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.
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
3.
Decimal
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This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal representations and other fractions have repeating decimal representations, Decimal notation is the writing of numbers in a base-ten numeral system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Roman numerals have symbols for the decimal powers and secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100, Chinese numerals have symbols for 1–9, and additional symbols for powers of ten, which in modern usage reach 1072. Positional decimal systems include a zero and use symbols for the ten values to represent any number, positional notation uses positions for each power of ten, units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier multiplied with that position has a value ten times that of the position to its right. There were at least two independent sources of positional decimal systems in ancient civilization, the Chinese counting rod system. Ten is the number which is the count of fingers and thumbs on both hands, the English word digit as well as its translation in many languages is also the anatomical term for fingers and toes. In English, decimal means tenth, decimate means reduce by a tenth, however, the symbols used in different areas are not identical, for instance, Western Arabic numerals differ from the forms used by other Arab cultures. A decimal fraction is a fraction the denominator of which is a power of ten. g, Decimal fractions 8/10, 1489/100, 24/100000, and 58900/10000 are expressed in decimal notation as 0.8,14.89,0.00024,5.8900 respectively. In English-speaking, some Latin American and many Asian countries, a period or raised period is used as the separator, in many other countries, particularly in Europe. The integer part, or integral part of a number is the part to the left of the decimal separator. The part from the separator to the right is the fractional part. It is usual for a number that consists only of a fractional part to have a leading zero in its notation. Any rational number with a denominator whose only prime factors are 2 and/or 5 may be expressed as a decimal fraction and has a finite decimal expansion. 1/2 =0.5 1/20 =0.05 1/5 =0.2 1/50 =0.02 1/4 =0.25 1/40 =0.025 1/25 =0.04 1/8 =0.125 1/125 =0.008 1/10 =0
4.
Warring States period
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The Warring States Period derives its name from the Record of the Warring States, a work compiled early in the Han dynasty. The political geography of the era was dominated by the Seven Warring States, namely, Qin, The State of Qin was in the far west, with its core in the Wei River Valley and Guanzhong. This geographical position offered protection from the states of the Central Plains, the Three Jins, Northeast of Qin, on the Shanxi plateau, were the three successor states of Jin. These were, Han, south, along the Yellow River, Zhao, the northernmost of the three. Qi, located in the east of China, centred on the Shandong Peninsula, described as east of Mount Tai, Chu, located in the south of China, with its core territory around the valleys of the Han River and, later, the Yangtze River. Yan, located in the northeast, centred on modern-day Beijing, late in the period Yan pushed northeast and began to occupy the Liaodong Peninsula Besides these seven major states, some minor states also survived into the period. Yue, On the southeast coast near Shanghai was the State of Yue, Sichuan, In the far southwest were the States of Ba and Shu. These were non-Zhou states that were conquered by Qin late in the period, in the Central Plains comprising much of modern-day Henan Province, many smaller city states survived as satellites of the larger states, though they were eventually to be absorbed as well. Zhongshan, Between the states of Zhao and Yan was the state of Zhongshan, the Spring and Autumn period was initiated by the eastward flight of the Zhou court. There is no one single incident or starting point for the Warring States era, some proposed starting points are as follows,481 BC, Proposed by Song-era historian Lü Zuqian, since it is the end of the Spring and Autumn Annals. 476–475 BC, The author, Sima Qian, of Records of the Grand Historian who chose the year of King Yuan of Zhou. 403 BC, The year when Han, Zhao and Wei were officially recognised as states by the Zhou court, author Sima Guang of Zizhi Tongjian tells us that the symbol of eroded Zhou authority should be taken as the start of the Warring States era. The Spring and Autumn period led to a few states gaining power at the expense of many others, during the Warring States period, many rulers claimed the Mandate of Heaven to justify their conquest of other states and spread their influence. Other major states also existed, such as Wu and Yue in the southeast, the last decades of the Spring and Autumn era were marked by increased stability, as the result of peace negotiations between Jin and Chu which established their respective spheres of influence. This situation ended with the partition of Jin, whereby the state was divided between the houses of Han, Zhao and Wei, and thus enabled the creation of the seven major warring states. This allowed other clans to gain fiefs and military authority, and decades of struggle led to the establishment of four major families. The Battle of Jinyang saw the allied Han, Zhao and Wei destroy the Zhi family, with this, they became the de facto rulers of most of Jins territory, though this situation would not be officially recognised until half a century later. The Jin division created a vacuum that enabled during the first 50 years expansion of Chu and Yue northward
5.
