Sejarah matematika: Perbedaan revisi

41.249 bita dihapus ,  7 tahun yang lalu
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[[Aryabhata]], pada tahun 499, memperkenalkan fungsi [[versinus]], menghasilkan tabel [[trigonometri]] India pertama tentang sinus, mengembangkan teknik-teknik dan [[algoritma]] [[aljabar]], [[infinitesimal]], dan [[persamaan diferensial]], dan memperoleh solusi seluruh bilangan untuk persamaan linear oleh sebuah metode yang setara dengan metode modern, bersama-sama dengan perhitungan [[astronomi] yang akurat berdasarkan sistem [[heliosentris]] [[gravitasi]].<ref name="sarma">{{citation | author=[[K. V. Sarma]] | journal=Indian Journal of History of Science | year=2001 | pages=105–115 | title=Āryabhaṭa: His name, time and provenance |volume=36 |issue=4 | url=http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005b67_105.pdf}}</ref> Sebuah terjemahan [[bahasa Arab]] dari karyanya ''Aryabhatiya'' tersedia sejak abad ke-8, diikuti oleh terjemahan bahasa Latin pada abad ke-13. Dia juga memberikan nilai π yang bersesuaian dengan 62832/20000 = 3,1416. Pada abad ke-14, [[Madhava dari Sangamagrama]] menemukan [[rumus Leibniz untuk pi]], dan, menggunakan 21 suku, untuk menghitung nilai π sebagai 3,14159265359.
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In the 7th century, [[Brahmagupta]] identified the [[Brahmagupta theorem]], [[Brahmagupta's identity]] and [[Brahmagupta's formula]], and for the first time, in ''[[Brahmasphutasiddhanta|Brahma-sphuta-siddhanta]]'', he lucidly explained the use of [[0 (number)|zero]] as both a [[placeholder]] and [[decimal digit]], and explained the [[Hindu-Arabic numeral system]].<ref name="Boyer Siddhanta">{{cite book|last=Boyer|authorlink=Carl Benjamin Boyer|title=|year=1991|chapter=The Arabic Hegemony|pages=226|quote=By 766 we learn that an astronomical-mathematical work, known to the Arabs as the ''Sindhind'', was brought to Baghdad from India. It is generally thought that this was the ''Brahmasphuta Siddhanta'', although it may have been the ''Surya Siddhanata''. A few years later, perhaps about 775, this ''Siddhanata'' was translated into Arabic, and it was not long afterwards (ca. 780) that Ptolemy's astrological ''Tetrabiblos'' was translated into Arabic from the Greek.}}</ref> It was from a translation of this Indian text on mathematics (c. 770) that Islamic mathematicians were introduced to this numeral system, which they adapted as [[Arabic numerals]]. Islamic scholars carried knowledge of this number system to Europe by the 12th century, and it has now displaced all older number systems throughout the world. In the 10th century, [[Halayudha]]'s commentary on [[Pingala]]'s work contains a study of the [[Fibonacci sequence]] and [[Pascal's triangle]], and describes the formation of a [[matrix (mathematics)|matrix]].
 
