German Central Uplands, the Weser Uplands, between these two lie the Lower Saxon Hills, a range of low ridges. Thus, Lower Saxony is the only Bundesland that encompasses both maritime and mountainous areas, Lower Saxonys major cities and economic centres are mainly situated in its central and southern parts, namely Hanover, Braunschweig, Osnabrück, Wolfsburg, Salzgitter, Hildesheim and Göttingen. Oldenburg, near the coastline, is another economic centre. To the north, the Elbe river separates Lower Saxony from Hamburg, Schleswig-Holstein, Mecklenburg-Western Pomerania, the banks just south of the Elbe are known as Altes Land. Due to its local climate and fertile soil, it is the states largest area of fruit farming. Most of the territory was part of the historic Kingdom of Hanover. It was created by the merger of the State of Hanover with several states in 1946. Lower Saxony has a boundary in the north in the North Sea. The state and city of Bremen is an enclave surrounded by Lower Saxony. The Bremen/Oldenburg Metropolitan Region is a body for the enclave area. To the southeast the state border runs through the Harz, low mountains that are part of the German Central Uplands, in northeast Lower Saxony is Lüneburg Heath
5.
Hanover
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At the end of the Napoleonic Wars, the Electorate was enlarged to become a Kingdom with Hanover as its capital. From 1868 to 1946 Hanover was the capital of the Prussian Province of Hanover and it is now the capital of the Land of Lower Saxony. Since 2001 it has been part of the Hanover district, which is a body made up from the former district. With a population of 518,000, Hanover is a centre of Northern Germany. Hanover also hosts annual commercial trade fairs such as the Hanover Fair, every year Hanover hosts the Schützenfest Hannover, the worlds largest marksmens festival, and the Oktoberfest Hannover, the second largest such festival in Germany. In 2000, Hanover hosted the world fair Expo 2000, the Hanover fairground, due to numerous extensions, especially for the Expo 2000, is the largest in the world. Hanover is of importance because of its universities and medical school, its international airport. The city is also a crossing point of railway lines and highways. Hanover is the traditional English spelling, the German spelling is becoming more popular in English, recent editions of encyclopaedias prefer the German spelling, and the local government uses the German spelling on English websites. The traditional English spelling is used in historical contexts, especially when referring to the British House of Hanover. Hanover was founded in times on the east bank of the River Leine. Its original name Honovere may mean high bank, though this is debated, Hanover was a small village of ferrymen and fishermen that became a comparatively large town in the 13th century due to its position at a natural crossroads. As overland travel was difficult, its position on the upper navigable reaches of the river helped it to grow by increasing trade. In the 14th century the churches of Hanover were built. The beginning of industrialization in Germany led to trade in iron and silver from the northern Harz Mountains, in 1636 George, Duke of Brunswick-Lüneburg, ruler of the Brunswick-Lüneburg principality of Calenberg, moved his residence to Hanover. The Dukes of Brunswick-Lüneburg were elevated by the Holy Roman Emperor to the rank of Prince-Elector in 1692, thus the principality was upgraded to the Electorate of Brunswick-Lüneburg, colloquially known as the Electorate of Hanover after Calenbergs capital. Its electors would later become monarchs of Great Britain, the first of these was George I Louis, who acceded to the British throne in 1714. The last British monarch who ruled in Hanover was William IV, semi-Salic law, which required succession by the male line if possible, forbade the accession of Queen Victoria in Hanover
6.
Deutsches Museum
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The museum was founded on 28 June 1903, at a meeting of the Association of German Engineers as an initiative of Oskar von Miller. Its official name is Deutsches Museum von Meisterwerken der Naturwissenschaft und Technik and it is the largest museum in Munich. The main site of the Deutsches Museum is an island in the Isar river. The island did not have any buildings before 1772 because it was regularly flooded prior to the building of the Sylvensteinspeicher, in 1772 the Isar barracks were built on the island and, after the flooding of 1899, the buildings were rebuilt with flood protection. In 1903 the city announced that they would donate the island for the newly built Deutsches Museum. The island formerly known as Kohleninsel was then renamed Museumsinsel, in addition to the main site on the Museumsinsel, the museum has two branches in and near Munich and one in Bonn. The Flugwerft Schleißheim branch is located some 18 kilometres north of Munichs city centre close to Schleißheim Palace and it is based on the premises of one of the first military airbases in Germany founded just before World War I. The Flugwerft Schleißheim displays various interesting airplanes for which there was insufficient room at the Museumsinsel site in downtown Munich, among the more prominent exhibits is a Horten flying wing glider built in the 1940s, restored from the few surviving parts. A collection of the German constructions of VTOL planes developed in the 1950s and 1960s is unique, a range of Vietnam era fighter planes as well as Russian planes taken over from East Germany after the reunification are on display. This outstation also features a dedicated to the restoration of all types of airplanes intended for static display. The latest branch of the Deutsches Museum, located at Theresienhöhe in Munich, the branch located in Bonn was opened in 1995 and focuses on German technology, science and research after 1945. In 1883 he joined AEG and founded an office in Munich. The Frankfurt electricity exhibition in 1891 and several power plants contributed to the reputation of Oskar von Miller, in the early years, the exhibition and the collection of the Deutsches Museum were strongly influenced personally by Oskar von Miller. A few months before the 1903 meeting of the Society of German Engineers, Oskar von Miller gathered a group who supported his desire to found a science. In a showing of support this group spontaneously donated 260,000 marks to the cause, in June 1903, Prince Ludwig agreed to act as patron of the museum and the city of Munich donated Coal Island as a site for the project. In addition, exhibits began to arrive from Munich, Germany, as no dedicated museum building existed, the exhibits were displayed in the National Museum. On 12 November 1906, the exhibits at the National Museum were ceremonially opened to the public. Oskar von Miller opened the new museum on his 70th birthday,2 May 1925, from the beginning, the museum displays are backed up by documents available in a public library and archives, which are open seven days a week to ensure access to the working public
7.