Pythagoras
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Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and the putative founder of the movement called Pythagoreanism. Most of the information about Pythagoras was written centuries after he lived. He was born on the island of Samos, and travelled, visiting Egypt and Greece, around 530 BC, he moved to Croton, in Magna Graecia, and there established some kind of school or guild. In 520 BC, he returned to Samos, Pythagoras made influential contributions to philosophy and religion in the late 6th century BC. He is often revered as a mathematician and scientist and is best known for the Pythagorean theorem which bears his name. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues, some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important. It was said that he was the first man to himself a philosopher, or lover of wisdom, and Pythagorean ideas exercised a marked influence on Plato. Burkert states that Aristoxenus and Dicaearchus are the most important accounts, Aristotle had written a separate work On the Pythagoreans, which is no longer extant. However, the Protrepticus possibly contains parts of On the Pythagoreans and his disciples Dicaearchus, Aristoxenus, and Heraclides Ponticus had written on the same subject. These writers, late as they are, were among the best sources from whom Porphyry and Iamblichus drew, while adding some legendary accounts. Herodotus, Isocrates, and other writers agree that Pythagoras was the son of Mnesarchus and born on the Greek island of Samos. His father is said to have been a gem-engraver or a wealthy merchant, a late source gives his mothers name as Pythais. As to the date of his birth, Aristoxenus stated that Pythagoras left Samos in the reign of Polycrates, at the age of 40, around 530 BC he arrived in the Greek colony of Croton in what was then Magna Graecia. There he founded his own school the members of which he engaged to a disciplined. He furthermore aquired some political influence, on Greeks and non-Greeks of the region, following a conflict with the neighbouring colony of Sybaris, internal discord drove most of the Pythagoreans out of Croton. Pythagoras left the city before the outbreak of civil unrest and moved to Metapontum, after his death, his house was transformed into a sanctuary of Demeter, out of veneration for the philosopher, by the local population. In ancient sources there was disagreement and inconsistency about the late life of Pythagoras. His tomb was shown at Metapontum in the time of Cicero, according to Walter Burkert, Most obvious is the contradiction between Aristoxenus and Dicaearchus, regarding the catastrophe that overwhelmed the Pythagorean society
6.
British Museum
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The British Museum is dedicated to human history, art and culture, and is located in the Bloomsbury area of London. The British Museum was established in 1753, largely based on the collections of the physician, the museum first opened to the public on 15 January 1759, in Montagu House, on the site of the current building. Although today principally a museum of art objects and antiquities. Its foundations lie in the will of the Irish-born British physician, on 7 June 1753, King George II gave his formal assent to the Act of Parliament which established the British Museum. They were joined in 1757 by the Old Royal Library, now the Royal manuscripts, together these four foundation collections included many of the most treasured books now in the British Library including the Lindisfarne Gospels and the sole surviving copy of Beowulf. The British Museum was the first of a new kind of museum – national, belonging to neither church nor king, freely open to the public, sloanes collection, while including a vast miscellany of objects, tended to reflect his scientific interests. The addition of the Cotton and Harley manuscripts introduced a literary, the body of trustees decided on a converted 17th-century mansion, Montagu House, as a location for the museum, which it bought from the Montagu family for £20,000. The Trustees rejected Buckingham House, on the now occupied by Buckingham Palace, on the grounds of cost. With the acquisition of Montagu House the first exhibition galleries and reading room for scholars opened on 15 January 1759. During the few years after its foundation the British Museum received several gifts, including the Thomason Collection of Civil War Tracts. A list of donations to the Museum, dated 31 January 1784, in the early 19th century the foundations for the extensive collection of sculpture began to be laid and Greek, Roman and Egyptian artefacts dominated the antiquities displays. Gifts and purchases from Henry Salt, British consul general in Egypt, beginning with the Colossal bust of Ramesses II in 1818, many Greek sculptures followed, notably the first purpose-built exhibition space, the Charles Towneley collection, much of it Roman Sculpture, in 1805. In 1816 these masterpieces of art, were acquired by The British Museum by Act of Parliament. The collections were supplemented by the Bassae frieze from Phigaleia, Greece in 1815, the Ancient Near Eastern collection also had its beginnings in 1825 with the purchase of Assyrian and Babylonian antiquities from the widow of Claudius James Rich. The neoclassical architect, Sir Robert Smirke, was asked to draw up plans for an extension to the Museum. For the reception of the Royal Library, and a Picture Gallery over it, and put forward plans for todays quadrangular building, much of which can be seen today. The dilapidated Old Montagu House was demolished and work on the Kings Library Gallery began in 1823, the extension, the East Wing, was completed by 1831. The Museum became a site as Sir Robert Smirkes grand neo-classical building gradually arose
7.
