1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
2.
Calculator
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An electronic calculator is a small, portable electronic device used to perform operations ranging from basic arithmetic to complex mathematics. The first solid state electronic calculator was created in the 1960s, building on the history of tools such as the abacus. It was developed in parallel with the computers of the day. The pocket sized devices became available in the 1970s, especially after the first microprocessor and they later became used commonly within the petroleum industry. Modern electronic calculators vary, from cheap, give-away, credit-card-sized models to sturdy desktop models with built-in printers and they became popular in the mid-1970s. By the end of decade, calculator prices had reduced to a point where a basic calculator was affordable to most. In addition to general purpose calculators, there are designed for specific markets. For example, there are scientific calculators which include trigonometric and statistical calculations, some calculators even have the ability to do computer algebra. Graphing calculators can be used to graph functions defined on the real line, as of 2016, basic calculators cost little, but the scientific and graphing models tend to cost more. In 1986, calculators still represented an estimated 41% of the worlds general-purpose hardware capacity to compute information, by 2007, this diminished to less than 0. 05%. Modern 2016 electronic calculators contain a keyboard with buttons for digits and arithmetical operations, most basic calculators assign only one digit or operation on each button, however, in more specific calculators, a button can perform multi-function working with key combinations. Large-sized figures and comma separators are used to improve readability. Various symbols for function commands may also be shown on the display, fractions such as 1⁄3 are displayed as decimal approximations, for example rounded to 0.33333333. Also, some fractions can be difficult to recognize in decimal form, as a result, Calculators also have the ability to store numbers into computer memory. Basic types of these only one number at a time. The variables can also be used for constructing formulas, some models have the ability to extend memory capacity to store more numbers, the extended memory address is termed an array index. Power sources of calculators are, batteries, solar cells or mains electricity, some models even have no turn-off button but they provide some way to put off. Crank-powered calculators were also common in the computer era
3.
Mathematical table
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Tables of logarithms and trigonometric functions were common in math and science textbooks. Specialized tables were published for such as astronomy, celestial navigation. However, this answer is only accurate to four decimal places, if one wanted greater accuracy, one could interpolate linearly as follows, From the Bernegger table, sin =0.9666746 sin =0.9666001 The difference between these values is 0.0000745. For tables with greater precision, higher order interpolation may be needed to get full accuracy, to understand the importance of accuracy in applications like navigation note that at sea level one minute of arc along the Earths equator or a meridian equals approximately one nautical mile. The first tables of trigonometric functions known to be made were by Hipparchus and Menelaus, along with the surviving table of Ptolemy, they were all tables of chords and not of half-chords, i. e. the sine function. The table produced by the Indian mathematician Āryabhaṭa is considered the first sine table ever constructed, Āryabhaṭas table remained the standard sine table of ancient India. Tables of common logarithms were used until the invention of computers and electronic calculators to do rapid multiplications, divisions and this was motivated mainly by errors in logarithmic tables made by the human computers of the time. Early digital computers were developed during World War II in part to produce specialized mathematical tables for aiming artillery, from 1972 onwards, with the launch and growing use of scientific calculators, most mathematical tables went out of use. In essence, one trades computing speed for the memory space required to store the tables. But same mantissa could be used for less than one by offsetting the characteristic. Thus a single table of common logarithms can be used for the range of positive decimal numbers. See common logarithm for details on the use of characteristics and mantissas, michael Stifel published Arithmetica integra in Nuremberg in 1544 which contains a table of integers and powers of 2 that has been considered an early version of a logarithmic table. The method of logarithms was publicly propounded by John Napier in 1614, the book contained fifty-seven pages of explanatory matter and ninety pages of tables related to natural logarithms. The English mathematician Henry Briggs visited Napier in 1615, and proposed a re-scaling of Napiers logarithms to form what is now known as the common or base-10 logarithms
4.