Munich
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Munich is the capital and largest city of the German state of Bavaria, on the banks of River Isar north of the Bavarian Alps. Munich is the third largest city in Germany, after Berlin and Hamburg, the Munich Metropolitan Region is home to 5.8 million people. According to the Globalization and World Rankings Research Institute Munich is considered an alpha-world city, the name of the city is derived from the Old/Middle High German term Munichen, meaning by the monks. It derives from the monks of the Benedictine order who ran a monastery at the place that was later to become the Old Town of Munich, Munich was first mentioned in 1158. From 1255 the city was seat of the Bavarian Dukes, black and gold—the colours of the Holy Roman Empire—have been the citys official colours since the time of Ludwig the Bavarian, when it was an imperial residence. Following a final reunification of the Wittelsbachian Duchy of Bavaria, previously divided and sub-divided for more than 200 years, like wide parts of the Holy Roman Empire, the area recovered slowly economically. In 1918, during the German Revolution, the house of Wittelsbach, which governed Bavaria since 1180, was forced to abdicate in Munich. In the 1920s, Munich became home to political factions, among them the NSDAP. During World War II, Munich was heavily bombed and more than 50% of the entire city, the postwar period was characterised by American occupation until 1949 and a strong increase of population and economic power during the years of the Wirtschaftswunder after 1949. The city is home to corporations like BMW, Siemens, MAN, Linde, Allianz and MunichRE as well as many small. Munich is home to national and international authorities, major universities, major museums. Its numerous architectural attractions, international events, exhibitions and conferences. Munich is one of the most prosperous and fastest growing cities in Germany and it is a top-ranked destination for migration and expatriate location, despite being the municipality with the highest density of population in Germany. Munich nowadays hosts more than 530,000 people of foreign background, the year 1158 is assumed to be the foundation date, which is the earliest date the city is mentioned in a document. The document was signed in Augsburg, by that time the Guelph Henry the Lion, Duke of Saxony and Bavaria, had built a bridge over the river Isar next to a settlement of Benedictine monks—this was on the Old Salt Route and a toll bridge. In 1175, Munich was officially granted city status and received fortification, in 1180, with the trial of Henry the Lion, Otto I Wittelsbach became Duke of Bavaria and Munich was handed over to the Bishop of Freising. In 1240, Munich was transferred to Otto II Wittelsbach and in 1255, Duke Louis IV, a native of Munich, was elected German king in 1314 and crowned as Holy Roman Emperor in 1328. He strengthened the position by granting it the salt monopoly
8.
Leibniz wheel
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Invented by Leibniz in 1673, it was used for three centuries until the advent of the electronic calculator in the mid-1970s. Leibniz built a machine called the Stepped Reckoner based on design in 1694. It was made famous by Thomas de Colmar when he used it, a century and a later, in his Arithmometer. It was also used in the Curta calculator, a popular portable calculator introduced in the second part of the 20th century. By coupling a Leibniz wheel with a counting wheel free to move up and down its length, the animation on the side shows a nine-tooth Leibniz wheel coupled to a red counting wheel. It is set to mesh with three teeth at each rotation and therefore would add or subtract 3 from the counter at each rotation, the computing engine of an Arithmometer has a set of linked Leibniz wheels coupled to a crank handle. Each turn of the handle rotates all the Leibniz wheels by one full turn. The input sliders move counting wheels up and down the Leibniz wheels which are linked by a carry mechanism. – Gottfried Leibniz built his first stepped reckoner in 1694 and another one in 1706, – Philipp-Matthäus Hahn, a German pastor, built two circular machines in 1770. – J. C. Schuster, Hahns brother in law, – Lord Stanhope designed a machine using Leibniz wheels in 1777. He also designed a pinwheel calculator in 1775, – Johann-Helfrich Müller built a machine very similar to Hahns machine in 1783. – Thomas de Colmar invented his Arithmometer in 1820 but it took him 30 years of development before it was commercialized in 1851, louis Payen, Veuve L. Payen and Darras were successive owners and distributors of the Arithmometer. – Timoleon Maurel invented his Arithmaurel in 1842, the complexity of its design limited its capacity and doomed its production, but it could multiply two numbers by the simple fact of setting them on its dials. These clones, often more sophisticated than the original arithmometer, were built until the beginning of World War II, – Joseph Edmondson invented and manufactured a circular calculator in 1885. – Friden and Monroe calculators used a variant of this mechanism. Both were made in numbers, Monroe started early in the 20th century. The variant mechanism worked with digits from 1 through 4 as shown in the animation and this made it unnecessary for the sliding gear to travel longer distances for the higher-number digits. Otherwise, pressing a 5.9 key would require either a stroke or excessive force combined with a gently sloping cam surface
9.