In the 12th century, [[Bhaskara]]<ref name="Plofker 418-419">{{cite book | first = Kim | last = Plofker | year = 2007 | pages = 418–419 | title = | quote = The ''Paitamahasiddhanta'' also directly inspired another major ''siddhanta'', written by a contemporary of Bhaskara: The ''Brahmasphutasiddhanta'' (''Corrected Treatise of Brahma'') completed by Brahmagupta in 628. This astronomer was born in 598 and apparently worked in Bhillamal (identified with modern Bhinmal in Rajasthan), during the reign (and possibly under the patronage) of King Vyaghramukha.<br />Although we do not know whether Brahmagupta encountered the work of his contemporary Bhaskara, he was certainly aware of the writings of other members of the tradition of the ''Aryabhatiya'', about which he has nothing good to say. This is almost the first trace we possess of the division of Indian astronomer-mathematicians into rival, sometimes antagonistic "schools." [...] it was in the application of mathematical models to the physical world - in this case, the choices of astronomical parameters and theories - that disagreements arose. [...]<br />Such critiques of rival works appear occasionally throughout the first ten astronomical chapters of the ''Brahmasphutasiddhanta'', and its eleventh chapter is entirely devoted to them. But they do not enter into the mathematical chapters that Brahmagupta devotes respectively to ''ganita'' (chapter 12) and the pulverizer (chapter 18). This division of mathematical subjects reflects a different twofold classification from Bhaskara's "mathematics of fields" and "mathematics of quantities." Instead, the first is concerned with arithmetic operations beginning with addition, proportion, interest, series, formulas for finding lengths, areas, and volumes in geometrical figures, and various procedures with fractions - in short, diverse rules for computing with known quantities. The second, on the other hand, deals with what Brahmagupta calls "the pulverizer, zero, negatives, positives, unknowns, elimination of the middle term, reduction to one [variable], ''bhavita'' [the product of two unknowns], and the nature of squares [second-degree indeterminate equations]" - that is, techniques for operating with unknown quantities. This distinction is more explicitely presented in later works as mathematics of the "manifest" and "unmanifest," respectively: i.e., what we will henceforth call "arithmetic" manipulations of known quantities and "algebraic" manipulation of so-called "seeds" or unknown quantities. The former, of course, may include geometric problems and other topics not covered by the modern definition of "arithmetic." (Like Aryabhata, Brahmagupta relegates his sine-table to an astronomical chapter where the computations require it, instead of lumping it in with other "mathematical" topics.}}</ref> first conceived [[differential calculus]], along with the concepts of the [[derivative]], [[differential]] coefficient, and differentiation. He also stated [[Rolle's theorem]] (a special case of the [[mean value theorem]]), studied [[Pell's equation]], and investigated the derivative of the sine function. From the 14th century, Madhava and other [[Kerala School]] mathematicians further developed his ideas. They developed the concepts of [[mathematical analysis]] and [[floating point]] numbers, and concepts fundamental to the overall development of [[calculus]], including the mean value theorem, term by term [[integral|integration]], the relationship of an area under a curve and its antiderivative or integral, the [[integral test for convergence]], [[iterative method]]s for solutions to [[non-linear]] equations, and a number of [[infinite series]], [[power series]], [[Taylor series]], and trigonometric series. In the 16th century, [[Jyeshtadeva]] consolidated many of the Kerala School's developments and theorems in the ''Yuktibhasa'', the world's first differential calculus text, which also introduced concepts of [[integral calculus]]. Mathematical progress in India stagnated from the late 16th century to the 20th century, due to political turmoil.
 
== Matematika Islam ==
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{{Main|Mathematics in medieval Islam}}
{{See also|History of the Hindu-Arabic numeral system}}
[[Image:Abu Abdullah Muhammad bin Musa al-Khwarizmi.jpg|thumb|[[Muhammad ibn Mūsā al-Khwārizmī|Muḥammad ibn Mūsā al-Ḵwārizmī]] ]]
The [[Islamic Empire]] established across [[Persia]], the [[Middle East]], [[Central Asia]], [[North Africa]], [[Iberian Peninsula|Iberia]], and in parts of [[History of India|India]] in the 8th century made significant contributions towards mathematics. Although most Islamic texts on mathematics were written in [[Arabic language|Arabic]], most of them were not written by [[Arab]]s, since much like the status of Greek in the Hellenistic world, Arabic was used as the written language of non-Arab scholars throughout the Islamic world at the time. [[Persian people|Persians]] contributed to the world of Mathematics alongside Arabs.
 
