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.
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
4.
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
5.
Tsinghua Bamboo Slips
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The Tsinghua Bamboo Slips are a collection of Chinese texts dating to the Warring States period and written in ink on strips of bamboo, that were acquired in 2008 by Tsinghua University, China. The texts were obtained by illegal excavation, probably of a tomb in the area of Hubei or Hunan province, the very large size of the collection and the significance of the texts for scholarship make it one of the most important discoveries of early Chinese texts to date. On 7 January 2014 the journal Nature announced that some Tsinghua Bamboo Slips represent the worlds oldest example of a multiplication table. The Tsinghua Bamboo Slips were donated to Tsinghua University in July 2008 by an alumnus of the university, the precise location and date of the illicit excavation that yielded the slips remain unknown. Li Xueqin, the director of the conservation and research project, has stated that the wishes of the alumnus to maintain his identity secret will be respected, a single radiocarbon date and the style of ornament on the accompanying box are in keeping with this conclusion. By the time reached the university, the slips were badly affected by mold. Conservation work on the slips was carried out, and a Center for Excavated Texts Research, there are 2388 slips altogether in the collection, including a number of fragments. A series of articles discussing the TBS, intended for an educated but non-specialist Chinese audience, the first volume of texts was published by the Tsinghua team in 2010. A2013 article in The New York Times reported on the TBSs importance to understanding the Chinese classics, the classics are all political, Li Xueqin commented, It would be like finding the original Bible or the original classics. It enables us to look at the classics before they were turned into classics, the questions now include, what were they in the beginning, and how they became what they are. In some cases, a TBS text can be found in the received Shang Shu, with variations in wording. Such examples include versions of the Jin Teng, Kang Gao, an annalistic history recording events from the beginning of the Western Zhou through to the early Warring States period is said to be similar in form and content to the received Bamboo Annals. Another text running across 14 slips recounts a celebratory gathering of the Zhou elite in the 8th year of the reign of King Wu of Zhou, prior to their conquest of the Shang dynasty. The gathering takes place in the temple of King Wen of Zhou, King Wus father, and consisted of beer drinking. Among the TBS texts in the style of the received Shang Shu, is one that has been titled The Admonition of Protection and this was the first text for which anything approaching a complete description and transcription was published. The text purports to be a record of a deathbed admonition by the Zhou king Wen Wang to his son and heir, Wu Wang. The content of the speech revolves around a concept of The Middle which seems to refer to an avoidance of extremes. Xinian 繫年, probably composed ca.370 BC, relates key events of Zhou history and it comprises 138 slips in a relatively well preserved condition
6.
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
7.
Babylonian mathematics
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Babylonian mathematics was any mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited, in respect of time they fall in two distinct groups, one from the Old Babylonian period, the other mainly Seleucid from the last three or four centuries BC. In respect of content there is any difference between the two groups of texts. Thus Babylonian mathematics remained constant, in character and content, for two millennia. In contrast to the scarcity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from some 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed while the clay was moist, the majority of recovered clay tablets date from 1800 to 1600 BCE, and cover topics that include fractions, algebra, quadratic and cubic equations and the Pythagorean theorem. The Babylonian tablet YBC7289 gives an approximation to 2 accurate to three significant sexagesimal digits, Babylonian mathematics is a range of numeric and more advanced mathematical practices in the ancient Near East, written in cuneiform script. Study has historically focused on the Old Babylonian period in the second millennium BC due to the wealth of data available. There has been debate over the earliest appearance of Babylonian mathematics, Babylonian mathematics was primarily written on clay tablets in cuneiform script in the Akkadian or Sumerian languages. Babylonian mathematics is perhaps an unhelpful term since the earliest suggested origins date to the use of accounting devices, such as bullae and tokens, the Babylonian system of mathematics was sexagesimal numeral system. From this we derive the modern day usage of 60 seconds in a minute,60 minutes in an hour, the Babylonians were able to make great advances in mathematics for two reasons. Firstly, the number 60 is a highly composite number, having factors of 1,2,3,4,5,6,10,12,15,20,30,60. Additionally, unlike the Egyptians and Romans, the Babylonians had a true place-value system, the ancient Sumerians of Mesopotamia developed a complex system of metrology from 3000 BC. From 2600 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises, the earliest traces of the Babylonian numerals also date back to this period. Most clay tablets that describe Babylonian mathematics belong to the Old Babylonian, some clay tablets contain mathematical lists and tables, others contain problems and worked solutions. The Babylonians used pre-calculated tables to assist with arithmetic, for example, two tablets found at Senkerah on the Euphrates in 1854, dating from 2000 BC, give lists of the squares of numbers up to 59 and the cubes of numbers up to 32. The Babylonians used the lists of squares together with the formulae a b =2 − a 2 − b 22 a b =2 −24 to simplify multiplication, the Babylonians did not have an algorithm for long division. Instead they based their method on the fact that a b = a ×1 b together with a table of reciprocals
8.