Slide rule
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The slide rule, also known colloquially in the United States as a slipstick, is a mechanical analog computer. The slide rule is used primarily for multiplication and division, and also for functions such as exponents, roots, logarithms and trigonometry, though similar in name and appearance to a standard ruler, the slide rule is not ordinarily used for measuring length or drawing straight lines. Slide rules exist in a range of styles and generally appear in a linear or circular form with a standardized set of markings essential to performing mathematical computations. Slide rules manufactured for specialized fields such as aviation or finance typically feature additional scales that aid in calculations common to those fields, at its simplest, each number to be multiplied is represented by a length on a sliding ruler. As the rulers each have a scale, it is possible to align them to read the sum of the logarithms. The Reverend William Oughtred and others developed the rule in the 17th century based on the emerging work on logarithms by John Napier. Before the advent of the calculator, it was the most commonly used calculation tool in science. In its most basic form, the slide rule uses two logarithmic scales to allow rapid multiplication and division of numbers and these common operations can be time-consuming and error-prone when done on paper. More elaborate slide rules allow other calculations, such as roots, exponentials, logarithms. Scales may be grouped in decades, which are numbers ranging from 1 to 10. Thus single decade scales C and D range from 1 to 10 across the width of the slide rule while double decade scales A and B range from 1 to 100 over the width of the slide rule. Numbers aligned with the marks give the value of the product, quotient. The user determines the location of the point in the result. Scientific notation is used to track the decimal point in more formal calculations, addition and subtraction steps in a calculation are generally done mentally or on paper, not on the slide rule. Most slide rules consist of three strips of the same length, aligned in parallel and interlocked so that the central strip can be moved lengthwise relative to the other two. The outer two strips are fixed so that their relative positions do not change. Some slide rules have scales on both sides of the rule and slide strip, others on one side of the outer strips and both sides of the slide strip, still others on one side only. A sliding cursor with a vertical alignment line is used to find corresponding points on scales that are not adjacent to other or
5.
Trigonometric functions
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In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples, in geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles
6.
John Napier
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John Napier of Merchiston, also signed as Neper, Nepair, nicknamed Marvellous Merchiston) was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8th Laird of Merchiston and his Latinized name was Ioannes Neper. John Napier is best known as the discoverer of logarithms and he also invented the so-called Napiers bones and made common the use of the decimal point in arithmetic and mathematics. Napiers birthplace, Merchiston Tower in Edinburgh, is now part of the facilities of Edinburgh Napier University, Napier died from the effects of gout at home at Merchiston Castle and his remains were buried in the kirkyard of St Giles. Following the loss of the kirkyard there to build Parliament House, archibald Napier was 16 years old when John Napier was born. As was the practice for members of the nobility at that time, he was privately tutored and did not have formal education until he was 13. He did not stay in very long. It is believed that he dropped out of school in Scotland, in 1571, Napier, aged 21, returned to Scotland, and bought a castle at Gartness in 1574. On the death of his father in 1608, Napier and his family moved into Merchiston Castle in Edinburgh and he died at the age of 67. In such conditions, it is surprising that many mathematicians were acutely aware of the issues of computation and were dedicated to relieving practitioners of the calculation burden. In particular, the Scottish mathematician John Napier was famous for his devices to assist with computation and he invented a well-known mathematical artifact, the ingenious numbering rods more quaintly known as “Napiers bones, ” that offered mechanical means for facilitating computation. He appreciated that, for the most part, practitioners who had laborious computations generally did them in the context of trigonometry, therefore, as well as developing the logarithmic relation, Napier set it in a trigonometric context so it would be even more relevant. His work, Mirifici Logarithmorum Canonis Descriptio contained fifty-seven pages of explanatory matter, the book also has an excellent discussion of theorems in spherical trigonometry, usually known as Napiers Rules of Circular Parts. Modern English translations of both Napiers books on logarithms, and their description can be found on the web, as well as a discussion of Napiers Bones and his invention of logarithms was quickly taken up at Gresham College, and prominent English mathematician Henry Briggs visited Napier in 1615. Among the matters discussed were a re-scaling of Napiers logarithms. Napier delegated to Briggs the computation of a revised table, the computational advance available via logarithms, the converse of powered numbers or exponential notation, was such that it made calculations by hand much quicker. The way was opened to later scientific advances, in astronomy, dynamics and he improved Simon Stevins decimal notation. Lattice multiplication, used by Fibonacci, was more convenient by his introduction of Napiers bones
7.
Leonhard Euler
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He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field and he spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Eulers influence on mathematics, Read Euler, read Euler, Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastors daughter. He had two sisters, Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Paul Euler was a friend of the Bernoulli family, Johann Bernoulli was then regarded as Europes foremost mathematician, and would eventually be the most important influence on young Leonhard. Eulers formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, during that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupils incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono, at that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition, Pierre Bouguer, who became known as the father of naval architecture, won and Euler took second place. Euler later won this annual prize twelve times, around this time Johann Bernoullis two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727 and he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian and settled life in Saint Petersburg. He also took on a job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia, as a result, it was made especially attractive to foreign scholars like Euler
8.