In the 9th century, {{Unicode|[[Muhammad ibn Mūsā al-Khwārizmī|Muḥammad ibn Mūsā al-Ḵwārizmī]]}} wrote several important books on the Hindu-Arabic numerals and on methods for solving equations. His book ''On the Calculation with Hindu Numerals'', written about 825, along with the work of [[Al-Kindi]], were instrumental in spreading [[Indian mathematics]] and [[Hindu-Arabic numeral system|Indian numerals]] to the West. The word ''[[algorithm]]'' is derived from the Latinization of his name, Algoritmi, and the word ''[[algebra]]'' from the title of one of his works, ''[[The Compendious Book on Calculation by Completion and Balancing|Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala]]'' (''The Compendious Book on Calculation by Completion and Balancing''). Al-Khwarizmi is often called the "father of algebra", for his fundamental contributions to the field.<ref>[http://www.ucs.louisiana.edu/~sxw8045/history.htm The History of Algebra]. [[Louisiana State University]].</ref> He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,<ref>{{Harv|Boyer|1991|loc="The Arabic Hegemony" p. 230}} "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwarizmi's exposition that his readers must have had little difficulty in mastering the solutions."</ref> and he was the first to teach algebra in an [[Elementary algebra|elementary form]] and for its own sake.<ref>Gandz and Saloman (1936), ''The sources of al-Khwarizmi's algebra'', Osiris i, pp. 263–77: "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".</ref> He also introduced the fundamental method of "[[Reduction (mathematics)|reduction]]" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which Al-Khwarizmi originally described as ''al-jabr''.<ref name=Boyer-229>{{Harv|Boyer|1991|loc="The Arabic Hegemony" p. 229}} "It is not certain just what the terms ''al-jabr'' and ''muqabalah'' mean, but the usual interpretation is similar to that implied in the translation above. The word ''al-jabr'' presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word ''muqabalah'' is said to refer to "reduction" or "balancing" - that is, the cancellation of like terms on opposite sides of the equation."</ref> His algebra was also no longer concerned "with a series of [[problem]]s to be resolved, but an [[Expository writing|exposition]] which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."<ref name=Rashed-Armstrong>{{Cite book | last1=Rashed | first1=R. | last2=Armstrong | first2=Angela | year=1994 | title=The Development of Arabic Mathematics | publisher=[[Springer Science+Business Media|Springer]] | isbn=0792325656 | oclc=29181926 | pages=11–12}}</ref>
 
Further developments in algebra were made by [[Al-Karaji]] in his treatise ''al-Fakhri'', where he extends the methodology to incorporate integer powers and integer roots of unknown quantities. The first known [[Mathematical proof|proof]] by [[mathematical induction]] appears in a book written by Al-Karaji around 1000 AD, who used it to prove the [[binomial theorem]], [[Pascal's triangle]], and the sum of [[integral]] [[Cube (algebra)|cubes]].<ref>Victor J. Katz (1998). ''History of Mathematics: An Introduction'', pp. 255–59. [[Addison-Wesley]]. ISBN 0-321-01618-1.</ref> The [[historian]] of mathematics, F. Woepcke,<ref>F. Woepcke (1853). ''Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi''. [[Paris]].</ref> praised Al-Karaji for being "the first who introduced the [[theory]] of [[algebra]]ic [[calculus]]." Also in the 10th century, [[Abul Wafa]] translated the works of [[Diophantus]] into Arabic and developed the [[tangent (trigonometry)|tangent]] function. [[Ibn al-Haytham]] was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. He performed an integration in order to find the volume of a [[paraboloid]], and was able to generalize his result for the integrals of [[polynomial]]s up to the [[Quartic polynomial|fourth degree]]. He thus came close to finding a general formula for the [[integral]]s of polynomials, but he was not concerned with any polynomials higher than the fourth degree.<ref name=Katz>Victor J. Katz (1995), "Ideas of Calculus in Islam and India", ''Mathematics Magazine'' '''68''' (3): 163–74.</ref>
 
In the late 11th century, [[Omar Khayyam]] wrote ''Discussions of the Difficulties in Euclid'', a book about flaws in [[Euclid's Elements|Euclid's ''Elements'']], especially the [[parallel postulate]], and laid the foundations for [[analytic geometry]] and [[non-Euclidean geometry]].{{Citation needed|date=March 2009}} He was also the first to find the general geometric solution to [[cubic equation]]s. He was also very influential in [[calendar reform]].{{Citation needed|date=March 2009}}
 