Curta
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The Curta is a small mechanical calculator developed by Curt Herzstark. The Curtas design is a descendant of Gottfried Leibnizs Stepped Reckoner and Charles Thomas Arithmometer, accumulating values on cogs and it has an extremely compact design, a small cylinder that fits in the palm of the hand. Curtas were considered the best portable calculators available until they were displaced by electronic calculators in the 1970s, the Curta was conceived by Curt Herzstark in the 1930s in Vienna, Austria. By 1938, he had filed a key patent, covering his complemented stepped drum, the nines complement math breakthrough eliminated the significant mechanical complexity created when borrowing during subtraction. This drum would prove to be the key to the small and his work on the pocket calculator stopped in 1938 when the Nazis forced him and his company to concentrate on manufacturing measuring instruments and distance gauges for the German army. Herzstark, the son of a Catholic mother and Jewish father, was taken into custody in 1943, the head of the department, Mr. Munich said, See, Herzstark, I understand youve been working on a new thing, a small calculating machine. Do you know, I can give you a tip and we will allow you to make and draw everything. If it is worth something, then we will give it to the Führer as a present after we win the war. Then, surely, you will be made an Aryan, for me, that was the first time I thought to myself, my God, if you do this, you can extend your life. And then and there I started to draw the CURTA, the way I had imagined it, Herzstark worked hard to move his invention from his knowing how to build the device in principle to concise working drawings for a manufacturable device. The department heads celebration plan did not materialize, but Herzstarks construction plans did, between April 11,1945, when Buchenwald was liberated by U. S. Soviet forces had arrived in July, and Herzstark feared being sent to Russia, so, later that same month and he began to look for financial backers, at the same time filing continuing patents as well as several additional patents to protect his work. The Prince of Liechtenstein eventually showed interest in the manufacture of the device and this ploy now backfired, without the patent rights, they could manufacture nothing. Herzstark was able to negotiate a new agreement, and money continued to flow to him, Curtas were considered the best portable calculators available until they were displaced by electronic calculators in the 1970s. Herzstark continued to make money from his invention until that time, although, like many inventors before him, he was not among those who profited the most from his invention. The Curta, however, lives on, being a popular collectible. In 2016 a Curta was designed that could be produced on a 3D printer, the fine tolerances needed are beyond present printer technology and this means the machine has to be much bigger, about the size of a coffee mug. The Curta Type I was sold for $125 in the years of production
10.
Meyers Konversations-Lexikon
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The first part of Das Grosse Conversations-Lexikon für die gebildeten Stände appeared in October 1839. In contrast to its contemporaries, it contained maps and illustrations with the text, there is no indication of the planned number of volumes or a time limit for this project, but little headway had been made by the otherwise dynamic Meyer. After six years,14 volumes had appeared, covering only one fifth of the alphabet, another six years passed before the last volume was published. Six supplementary volumes finally finished the work in 1855, ultimately numbering 52 volumes, Das Grosse Conversations-Lexikon für die gebildeten Stände was the most comprehensive completed German encyclopedia of the 19th century, also called der Wunder-Meyer. The complete set was reprinted 1858-59, the son of Joseph Meyer, Hermann Julius, published the next edition, entitled Neues Conversations-Lexikon für alle Stände, 1857–60, that would only count 15 volumes. To avoid a long-time project, subscribers were promised it would be completed three years, and all volumes appearing later would be given free. Of course, it was finished right on time, the 2nd edition, Neues Konversations-Lexikon, ein Wörterbuch des allgemeinen Wissens, appeared 1861-67, the 3rd edition, now from Leipzig, was issued as Meyers Konversations-Lexikon. Eine Encyklopädie des allgemeinen Wissens 1874-78, both had 15 volumes, the 4th edition, consisting of 16 volumes, appeared in 1885-90, with 2 supplements of update pages, vol.17 and vol. The 5th edition had 17 volumes, Meyers Konversations-Lexikon, ein Nachschlagewerk des allgemeinen Wissens,1893 to 1897. This edition sold no less than 233,000 sets, the 6th edition, entitled Meyers Großes Konversations-Lexikon, was published 1902-08. It had 20 volumes, and the largest sale of all Meyer editions, the First World War prevented an even bigger success. There was also the small 1908 edition, Meyers Kleines Konversations-Lexikon, the 7th and 8th editions were both briefly named Meyers Lexikon. The 7th edition counted only 12 volumes, due to the depression of the twenties. It came with a condensed single-volume version known as the Blitz-Lexikon, the 8th edition remained incomplete due to wartime circumstances, out of 12 planned volumes only volumes number 1 through 9 plus the atlas volume number 12 could be issued. The buildings of the company were destroyed by the bombing raids on Leipzig in 1943/44. In 1953 the place of business was moved to Mannheim in West Germany, the 9th edition, now from Mannheim, entitled Meyers Enzyklopädisches Lexikon in 25 Bänden, appeared in 1971-79. Just like the very first, this final Mannheim edition was the most comprehensive German encyclopaedia of the century, from Leipzig came Meyers Neues Lexikon, embedded in the Marxist ideology. In 1984, the Bibliographisches Institut Mannheim amalgamated with its biggest competitor in field of reference books F. A. Brockhaus of Wiesbaden, in a so-called Elefantenhochzeit
11.