Roman numerals
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The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet, Roman numerals, as used today, are based on seven symbols, The use of Roman numerals continued long after the decline of the Roman Empire. The numbers 1 to 10 are usually expressed in Roman numerals as follows, I, II, III, IV, V, VI, VII, VIII, IX, Numbers are formed by combining symbols and adding the values, so II is two and XIII is thirteen. Symbols are placed left to right in order of value. Named after the year of its release,2014 as MMXIV, the year of the games of the XXII Olympic Winter Games The standard forms described above reflect typical modern usage rather than a universally accepted convention. Usage in ancient Rome varied greatly and remained inconsistent in medieval, Roman inscriptions, especially in official contexts, seem to show a preference for additive forms such as IIII and VIIII instead of subtractive forms such as IV and IX. Both methods appear in documents from the Roman era, even within the same document, double subtractives also occur, such as XIIX or even IIXX instead of XVIII. Sometimes V and L are not used, with such as IIIIII. Such variation and inconsistency continued through the period and into modern times. Clock faces that use Roman numerals normally show IIII for four o’clock but IX for nine o’clock, however, this is far from universal, for example, the clock on the Palace of Westminster in London uses IV. Similarly, at the beginning of the 20th century, different representations of 900 appeared in several inscribed dates. For instance,1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, although Roman numerals came to be written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used
8.
John Leslie (physicist)
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Sir John Leslie, FRSE KH was a Scottish mathematician and physicist best remembered for his research into heat. Leslie gave the first modern account of action in 1802 and froze water using an air-pump in 1810. In 1804, he experimented with radiant heat using a vessel filled with boiling water. One side of the cube is composed of polished metal. He showed that radiation was greatest from the side and negligible from the polished side. The apparatus is known as a Leslie cube, Leslie was born the son of Robert Leslie, a joiner and cabinetmaker, and his wife Anne Carstairs, in Largo in Fife. He received his education there and at Leven. In his thirteenth year, encouraged by friends who had even then remarked his aptitude for mathematical and physical science, on the completion of his course in 1784, he nominally studied Divinity at Edinburgh University but gained no further degrees. In 1805 he was elected to succeed John Playfair in the chair of mathematics at Edinburgh and this despite violent opposition on the part of a party who accused him of heresy. In 1807 he became a member of the Royal Society of Edinburgh and his proposers were John Playfair, Thomas Charles Hope and George Dunbar. When John Playfair died in 1819, Leslie was promoted to the more congenial chair of natural philosophy and he published a famous book about multiplication table The Philosophy of Arithmetic in 1820. In 1823 he published, chiefly for the use of his class, leslies main contributions to physics were made by the help of the differential thermometer, an instrument whose invention was contested with him by Count Rumford. In 1820 he was elected a member of the Institute of France, the only distinction of the kind which he valued. In his final years he is listed as living at 62 Queen Street, Leslie died of typhus in November 1832 at Coates, a small property he had acquired near Largo in Fife, at the age of 66. John Leslie did not marry and had no children and his nephew was the civil engineer, James Leslie, son of his brother, Alexander Leslie, an architect-builder in Largo. His great nephew was Alexander Leslie, Second edition Geometrical Analysis and Geometry of Curve Lines being Volume the Second of A Course of Mathematics and designed as an Introduction to the Study of Natural Philosophy E. M. Horsburgh. The Works of Sir John Leslie, mathematical Notes,28, pp i-v. doi,10. 1017/S1757748900002279. Atmometer Timeline of low-temperature technology Olson, Richard G, Sir John Leslie and the Laws of Electrical Conduction in Solids
9.