In the late 12th century, [[Sharaf al-Dīn al-Tūsī]] introduced the concept of a [[Function (mathematics)|function]],<ref>{{Cite journal|last=Victor J. Katz|first=Bill Barton|title=Stages in the History of Algebra with Implications for Teaching|journal=Educational Studies in Mathematics|publisher=[[Springer Science+Business Media|Springer Netherlands]]|volume=66|issue=2|date=October 2007|doi=10.1007/s10649-006-9023-7|pages=185–201 [192]}}</ref> and he was the first to discover the [[derivative]] of [[Cubic function|cubic polynomials]].<ref>J. L. Berggren (1990). "Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat", ''Journal of the American Oriental Society'' '''110''' (2), pp. 304–09.</ref> His ''Treatise on Equations'' developed concepts related to differential calculus, such as the derivative function and the [[maxima and minima]] of curves, in order to solve cubic equations which may not have positive solutions.<ref name=Sharaf>{{MacTutor|id=Al-Tusi_Sharaf|title=Sharaf al-Din al-Muzaffar al-Tusi}}</ref>
 
In the 13th century, [[Nasir al-Din Tusi]] (Nasireddin) made advances in [[spherical trigonometry]]. He also wrote influential work on [[Euclid]]'s [[parallel postulate]]. In the 15th century, [[Ghiyath al-Kashi]] computed the value of [[π]] to the 16th decimal place. Kashi also had an algorithm for calculating ''n''th roots, which was a special case of the methods given many centuries later by [[Ruffini]] and [[Horner]].
 
Other notable Muslim mathematicians included [[al-Samawal]], [[Abu'l-Hasan al-Uqlidisi]], [[Jamshid al-Kashi]], [[Thabit ibn Qurra]], [[Abu Kamil]] and [[Abu Sahl al-Kuhi]].
 
Other achievements of Muslim mathematicians during this period include the development of [[algebra]] and [[algorithm]]s, the development of [[spherical trigonometry]],<ref>{{cite book |last=Syed |first=M. H. |title=Islam and Science |year=2005 |publisher=Anmol Publications PVT. LTD. |isbn=8-1261-1345-6 |page=71}}</ref> the addition of the [[decimal point]] notation to the [[Arabic numerals]], the discovery of all the modern [[trigonometric function]]s besides the sine, [[al-Kindi]]'s introduction of [[cryptanalysis]] and [[frequency analysis]], the development of [[analytic geometry]] by [[Ibn al-Haytham]], the beginning of [[algebraic geometry]] by [[Omar Khayyam]], the first refutations of [[Euclidean geometry]] and the [[parallel postulate]] by [[Nasīr al-Dīn al-Tūsī]], the first attempt at a [[non-Euclidean geometry]] by Sadr al-Din, the development of an [[Mathematical notation|algebraic notation]] by [[Abū al-Hasan ibn Alī al-Qalasādī|al-Qalasādī]],<ref name=Qalasadi>{{MacTutor Biography|id=Al-Qalasadi|title= Abu'l Hasan ibn Ali al Qalasadi}}</ref> and many other advances in algebra, [[arithmetic]], calculus, [[cryptography]], [[geometry]], [[number theory]] and [[trigonometry]].
 
During the time of the [[Ottoman Empire]] from the 15th century, the development of Islamic mathematics became stagnant.
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== Matematika Eropa Pertengahan ==
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Medieval European interest in mathematics was driven by concerns quite different from those of modern mathematicians. One driving element was the belief that mathematics provided the key to understanding the created order of nature, frequently justified by [[Plato]]'s ''[[Timaeus (dialogue)|Timaeus]]'' and the [[Biblical apocrypha|apocryphal]] biblical passage (in the ''[[Book of Wisdom]]'') that God had ''ordered all things in measure, and number, and weight''<ref>''Wisdom'', 11:21</ref>.
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=== Zaman Pertengahan Dini ===
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[[Boethius]] provided a place for mathematics in the curriculum when he coined the term ''[[quadrivium]]'' to describe the study of arithmetic, geometry, astronomy, and music. He wrote ''De institutione arithmetica'', a free translation from the Greek of [[Nicomachus]]'s ''Introduction to Arithmetic''; ''De institutione musica'', also derived from Greek sources; and a series of excerpts from [[Euclid]]'s [[Euclid's Elements|''Elements'']]. His works were theoretical, rather than practical, and were the basis of mathematical study until the recovery of Greek and Arabic mathematical works.<ref>Caldwell, John (1981) "The ''De Institutione Arithmetica'' and the ''De Institutione Musica''", pp. 135–54 in Margaret Gibson, ed., ''Boethius: His Life, Thought, and Influence,'' (Oxford: Basil Blackwell).</ref><ref>Folkerts, Menso, ''"Boethius" Geometrie II'', (Wiesbaden: Franz Steiner Verlag, 1970).</ref>
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=== Kelahiran kembali ===
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In the 12th century, European scholars traveled to Spain and Sicily [[Latin translations of the 12th century|seeking scientific Arabic texts]], including [[al-Khwarizmi]]'s ''[[The Compendious Book on Calculation by Completion and Balancing]]'', translated into Latin by [[Robert of Chester]], and the complete text of [[Euclid's Elements|Euclid's ''Elements'']], translated in various versions by [[Adelard of Bath]], [[Herman of Carinthia]], and [[Gerard of Cremona]].<ref>Marie-Thérèse d'Alverny, "Translations and Translators", pp. 421–62 in Robert L. Benson and Giles Constable, ''Renaissance and Renewal in the Twelfth Century'', (Cambridge: Harvard University Press, 1982).</ref><ref>Guy Beaujouan, "The Transformation of the Quadrivium", pp. 463–87 in Robert L. Benson and Giles Constable, ''Renaissance and Renewal in the Twelfth Century'', (Cambridge: Harvard University Press, 1982).</ref>
 