Accumulator (computing)
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In a computers central processing unit, an accumulator is a register in which intermediate arithmetic and logic results are stored. Without a register like an accumulator, it would be necessary to write the result of each calculation to main memory, perhaps only to be read right back again for use in the next operation. Access to main memory is slower than access to a register like the accumulator because the used for the large main memory is slower than that used for a register. Early electronic computer systems were often split into two groups, those with accumulators and those without, modern computer systems often have multiple general purpose registers that operate as accumulators, and the term is no longer as common as it once was. However, a number of special-purpose processors still use a single accumulator for their work, mathematical operations often take place in a stepwise fashion, using the results from one operation as the input to the next. In early computers the number of hours would likely be held on a punch card, once the multiplication is complete, the result needs to be placed somewhere. On a drum machine this would likely be back to the drum, and then the very next operation has to read that value back in, which introduces another considerable delay. Accumulators dramatically improve performance in systems like these by providing an area where the results of one operation can be fed to the next one for little or no performance penalty. In the example above, the weekly pay would be calculated and placed in the accumulator. This removes one save and one read operation from the sequence, almost all early computers were accumulator machines with only the high-performance supercomputers having multiple registers. Then as mainframe systems gave way to microcomputers, accumulator architectures were again popular with the MOS6502 being a notable example, many 8-bit microcontrollers that are still popular as of 2014, such as the PICmicro and 8051, are accumulator-based machines. Modern CPUs are typically 2-operand or 3-operand machines, the additional operands specify which one of many general purpose registers are used as the source and destination for calculations. These CPUs are not considered accumulator machines, the characteristic which distinguishes one register as being the accumulator of a computer architecture is that the accumulator would be used as an implicit operand for arithmetic instructions. The accumulator is not identified in the instruction by a number, it is implicit in the instruction. Some architectures use a particular register as an accumulator in some instructions, any system that uses a single memory to store the result of multiple operations can be considered an accumulator. J. Presper Eckert refers to even the earliest adding machines of Gottfried Leibniz, should be a parallel storage organ which can receive a number and add it to the one already in it, which is also able to clear its contents and which can store what it contains. We will call such an organ an Accumulator and it is quite conventional in principle in past and present computing machines of the most varied types, e. g. desk multipliers, standard IBM counters, more modern relay machines, the ENIAC. Tradition, for example, uses two instructions called load accumulator from register/memory and store accumulator to register/memory, knuths model has many other instructions as well
12.
Worm drive
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A worm drive is a gear arrangement in which a worm meshes with a worm gear. The two elements are called the worm screw and worm wheel. The terminology is confused by imprecise use of the term worm gear to refer to the worm. Like other gear arrangements, a drive can reduce rotational speed or transmit higher torque. A worm is an example of a screw, one of the six simple machines, a gearbox designed using a worm and worm-wheel is considerably smaller than one made from plain spur gears, and has its drive axes at 90° to each other. With a single start worm, for each 360° turn of the worm, therefore, regardless of the worms size, the gear ratio is the size of the worm gear - to -1. Given a single start worm, a 20 tooth worm gear reduces the speed by the ratio of 20,1, with spur gears, a gear of 12 teeth must match with a 240 tooth gear to achieve the same 20,1 ratio. There are three different types of gears that can be used in a worm drive, the first are non-throated worm gears. These dont have a throat, or groove, machined around the circumference of either the worm or worm wheel, the second are single-throated worm gears, in which the worm wheel is throated. The final type are double-throated worm gears, which have both gears throated and this type of gearing can support the highest loading. An enveloping worm has one or more teeth and increases in diameter from its middle portion toward both ends, double-enveloping wormgearing comprises enveloping worms mated with fully enveloping wormgears. It is also known as globoidal wormgearing and this can be an advantage when it is desired to eliminate any possibility of the output driving the input. If a multistart worm is used then the ratio reduces accordingly, Worm gear configurations in which the gear cannot drive the worm are called self-locking. Whether a worm and gear is self-locking depends on the angle, the pressure angle. The use of a worm screw reduced this effect, further worm drive development led to recirculating ball bearings to reduce frictional forces, which transmitted some steering force to the wheel. This aids vehicle control and reduces wear that could cause difficulties in steering precisely, Worm drives are a compact means of substantially decreasing speed and increasing torque. Worm drives are used in presses, rolling mills, conveying engineering, mining machines, on rudders. In addition, milling heads and rotary tables are positioned using high-precision duplex worm drives with adjustable backlash, Worm gears are used on many lift/elevator and escalator-drive applications due to their compact size and the non-reversibility of the gear
13.
Multiplication algorithm
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A multiplication algorithm is an algorithm to multiply two numbers. Depending on the size of the numbers, different algorithms are in use, efficient multiplication algorithms have existed since the advent of the decimal system. The grid method is a method for multiple-digit multiplication that is often taught to pupils at primary school or elementary school level. It has been a part of the national primary-school mathematics curriculum in England. The calculation 34 ×13, for example, could be computed using the grid, followed by addition to obtain 442, either in a single sum and this calculation approach is also known as the partial products algorithm. Its essence is the calculation of the simple multiplications separately, with all addition being left to the final gathering-up stage, the grid method can in principle be applied to factors of any size, although the number of sub-products becomes cumbersome as the number of digits increases. It requires memorization of the table for single digits. This is the algorithm for multiplying larger numbers by hand in base 10. A person doing long multiplication on paper will write all the products and then add them together. This example uses long multiplication to multiply 23,958,233 by 5,830,23958233 ×5830 ———————————————0000000071874699191665864 +119791165 ———————————————139676498390 Below pseudocode describes the process of above multiplication. It keeps only one row to maintain sum which finally becomes the result, note that the += operator is used to denote sum to existing value and store operation for compactness. Let n be the number of digits in the two input numbers in base D. If the result must be kept in memory then the space complexity is trivially Θ, however, in certain applications, the entire result need not be kept in memory and instead the digits of the result can be streamed out as they are computed. This is the logarithm of the numbers being multiplied themselves. Note that operands themselves still need to be kept in memory, the method is based on the observation that each digit of the result can be computed from right to left with only knowing the carry from the previous step. The sum is at most D * n, and the carry to the column is at most D * n / D. Thus both these values can be stored in O digits, in pseudocode, the log-space algorithm is, Some chips implement this algorithm for various integer and floating-point sizes in computer hardware or in microcode. In arbitrary-precision arithmetic, its common to use long multiplication with the set to 2w
14.