IBM 1620
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The IBM1620 was announced by IBM on October 21,1959, and marketed as an inexpensive scientific computer. After a total production of two thousand machines, it was withdrawn on November 19,1970. Modified versions of the 1620 were used as the CPU of the IBM1710, core memory cycle times were 20 microseconds for the Model I,10 microseconds for the Model II. For an explanation of all three known interpretations of the code name see the section on the machines development history. It was a word length decimal computer with a memory that could hold anything from 20,000 to 60,000 decimal digits increasing in 20,000 decimal digit increments. Memory was accessed two decimal digits at the same time and it was set to mark the most significant digit of a number. In the least significant digit of 5-digit addresses it was set for indirect addressing, in the middle 3 digits of 5-digit addresses they were set to select one of 7 index registers. Some instructions, such as the B instruction, only used the P Address, fixed-point data words could be any size from two decimal digits up to all of memory not used for other purposes. Floating-point data words could be any size from 4 decimal digits up to 102 decimal digits, the machine had no programmer-accessible registers, all operations were memory to memory. The table below lists Alphameric mode characters, the table below lists numeric mode characters. The Model I used the Cyrillic character Ж on the typewriter as a general purpose invalid character with correct parity, in some 1620 installations it was called a SMERSH, as used in the James Bond novels that had become popular in the late 1960s. The Model II used a new character ❚ as a general purpose invalid character with correct parity and he also showed how the machines paper tape reading support could not properly read tapes containing record marks, since record marks are used to terminate the characters read in storage. Most 1620 installations used the more convenient punched card input/output, rather than paper tape, the successor to the 1620, the IBM1130 was based on a totally different, 16-bit binary architecture. The Monitors provided disk based versions of 1620 SPS IId, FORTRAN IId as well as a DUP, both Monitor systems required 20,000 digits or more of memory and 1 or more 1311 disk drives. A standard preliminary was to clear the computer memory of any previous users detritus - being magnetic cores and this was effected by using the console facilities to load a simple computer program via typing its machine code at the console typewriter, running it, and stopping it. This was not challenging as only one instruction was needed such as 160001000000, loaded at address zero and this was the normal machine code means of copying a constant of up to five digits. The digit string was addressed at its end and extended through lower addresses until a digit with a flag marked its end. But for this instruction, no flag would ever be found because the source digits had shortly before been overwritten by digits lacking a flag, each 20,000 digit module of memory took just under one second to clear
10.
Chinese mathematics
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Mathematics in China emerged independently by the 11th century BC. The Chinese independently developed very large and negative numbers, decimals, a place value system, a binary system, algebra, geometry. Knowledge of Chinese mathematics before 254 BC is somewhat fragmentary, as in other early societies the focus was on astronomy in order to perfect the agricultural calendar, and other practical tasks, and not on establishing formal systems. Ancient Chinese mathematicians did not develop an approach, but made advances in algorithm development. Some exchange of ideas across Asia through known cultural exchanges from at least Roman times is likely, frequently, elements of the mathematics of early societies correspond to rudimentary results found later in branches of modern mathematics such as geometry or number theory. The Pythagorean theorem for example, has been attested to the time of the Duke of Zhou, knowledge of Pascals triangle has also been shown to have existed in China centuries before Pascal, such as by Shen Kuo. Simple mathematics on Oracle bone script date back to the Shang Dynasty, one of the oldest surviving mathematical works is the Yi Jing, which greatly influenced written literature during the Zhou Dynasty. For mathematics, the book included a sophisticated use of hexagrams, leibniz pointed out, the I Ching contained elements of binary numbers. Since the Shang period, the Chinese had already developed a decimal system. Since early times, Chinese understood basic arithmetic, algebra, equations, although the Chinese were more focused on arithmetic and advanced algebra for astronomical uses, they were also the first to develop negative numbers, algebraic geometry and the usage of decimals. Math was one of the Liù Yì or Six Arts, students were required to master during the Zhou Dynasty, learning them all perfectly was required to be a perfect gentleman, or in the Chinese sense, a Renaissance Man. Six Arts have their roots in the Confucian philosophy, the oldest existent work on geometry in China comes from the philosophical Mohist canon of c.330 BC, compiled by the followers of Mozi. The Mo Jing described various aspects of many associated with physical science. It provided a definition of the geometric point, stating that a line is separated into parts. Much like Euclids first and third definitions and Platos beginning of a line, there is nothing similar to it. Similar to the atomists of Democritus, the Mo Jing stated that a point is the smallest unit and it also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch. The book provided word recognition for circumference, diameter, and radius, the history of mathematical development lacks some evidence. There are still debates about certain mathematical classics, for example, the Zhoubi Suanjing dates around 1200–1000 BC, yet many scholars believed it was written between 300–250 BC