Bibliotheca Teubneriana
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The Bibliotheca Teubneriana, or Teubner editions of Greek and Latin texts, comprise the most thorough modern collection ever published of ancient Greco-Roman literature. Today, the only comparable publishing ventures, producing authoritative scholarly reference editions of ancient authors, are the Oxford Classical Texts. In 1811, Benedictus Gotthelf Teubner refounded in his own name an operation he had directed since 1806. The volumes of the Bibliotheca Teubneriana began to appear in 1849, although today Teubner editions are relatively expensive, they were originally introduced to fill the need, then unmet, for low-priced but high-quality editions. Prior to the introduction of the Teubner series, accurate editions of antique authors could only be purchased by libraries, students and other individuals of modest means had to rely on editions which were affordable but also filled with errors. To satisfy the need for accurate and affordable editions Teubner introduced the Bibliotheca Teubneriana, in the 19th century, Teubner offered both affordable editiones maiores for scholars, and low-priced editiones minores for students. Eventually, editiones minores were dropped from the series and Teubner began to offer only scholarly reference editions of ancient authors, during the period between the end of World War II and German reunification, the publishing house of B. G. Teubner split into two firms, Teubner KG, later BSB B. G. Teubner Verlagsgesellschaft, in Leipzig in East Germany, both offered volumes in the Bibliotheca Teubneriana. After the fall of the Berlin wall and the reunification of Germany, Teubner was also reunited and subsequently consolidated its headquarters at Wiesbaden. Teubner Verlag announced their intention to concentrate on scientific and technical publishing, all their Classical Studies titles, including the Biblotheca Teubneriana, were sold to K. G. Saur, a publisher based in Munich. Although new volumes began to appear with the imprint in aedibus K. G. Saur, in 2006, the publishing firm of Walter de Gruyter acquired K. G. Saur and their entire publishing range, including the Bibliotheca Teubneriana. Since January 2007, the Bibliotheca Teubneriana is being published by Walter de Gruyter GmbH & Co. As of 1 May 2007, the new North American distributor of titles from the Bibliotheca Teubneriana is Walter de Gruyter, while the typography of the Greek Teubners has been subject to innovations over the years, an overview of the whole series shows a great deal of consistency. The old-fashioned, cursive font used in most of the volumes is instantly recognized by classicists. This type was in use at least from the 1870s to the 1970s. Teubner used a type, designed to match the original cursive type. In the example shown, the type is still used in the critical apparatus. In other editions, this font is used throughout
9.
Logarithm
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In mathematics, the logarithm is the inverse operation to exponentiation. That means the logarithm of a number is the exponent to which another fixed number, in simple cases the logarithm counts factors in multiplication. For example, the base 10 logarithm of 1000 is 3, the logarithm of x to base b, denoted logb, is the unique real number y such that by = x. For example, log2 =6, as 64 =26, the logarithm to base 10 is called the common logarithm and has many applications in science and engineering. The natural logarithm has the e as its base, its use is widespread in mathematics and physics. The binary logarithm uses base 2 and is used in computer science. Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations and they were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the function in the 18th century. Logarithmic scales reduce wide-ranging quantities to tiny scopes, for example, the decibel is a unit quantifying signal power log-ratios and amplitude log-ratios. In chemistry, pH is a measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and they describe musical intervals, appear in formulas counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting. In the same way as the logarithm reverses exponentiation, the logarithm is the inverse function of the exponential function applied to complex numbers. The discrete logarithm is another variant, it has uses in public-key cryptography, the idea of logarithms is to reverse the operation of exponentiation, that is, raising a number to a power. For example, the power of 2 is 8, because 8 is the product of three factors of 2,23 =2 ×2 ×2 =8. It follows that the logarithm of 8 with respect to base 2 is 3, the third power of some number b is the product of three factors equal to b. More generally, raising b to the power, where n is a natural number, is done by multiplying n factors equal to b. The n-th power of b is written bn, so that b n = b × b × ⋯ × b ⏟ n factors, exponentiation may be extended to by, where b is a positive number and the exponent y is any real number. For example, b−1 is the reciprocal of b, that is, the logarithm of a positive real number x with respect to base b, a positive real number not equal to 1, is the exponent by which b must be raised to yield x
10.