Long division
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In arithmetic, long division is a standard division algorithm suitable for dividing multidigit numbers that is simple enough to perform by hand. It breaks down a division problem into a series of easier steps, as in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a result called the quotient. It enables computations involving arbitrarily large numbers to be performed by following a series of simple steps, the abbreviated form of long division is called short division, which is almost always used instead of long division when the divisor has only one digit. Chunking is a less-efficient form of long division which may be easier to understand, while related algorithms have existed since the 12th century AD, the specific algorithm in modern use was introduced by Henry Briggs c.1600 AD. In English-speaking countries, long division does not use the division slash ⟨∕⟩ or obelus ⟨÷⟩ signs, the divisor is separated from the dividend by a right parenthesis ⟨)⟩ or vertical bar ⟨|⟩, the dividend is separated from the quotient by a vinculum. The combination of two symbols is sometimes known as a long division symbol or division bracket. It developed in the 18th century from an earlier single-line notation separating the dividend from the quotient by a left parenthesis, the process is begun by dividing the left-most digit of the dividend by the divisor. The quotient becomes the first digit of the result, and the remainder is calculated and this remainder carries forward when the process is repeated on the following digit of the dividend. When all digits have been processed and no remainder is left, an example is shown below, representing the division of 500 by 4. The largest number that the divisor 4 can be multiplied by without exceeding 5 is 1, next the 4 under the 5 is subtracted from the 5 to get the remainder,1, which is placed under the 4 under the 5. This remainder 1 is necessarily smaller than the divisor 4, next the first as-yet unused digit in the dividend, in this case the first digit 0 after the 5, is copied directly underneath itself and next to the remainder 1, to form the number 10. This remainder 2 is necessarily smaller than the divisor 4, the next digit of the dividend is copied directly below itself and next to the remainder 2, to form 20. Then the largest number by which the divisor 4 can be multiplied without exceeding 20 is ascertained, then this new quotient digit 5 is multiplied by the divisor 4 to get 20, which is written at the bottom below the existing 20. Then 20 is subtracted from 20, yielding 0, which is written below the 20 and we know we are done now because two things are true, there are no more digits to bring down from the dividend, and the last subtraction result was 0. If the last remainder when we ran out of dividend digits had been something other than 0, there would have been two possible courses of action. Or, we could extend the dividend by writing it as, say,500.000. and continue the process, in order to get a decimal answer, as in the following example. This example also illustrates that, at the beginning of the process, since the first digit 1 is less than the divisor 4, the first step is instead performed on the first two digits 12. Similarly, if the divisor were 13, one would perform the first step on 127 rather than 12 or 1, find the location of all decimal points in the dividend n and divisor m
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Nth root
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A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred by using numbers, as in fourth root, twentieth root. For example,2 is a root of 4, since 22 =4. −2 is also a root of 4, since 2 =4. A real number or complex number has n roots of degree n. While the roots of 0 are not distinct, the n nth roots of any real or complex number are all distinct. If n is odd and x is real, one nth root is real and has the sign as x. Finally, if x is not real, then none of its nth roots is real. Roots are usually using the radical symbol or radix or √, with x or √ x denoting the square root, x 3 denoting the cube root, x 4 denoting the fourth root. In the expression x n, n is called the index, is the sign or radix. For example, −8 has three roots, −2,1 + i √3 and 1 − i √3. Out of these,1 + i √3 has the least argument,4 has two square roots,2 and −2, having arguments 0 and π respectively. So 2 is considered the root on account of having the lesser argument. An unresolved root, especially one using the symbol, is often referred to as a surd or a radical. Nth roots can also be defined for complex numbers, and the roots of 1 play an important role in higher mathematics. The origin of the root symbol √ is largely speculative, some sources imply that the symbol was first used by Arab mathematicians. One of those mathematicians was Abū al-Hasan ibn Alī al-Qalasādī, legend has it that it was taken from the Arabic letter ج, which is the first letter in the Arabic word جذر. However, many scholars, including Leonhard Euler, believe it originates from the letter r, the symbol was first seen in print without the vinculum in the year 1525 in Die Coss by Christoff Rudolff, a German mathematician
16.
Pascal's calculator
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Pascals calculator is a mechanical calculator invented by Blaise Pascal in the early 17th century. Pascal was led to develop a calculator by the laborious arithmetical calculations required by his fathers work as supervisor of taxes in Rouen and he designed the machine to add and subtract two numbers directly and to perform multiplication and division through repeated addition or subtraction. Pascals calculator was successful in the design of its carry mechanism, which adds 1 to 9 on one dial. His innovation made each digit independent of the state of the others, Pascal designed the machine in 1642, and after 50 prototypes, he presented it to the public in 1645, dedicating it to Pierre Séguier, then chancellor of France. Pascal built around twenty more machines during the decade, many of which improved on his original design. In 1649, King Louis XIV of France gave Pascal a royal privilege, gottfried Leibniz invented his Leibniz wheels after 1671 after trying to add an automatic multiplication feature to the Pascaline. In 1820, Thomas de Colmar designed his arithmometer and it is not at all clear whether he ever saw Leibnizs device, but he either re-invented it or utilised Leibnizs invention of the step drum. The arithmometer was the first mechanical calculator strong enough and reliable enough to be used daily in an office environment, Pascal was also influential in seeking to commercialise his machine. Nine Pascal calculators have been made, including the first surviving mechanical calculator from the 17th century, most of them are on display in European museums. Pascal has been called and celebrated as the inventor of the mechanical calculator, Pascal began to work on his calculator in 1642, when he was 19 years old. He had been assisting his father, who worked as a tax commissioner, Pascal received a Royal Privilege in 1649 that granted him exclusive rights to make and sell calculating machines in France. By 1654 he had sold about twenty machines, but the cost and complexity of the Pascaline was a barrier to further sales, by that time Pascal had moved on to the study of religion and philosophy, which gave us both the Lettres provinciales and the Pensées. The tercentenary celebration of Pascals invention of the mechanical calculator occurred during WWII when France was occupied by Germany and therefore the main celebration was held in London, the calculator had spoked metal wheel dials, with the digit 0 through 9 displayed around the circumference of each wheel. This displayed the number in the windows at the top of the calculator, then, one simply redialed the second number to be added, causing the sum of both numbers to appear in the accumulator. Each dial is associated with a display window located directly above it. The complement of this digit, in the base of the wheel, is displayed just above this digit. A horizontal bar hides either all the complement numbers when it is slid to the top and it thereby displays either the content of the accumulator or the complement of its value. Since the gears of the calculator rotated in one direction
17.