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
9.
Napier's bones
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Napiers bones is a manually-operated calculating device created by John Napier of Merchiston for calculation of products and quotients of numbers. The method was based on Arab mathematics and the lattice multiplication used by Matrakci Nasuh in the Umdet-ul Hisab, the technique was also called Rabdology. Napier published his version in 1617 in Rabdology, printed in Edinburgh, Scotland, dedicated to his patron Alexander Seton. Using the multiplication tables embedded in the rods, multiplication can be reduced to addition operations, more advanced use of the rods can even extract square roots. Note that Napiers bones are not the same as logarithms, with which Napiers name is also associated, the complete device usually includes a base board with a rim, the user places Napiers rods inside the rim to conduct multiplication or division. The boards left edge is divided into 9 squares, holding the numbers 1 to 9, the Napiers rods consist of strips of wood, metal or heavy cardboard. Napiers bones are three-dimensional, square in section, with four different rods engraved on each one. A set of such bones might be enclosed in a convenient carrying case, a rods surface comprises 9 squares, and each square, except for the top one, comprises two halves divided by a diagonal line. The digits of each product are written one to side of the diagonal, numbers less than 10 occupy the lower triangle. A set consists of 10 rods corresponding to digits 0 to 9, the rod 0, although it may look unnecessary, is needed for multipliers or multiplicands having 0 in them. To demonstrate how to use Napier’s Bones for multiplication, three examples of increasing difficulty are explained below, problem, Multiply 425 by 6 Start by placing the bones corresponding to the leading number of the problem into the boards. If a 0 is used in number, a space is left between the bones corresponding to where the 0 digit would be. In this example, the bones 4,2, and 5 are placed in the order as shown below. Looking at the first column, choose the number wishing to multiply by, in this example, that number is 6. The row this number is located in is the only row needed to perform the remaining calculations, starting at the right side of the row, evaluate the diagonal columns by adding the numbers that share the same diagonal column. Single numbers simply remain that number, once the diagonal columns have been evaluated, one must simply read from left to right the numbers calculated for each diagonal column. For this example, reading the results of the summations from left to right produces the answer of 2550. Therefore, The solution to multiplying 425 by 6 is 2550, when multiplying by larger single digits, it is common that upon adding a diagonal column, the sum of the numbers result in a number that is 10 or greater
10.
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
11.
Greco-Roman world
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Certainly, all men of note and accomplishment, whatever their ethnic extractions, spoke and wrote in Greek and/or Latin. The historian Josephus Flavius was Jewish but he wrote and spoke in Greek and was a Roman citizen. Occupying the periphery of this world were Roman Germany, Illyria and Pannonia, also included was Dacia, Nubia, Mauretania, Arabia Petraea, the Tauric Chersonesus. The above seems to ignore the major rivalry between the Greco-Romans, during their period of ascendancy, and the empire to the east. In the schools of art, philosophy and rhetoric, the foundations of education were transmitted throughout the lands of Greek, within its educated class spanning all of the Greco-Roman era, the testimony of literary borrowings and influences is overwhelming proof of a mantle of mutual knowledge. For example, several hundred papyrus volumes found in a Roman villa at Herculaneum are in Greek, from the lives of Cicero and Julius Caesar, it is known that Romans frequented the schools in Greece. The installation, both in Greek and Latin, of Augustuss monumental eulogy, the Res Gestae, is a proof of recognition for the dual vehicles of the common culture. Most educated Romans were likely bilingual in Greek and Latin, Greco-Roman architecture is architecture of the Roman world that followed the principles and style established in ancient Greece. The most representative building of that era was the temple, other prominent structures that represented the style included government buildings, like the Roman Senate, and cultural structures, like the Colosseum. The three primary styles of column design used in temples in classical Greece were Doric, Ionic, examples of Doric architecture are the Parthenon and the Temple of Hephaestus in Athens, while the Erechtheum, which is located right next to the Parthenon is Ionic. Ionic Greco-Roman architecture tend to be more decorative than the formal Doric styles, by AD211, with Caracallas edict known as the Constitutio Antoniniana, all free inhabitants of the Empire became citizens. As a result, even after the Fall of Rome, the people of the empire that remained continued to call themselves Romans even though Greek became the language of the Empire. Rhomaioi is what they continued to call themselves through the Ottoman era, Classical Antiquity Classical mythology Legacy of the Roman Empire Greco-Roman mysteries Hellenistic Greece Magic in the Greco-Roman world List of Greco-Roman geographers Sir William Smith. Dictionary of Greek and Roman Geography
12.