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
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Edinburgh
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Edinburgh is the capital city of Scotland and one of its 32 local government council areas. Located in Lothian on the Firth of Forths southern shore, it is Scotlands second most populous city and the seventh most populous in the United Kingdom. The 2014 official population estimates are 464,990 for the city of Edinburgh,492,680 for the authority area. Recognised as the capital of Scotland since at least the 15th century, Edinburgh is home to the Scottish Parliament and it is the largest financial centre in the UK after London. Historically part of Midlothian, the city has long been a centre of education, particularly in the fields of medicine, Scots law, literature, the sciences and engineering. The University of Edinburgh, founded in 1582 and now one of four in the city, was placed 17th in the QS World University Rankings in 2013 and 2014. The city is famous for the Edinburgh International Festival and the Fringe. The citys historical and cultural attractions have made it the United Kingdoms second most popular tourist destination after London, attracting over one million overseas visitors each year. Historic sites in Edinburgh include Edinburgh Castle, Holyrood Palace, the churches of St. Giles, Greyfriars and the Canongate, Edinburghs Old Town and New Town together are listed as a UNESCO World Heritage Site, which has been managed by Edinburgh World Heritage since 1999. It appears to derive from the place name Eidyn mentioned in the Old Welsh epic poem Y Gododdin, the poem names Din Eidyn as a hill fort in the territory of the Gododdin. The Celtic element din was dropped and replaced by the Old English burh, the first documentary evidence of the medieval burgh is a royal charter, c. 1124–1127, by King David I granting a toft in burgo meo de Edenesburg to the Priory of Dunfermline. In modern Gaelic, the city is called Dùn Èideann, the earliest known human habitation in the Edinburgh area was at Cramond, where evidence was found of a Mesolithic camp site dated to c.8500 BC. Traces of later Bronze Age and Iron Age settlements have found on Castle Rock, Arthurs Seat, Craiglockhart Hill. When the Romans arrived in Lothian at the end of the 1st century AD, at some point before the 7th century AD, the Gododdin, who were presumably descendants of the Votadini, built the hill fort of Din Eidyn or Etin. Although its location has not been identified, it likely they would have chosen a commanding position like the Castle Rock, Arthurs Seat. In 638, the Gododdin stronghold was besieged by forces loyal to King Oswald of Northumbria and it thenceforth remained under their jurisdiction. The royal burgh was founded by King David I in the early 12th century on land belonging to the Crown, in 1638, King Charles Is attempt to introduce Anglican church forms in Scotland encountered stiff Presbyterian opposition culminating in the conflicts of the Wars of the Three Kingdoms. In the 17th century, Edinburghs boundaries were defined by the citys defensive town walls
12.
Floating-point arithmetic
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In computing, floating-point arithmetic is arithmetic using formulaic representation of real numbers as an approximation so as to support a trade-off between range and precision. A number is, in general, represented approximately to a number of significant digits and scaled using an exponent in some fixed base. For example,1.2345 =12345 ⏟ significand ×10 ⏟ base −4 ⏞ exponent, the term floating point refers to the fact that a numbers radix point can float, that is, it can be placed anywhere relative to the significant digits of the number. This position is indicated as the exponent component, and thus the floating-point representation can be thought of as a kind of scientific notation. The result of dynamic range is that the numbers that can be represented are not uniformly spaced. Over the years, a variety of floating-point representations have been used in computers, however, since the 1990s, the most commonly encountered representation is that defined by the IEEE754 Standard. A floating-point unit is a part of a computer system designed to carry out operations on floating point numbers. A number representation specifies some way of encoding a number, usually as a string of digits, there are several mechanisms by which strings of digits can represent numbers. In common mathematical notation, the string can be of any length. If the radix point is not specified, then the string implicitly represents an integer, in fixed-point systems, a position in the string is specified for the radix point. So a fixed-point scheme might be to use a string of 8 decimal digits with the point in the middle. The scaling factor, as a power of ten, is then indicated separately at the end of the number, floating-point representation is similar in concept to scientific notation. Logically, a floating-point number consists of, A signed digit string of a length in a given base. This digit string is referred to as the significand, mantissa, the length of the significand determines the precision to which numbers can be represented. The radix point position is assumed always to be somewhere within the significand—often just after or just before the most significant digit and this article generally follows the convention that the radix point is set just after the most significant digit. A signed integer exponent, which modifies the magnitude of the number, using base-10 as an example, the number 7005152853504700000♠152853.5047, which has ten decimal digits of precision, is represented as the significand 1528535047 together with 5 as the exponent. In storing such a number, the base need not be stored, since it will be the same for the range of supported numbers. Symbolically, this value is, s b p −1 × b e, where s is the significand, p is the precision, b is the base