Pedometer
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A pedometer is a device, usually portable and electronic or electromechanical, that counts each step a person takes by detecting the motion of the persons hands or hips. Distance traveled can be measured directly by a GPS receiver, used originally by sports and physical fitness enthusiasts, pedometers are now becoming popular as an everyday exercise counter and motivator. Often worn on the belt and kept on all day, it can record how many steps the wearer has walked that day, Step counters can give encouragement to compete with oneself in getting fit and losing weight. A total of 10,000 steps per day, equivalent to 8 kilometres, is recommended by some to be the benchmark for an active lifestyle, thirty minutes of moderate walking are equivalent to 3, 000-4,000 steps as determined by a pedometer. Step counters are being integrated into a number of portable consumer electronic devices such as music players, smartphones. Pedometers can be a tool for people wanting to increase their physical activity. Various websites exist to allow people to track their progress, however, many also find entering their daily step count. Pedometers have been shown in studies to increase physical activity. A daily target of 10,000 steps was first proposed, the target has been recommended by the US Surgeon General and by the UK Department of Health. One criticism of the pedometer is that it not record intensity. Leonardo da Vinci envisioned a mechanical pedometer as a device with military applications, in 1780 Abraham-Louis Perrelet of Switzerland created the first pedometer, measuring the steps and distance while walking, it was based on a 1770 mechanism of his to power a self-winding watch. A mechanical pedometer obtained from France was introduced in the US by Thomas Jefferson and it is not known if he modified the design, although this pedometer is widely attributed to Jefferson, proof is difficult to obtain as he did not apply for patents on any of his inventions. In 1965 a pedometer called a manpo-kei was marketed in Japan by Y, the technology for a pedometer includes a mechanical sensor and software to count steps. Early forms used a switch to detect steps together with a simple counter. If one shakes these devices, one hears a lead ball sliding back and forth, today advanced step counters rely on MEMS inertial sensors and sophisticated software to detect steps. These MEMS sensors have either 1-, 2- or 3-axis detection of acceleration, the use of MEMS inertial sensors permits more accurate detection of steps and fewer false positives. The software technology used to interpret the output of the inertial sensor, the problem is compounded by the fact that in modern day-to-day life, such step-counters are expected to count accurately on locations where users frequently carry their devices. The accuracy of step counters varies widely between devices, typically, step counters are reasonably accurate at a walking pace on a flat surface if the device is placed in its optimal position
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Blaise Pascal
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Blaise Pascal was a French mathematician, physicist, inventor, writer and Christian philosopher. He was a prodigy who was educated by his father. Pascal also wrote in defence of the scientific method, in 1642, while still a teenager, he started some pioneering work on calculating machines. After three years of effort and 50 prototypes, he built 20 finished machines over the following 10 years, following Galileo Galilei and Torricelli, in 1647, he rebutted Aristotles followers who insisted that nature abhors a vacuum. Pascals results caused many disputes before being accepted, in 1646, he and his sister Jacqueline identified with the religious movement within Catholicism known by its detractors as Jansenism. Following a religious experience in late 1654, he began writing works on philosophy. His two most famous works date from this period, the Lettres provinciales and the Pensées, the set in the conflict between Jansenists and Jesuits. In that year, he wrote an important treatise on the arithmetical triangle. Between 1658 and 1659 he wrote on the cycloid and its use in calculating the volume of solids, Pascal had poor health, especially after the age of 18, and he died just two months after his 39th birthday. Pascal was born in Clermont-Ferrand, which is in Frances Auvergne region and he lost his mother, Antoinette Begon, at the age of three. His father, Étienne Pascal, who also had an interest in science and mathematics, was a local judge, Pascal had two sisters, the younger Jacqueline and the elder Gilberte. In 1631, five years after the death of his wife, the newly arrived family soon hired Louise Delfault, a maid who eventually became an instrumental member of the family. Étienne, who never remarried, decided that he alone would educate his children, for they all showed extraordinary intellectual ability, the young Pascal showed an amazing aptitude for mathematics and science. Particularly of interest to Pascal was a work of Desargues on conic sections and it states that if a hexagon is inscribed in a circle then the three intersection points of opposite sides lie on a line. Pascals work was so precocious that Descartes was convinced that Pascals father had written it, in France at that time offices and positions could be—and were—bought and sold. In 1631 Étienne sold his position as president of the Cour des Aides for 65,665 livres. The money was invested in a government bond which provided, if not a lavish, then certainly a comfortable income which allowed the Pascal family to move to, but in 1638 Richelieu, desperate for money to carry on the Thirty Years War, defaulted on the governments bonds. Suddenly Étienne Pascals worth had dropped from nearly 66,000 livres to less than 7,300 and it was only when Jacqueline performed well in a childrens play with Richelieu in attendance that Étienne was pardoned
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Royal Society