Neopythagoreanism
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Neopythagoreanism was a school of Hellenistic philosophy which revived Pythagorean doctrines. Neopythagoreanism was influenced by Middle Platonism and in turn influenced Neoplatonism and it originated in the 1st century BCE and flourished during the 1st and 2nd centuries CE. The 1911 Britannica describes Neopythagoreanism as a link in the chain between the old and the new within Hellenistic philosophy, as such, it contributed to the doctrine of monotheism as it emerged during Late Antiquity. Central to Neopythagorean thought was the concept of a soul and its inherent desire for a unio mystica with the divine, the word Neopythagoreanism is a modern term, coined as a parallel of Neoplatonism. Other important Neopythagoreans include the mathematician Nicomachus of Gerasa, who wrote about the properties of numbers. In the 2nd century, Numenius of Apamea sought to fuse elements of Platonism into Neopythagoreanism. Neopythagoreanism was an attempt to re-introduce a mystical religious element into Hellenistic philosophy in place of what had come to be regarded as an arid formalism, the founders of the school sought to invest their doctrines with the halo of tradition by ascribing them to Pythagoras and Plato. They emphasized the distinction between the soul and the body. God must be worshipped spiritually by prayer and the will to be good, the soul must be freed from its material surrounding, the muddy vesture of decay, by an ascetic habit of life. Bodily pleasures and all sensuous impulses must be abandoned as detrimental to the purity of the soul. God is the principle of good, Matter the groundwork of Evil, in this system can be distinguished not only the asceticism of Pythagoras and the later mysticism of Plato, but also the influence of the Orphic mysteries and of Oriental philosophy. The Ideas of Plato are no longer self-subsistent entities but are the elements which constitute the content of spiritual activity, the non-material universe is regarded as the sphere of mind or spirit. A basilica where Neopythagoreans held their meetings in the 1st century was found near Porta Maggiore on Via Praenestina in Rome, pythagoreanism School of the Sextii Allegorical interpretations of Plato Charles H. Neopythagoreanism
13.
Greek mathematics
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Greek mathematics, as the term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture, Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. The word mathematics itself derives from the ancient Greek μάθημα, meaning subject of instruction, the study of mathematics for its own sake and the use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations. The origin of Greek mathematics is not well documented, the earliest advanced civilizations in Greece and in Europe were the Minoan and later Mycenaean civilization, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs, though no direct evidence is available, it is generally thought that the neighboring Babylonian and Egyptian civilizations had an influence on the younger Greek tradition. Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus. Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of 585 BC, despite this, it is generally agreed that Thales is the first of the seven wise men of Greece. The two earliest mathematical theorems, Thales theorem and Intercept theorem are attributed to Thales. The former, which states that an angle inscribed in a semicircle is a right angle and it is for this reason that Thales is often hailed as the father of the deductive organization of mathematics and as the first true mathematician. Thales is also thought to be the earliest known man in history to whom specific mathematical discoveries have been attributed, another important figure in the development of Greek mathematics is Pythagoras of Samos. Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of Nebuchadnezzar, Pythagoras established an order called the Pythagoreans, which held knowledge and property in common and hence all of the discoveries by individual Pythagoreans were attributed to the order. And since in antiquity it was customary to give all credit to the master, aristotle for one refused to attribute anything specifically to Pythagoras as an individual and only discussed the work of the Pythagoreans as a group. One of the most important characteristics of the Pythagorean order was that it maintained that the pursuit of philosophical and mathematical studies was a basis for the conduct of life. Indeed, the philosophy and mathematics are said to have been coined by Pythagoras. From this love of knowledge came many achievements and it has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclids Elements. The reason it is not clear exactly what either Thales or Pythagoras actually did is that almost no documentation has survived. The only evidence comes from traditions recorded in such as Proclus’ commentary on Euclid written centuries later. Some of these works, such as Aristotle’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments
14.