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Founded in November 1660, it was granted a royal charter by King Charles II as The Royal Society. The society is governed by its Council, which is chaired by the Societys President, according to a set of statutes and standing orders. The members of Council and the President are elected from and by its Fellows, the members of the society. As of 2016, there are about 1,600 fellows, allowed to use the postnominal title FRS, there are also royal fellows, honorary fellows and foreign members, the last of which are allowed to use the postnominal title ForMemRS. The Royal Society President is Venkatraman Ramakrishnan, who took up the post on 30 November 2015, since 1967, the society has been based at 6–9 Carlton House Terrace, a Grade I listed building in central London which was previously used by the Embassy of Germany, London. The Royal Society started from groups of physicians and natural philosophers, meeting at variety of locations and they were influenced by the new science, as promoted by Francis Bacon in his New Atlantis, from approximately 1645 onwards. A group known as The Philosophical Society of Oxford was run under a set of rules still retained by the Bodleian Library, after the English Restoration, there were regular meetings at Gresham College. It is widely held that these groups were the inspiration for the foundation of the Royal Society, I will not say, that Mr Oldenburg did rather inspire the French to follow the English, or, at least, did help them, and hinder us. But tis well known who were the men that began and promoted that design. This initial royal favour has continued and, since then, every monarch has been the patron of the society, the societys early meetings included experiments performed first by Hooke and then by Denis Papin, who was appointed in 1684. These experiments varied in their area, and were both important in some cases and trivial in others. The Society returned to Gresham in 1673, there had been an attempt in 1667 to establish a permanent college for the society. Michael Hunter argues that this was influenced by Solomons House in Bacons New Atlantis and, to a lesser extent, by J. V. The first proposal was given by John Evelyn to Robert Boyle in a letter dated 3 September 1659, he suggested a scheme, with apartments for members. The societys ideas were simpler and only included residences for a handful of staff and these plans were progressing by November 1667, but never came to anything, given the lack of contributions from members and the unrealised—perhaps unrealistic—aspirations of the society. During the 18th century, the gusto that had characterised the early years of the society faded, with a number of scientific greats compared to other periods. The pointed lightning conductor had been invented by Benjamin Franklin in 1749, during the same time period, it became customary to appoint society fellows to serve on government committees where science was concerned, something that still continues. The 18th century featured remedies to many of the early problems
20.
Rotary dial
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A rotary dial is a component of a telephone or a telephone switchboard that implements a signaling technology in telecommunications known as pulse dialing. It is used when initiating a call to transmit the destination telephone number to a telephone exchange. When released at the stop, the wheel returns to its home position by spring action at a speed regulated by a governor device. Each of the ten digits are encoded in sequences of up to ten pulses, for this reason, the method is sometimes called decadic dialling. The first patent for a dial is due to Almon Brown Strowger as U. S. Patent 486,909, but the known form with holes in the finger wheel was not introduced until ca. Touch-tone technology primarily used a keypad in form of an array of push-buttons for dialing. From as early as 1836 onward, various suggestions and inventions of dials for sending telegraph signals were reported, after the first commercial telephone exchange was installed in 1878, the need for an automated, user-controlled method of directing a telephone call became apparent. Addressing the technical shortcomings, Almon Brown Strowger invented a dial in 1891. Most inventions involved costly, intricate mechanisms and required the user to perform complex manipulations, the original dials required complex operational sequences. A workable, albeit error-prone, system was invented by the Automatic Electric Company using three push-buttons on the telephone and these buttons represented the hundreds, tens, and single units of a telephone number. When calling the subscriber number 163, for example, the user had to push the button once, followed by six presses of the tens button. In 1896, this system was supplanted by an automatic contact-making machine, further development continued during the 1890s and the early 1900s in conjunction with improvements in switching technology. Almon Brown Strowger was the first to file a patent for a dial on December 21,1891. In the 1950s, plastic materials were introduced in dial construction, replacing metal which was heavier, despite their lack of modern features, rotary phones occasionally find special uses. They are also retained for authenticity in historic properties such as the U. S. Route 66 Blue Swallow Motel which date from an era of named exchanges, a rotary dial typically features a circular construction. The shaft that actuates the mechanical switching mechanism is driven by the finger wheel, the finger wheel may be transparent or opaque permitting the viewing of the face plate, or number plate below, either in whole, or only showing the number assignment for each finger hole. The faceplate is printed with numbers, and sometimes letters, corresponding to each finger hole, a curved device called a finger stop sits above the dial at approximately the 4 oclock position
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Multiplication
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Multiplication is one of the four elementary, mathematical operations of arithmetic, with the others being addition, subtraction and division. Multiplication can also be visualized as counting objects arranged in a rectangle or as finding the area of a rectangle whose sides have given lengths, the area of a rectangle does not depend on which side is measured first, which illustrates the commutative property. The product of two measurements is a new type of measurement, for multiplying the lengths of the two sides of a rectangle gives its area, this is the subject of dimensional analysis. The inverse operation of multiplication is division, for example, since 4 multiplied by 3 equals 12, then 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number, Multiplication is also defined for other types of numbers, such as complex numbers, and more abstract constructs, like matrices. For these more abstract constructs, the order that the operands are multiplied sometimes does matter, a listing of the many different kinds of products that are used in mathematics is given in the product page. In arithmetic, multiplication is often written using the sign × between the terms, that is, in infix notation, there are other mathematical notations for multiplication, Multiplication is also denoted by dot signs, usually a middle-position dot,5 ⋅2 or 5. 