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
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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
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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
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Group (mathematics)
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In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure and it allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a kinship with the notion of symmetry. The concept of a group arose from the study of polynomial equations, after contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right, to explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. A theory has developed for finite groups, which culminated with the classification of finite simple groups. Since the mid-1980s, geometric group theory, which studies finitely generated groups as objects, has become a particularly active area in group theory. One of the most familiar groups is the set of integers Z which consists of the numbers, −4, −3, −2, −1,0,1,2,3,4. The following properties of integer addition serve as a model for the group axioms given in the definition below. For any two integers a and b, the sum a + b is also an integer and that is, addition of integers always yields an integer. This property is known as closure under addition, for all integers a, b and c, + c = a +. Expressed in words, adding a to b first, and then adding the result to c gives the final result as adding a to the sum of b and c. If a is any integer, then 0 + a = a +0 = a, zero is called the identity element of addition because adding it to any integer returns the same integer. For every integer a, there is a b such that a + b = b + a =0. The integer b is called the element of the integer a and is denoted −a. The integers, together with the operation +, form a mathematical object belonging to a class sharing similar structural aspects. To appropriately understand these structures as a collective, the abstract definition is developed
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Field (mathematics)
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In mathematics, a field is a set on which are defined addition, subtraction, multiplication, and division, which behave as they do when applied to rational and real numbers. A field is thus an algebraic structure, which is widely used in algebra, number theory. The best known fields are the field of numbers. In addition, the field of numbers is widely used, not only in mathematics. Finite fields are used in most cryptographic protocols used for computer security, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Formally, a field is a set together with two operations the addition and the multiplication, which have the properties, called axioms of fields. An operation is a mapping that associates an element of the set to every pair of its elements, the result of the addition of a and b is called the sum of a and b and denoted a + b. Similarly, the result of the multiplication of a and b is called the product of a and b, associativity of addition and multiplication For all a, b and c in F, one has a + = + c and a · = · c. Commutativity of addition and multiplication For all a and b in F one has a + b = b + a and a · b = b · a. Existence of additive and multiplicative identity elements There exists an element 0 in F, called the identity, such that for all a in F. There is an element 1, different from 0 and called the identity, such that for all a in F. Existence of additive inverses and multiplicative inverses For every a in F, there exists an element in F, denoted −a, such that a + =0. For every a ≠0 in F, there exists an element in F, denoted a−1, 1/a, or 1/a, distributivity of multiplication over addition For all a, b and c in F, one has a · = +. The elements 0 and 1 being required to be distinct, a field has, at least, for every a in F, one has − a = ⋅ a. Thus, the inverse of every element is known as soon as one knows the additive inverse of 1. A subtraction and a division are defined in every field by a − b = a +, a subfield E of a field F is a subset of F that contains 1, and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. It is straightforward to verify that a subfield is indeed a field, two groups are associated to every field. The field itself is a group under addition, when considering this group structure rather the field structure, one talks of the additive group of the field
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Ring (mathematics)
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In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices, the conceptualization of rings started in the 1870s and completed in the 1920s. Key contributors include Dedekind, Hilbert, Fraenkel, and Noether, rings were first formalized as a generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. Afterward, they proved to be useful in other branches of mathematics such as geometry. A ring is a group with a second binary operation that is associative, is distributive over the abelian group operation. By extension from the integers, the group operation is called addition. Whether a ring is commutative or not has profound implications on its behavior as an abstract object, as a result, commutative ring theory, commonly known as commutative algebra, is a key topic in ring theory. Its development has greatly influenced by problems and ideas occurring naturally in algebraic number theory. The most familiar example of a ring is the set of all integers, Z, −5, −4, −3, −2, −1,0,1,2,3,4,5. The familiar properties for addition and multiplication of integers serve as a model for the axioms for rings, a ring is a set R equipped with two binary operations + and · satisfying the following three sets of axioms, called the ring axioms 1. R is a group under addition, meaning that, + c = a + for all a, b, c in R. a + b = b + a for all a, b in R. There is an element 0 in R such that a +0 = a for all a in R, for each a in R there exists −a in R such that a + =0. R is a monoid under multiplication, meaning that, · c = a · for all a, b, c in R. There is an element 1 in R such that a ·1 = a and 1 · a = a for all a in R.3. Multiplication is distributive with respect to addition, a ⋅ = + for all a, b, c in R. · a = + for all a, b, c in R. As explained in § History below, many follow a alternative convention in which a ring is not defined to have a multiplicative identity. This article adopts the convention that, unless stated, a ring is assumed to have such an identity