2 The middle dot notation, encoded in Unicode as U+22C5 ⋅ dot operator, is standard in the United States, the United Kingdom, when the dot operator character is not accessible, the interpunct is used. In other countries use a comma as a decimal mark. In algebra, multiplication involving variables is often written as a juxtaposition, the notation can also be used for quantities that are surrounded by parentheses. In matrix multiplication, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a product of two vectors, yielding a vector as the result, while the dot denotes taking the dot product of two vectors, resulting in a scalar. In computer programming, the asterisk is still the most common notation and this is due to the fact that most computers historically were limited to small character sets that lacked a multiplication sign, while the asterisk appeared on every keyboard. This usage originated in the FORTRAN programming language, the numbers to be multiplied are generally called the factors. The number to be multiplied is called the multiplicand, while the number of times the multiplicand is to be multiplied comes from the multiplier. Usually the multiplier is placed first and the multiplicand is placed second, however sometimes the first factor is the multiplicand, additionally, there are some sources in which the term multiplicand is regarded as a synonym for factor. In algebra, a number that is the multiplier of a variable or expression is called a coefficient, the result of a multiplication is called a product. A product of integers is a multiple of each factor, for example,15 is the product of 3 and 5, and is both a multiple of 3 and a multiple of 5
22.
Division (mathematics)
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Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication. The division of two numbers is the process of calculating the number of times one number is contained within one another. For example, in the picture on the right, the 20 apples are divided into groups of five apples, Division can also be thought of as the process of evaluating a fraction, and fractional notation is commonly used to represent division. Division is the inverse of multiplication, if a × b = c, then a = c ÷ b, as long as b is not zero. Division by zero is undefined for the numbers and most other contexts, because if b =0, then a cannot be deduced from b and c. In some contexts, division by zero can be defined although to a limited extent, in division, the dividend is divided by the divisor to get a quotient. In the above example,20 is the dividend, five is the divisor, in some cases, the divisor may not be contained fully by the dividend, for example,10 ÷3 leaves a remainder of one, as 10 is not a multiple of three. Sometimes this remainder is added to the quotient as a fractional part, but in the context of integer division, where numbers have no fractional part, the remainder is kept separately or discarded. Besides dividing apples, division can be applied to other physical, Division has been defined in several contexts, such as for the real and complex numbers and for more abstract contexts such as for vector spaces and fields. Division is the most mentally difficult of the four operations of arithmetic. Teaching the objective concept of dividing integers introduces students to the arithmetic of fractions, unlike addition, subtraction, and multiplication, the set of all integers is not closed under division. Dividing two integers may result in a remainder, to complete the division of the remainder, the number system is extended to include fractions or rational numbers as they are more generally called. When students advance to algebra, the theory of division intuited from arithmetic naturally extends to algebraic division of variables, polynomials. Division is often shown in algebra and science by placing the dividend over the divisor with a line, also called a fraction bar. For example, a divided by b is written a b This can be read out loud as a divided by b, a fraction is a division expression where both dividend and divisor are integers, and there is no implication that the division must be evaluated further. A second way to show division is to use the obelus, common in arithmetic, in this manner, ISO 80000-2-9.6 states it should not be used. The obelus is also used alone to represent the operation itself. In some non-English-speaking cultures, a divided by b is written a, b and this notation was introduced in 1631 by William Oughtred in his Clavis Mathematicae and later popularized by Gottfried Wilhelm Leibniz
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Divisor
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In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some other integer to produce n. In this case one says also that n is a multiple of m, an integer n is divisible by another integer m if m is a divisor of n, this implies dividing n by m leaves no remainder. Under this definition, the statement m ∣0 holds for every m, as before, but with the additional constraint k ≠0. Under this definition, the statement m ∣0 does not hold for m ≠0, in the remainder of this article, which definition is applied is indicated where this is significant. Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4, they are 1,2,4, −1, −2, and −4,1 and −1 divide every integer. Every integer is a divisor of itself, every integer is a divisor of 0. Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd,1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a divisor is known as a non-trivial divisor. A non-zero integer with at least one divisor is known as a composite number, while the units −1 and 1. There are divisibility rules which allow one to recognize certain divisors of a number from the numbers digits, the generalization can be said to be the concept of divisibility in any integral domain. 7 is a divisor of 42 because 7 ×6 =42 and it can also be said that 42 is divisible by 7,42 is a multiple of 7,7 divides 42, or 7 is a factor of 42. The non-trivial divisors of 6 are 2, −2,3, the positive divisors of 42 are 1,2,3,6,7,14,21,42. 5 ∣0, because 5 ×0 =0, if a ∣ b and b ∣ a, then a = b or a = − b. If a ∣ b and a ∣ c, then a ∣ holds, however, if a ∣ b and c ∣ b, then ∣ b does not always hold. If a ∣ b c, and gcd =1, then a ∣ c, if p is a prime number and p ∣ a b then p ∣ a or p ∣ b. A positive divisor of n which is different from n is called a proper divisor or a part of n. A number that does not evenly divide n but leaves a remainder is called an aliquant part of n, an integer n >1 whose only proper divisor is 1 is called a prime number
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International Standard Book Number
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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