These new sources sparked a renewal of mathematics. [[Fibonacci]], writing in the ''[[Liber Abaci]]'', in 1202 and updated in 1254, produced the first significant mathematics in Europe since the time of [[Eratosthenes]], a gap of more than a thousand years. The work introduced [[Hindu-Arabic numerals]] to Europe, and discussed many other mathematical problems.
 
The fourteenth century saw the development of new mathematical concepts to investigate a wide range of problems.<ref>Grant, Edward and John E. Murdoch (1987), eds., ''Mathematics and Its Applications to Science and Natural Philosophy in the Middle Ages,'' (Cambridge: Cambridge University Press) ISBN 0-521-32260-X.</ref> One important contribution was development of mathematics of local motion.
 
[[Thomas Bradwardine]] proposed that speed (V) increases in arithmetic proportion as the ratio of force (F) to resistance (R) increases in geometric proportion. Bradwardine expressed this by a series of specific examples, but although the logarithm had not yet been conceived, we can express his conclusion anachronistically by writing:
V = log (F/R).<ref>Clagett, Marshall (1961) ''The Science of Mechanics in the Middle Ages,'' (Madison: University of Wisconsin Press), pp. 421–40.</ref> Bradwardine's analysis is an example of transferring a mathematical technique used by [[al-Kindi]] and [[Arnald of Villanova]] to quantify the nature of compound medicines to a different physical problem.<ref>Murdoch, John E. (1969) "''Mathesis in Philosophiam Scholasticam Introducta:'' The Rise and Development of the Application of Mathematics in Fourteenth Century Philosophy and Theology", in ''Arts libéraux et philosophie au Moyen Âge'' (Montréal: Institut d'Études Médiévales), at pp. 224–27.</ref>
 
One of the 14th-century [[Oxford Calculators]], [[William Heytesbury]], lacking [[differential calculus]] and the concept of [[Limit of a function|limits]], proposed to measure instantaneous speed "by the path that '''would''' be described by [a body] '''if'''... it were moved uniformly at the same degree of speed with which it is moved in that given instant".<ref>Clagett, Marshall (1961) ''The Science of Mechanics in the Middle Ages,'' (Madison: University of Wisconsin Press), pp. 210, 214–15, 236.</ref>
 
Heytesbury and others mathematically determined the distance covered by a body undergoing uniformly accelerated motion (today solved by [[Integral|integration]]), stating that "a moving body uniformly acquiring or losing that increment [of speed] will traverse in some given time a [distance] completely equal to that which it would traverse if it were moving continuously through the same time with the mean degree [of speed]".<ref>Clagett, Marshall (1961) ''The Science of Mechanics in the Middle Ages,'' (Madison: University of Wisconsin Press), p. 284.</ref>
 
[[Nicole Oresme]] at the [[University of Paris]] and the Italian [[Giovanni di Casali]] independently provided graphical demonstrations of this relationship, asserting that the area under the line depicting the constant acceleration, represented the total distance traveled.<ref>Clagett, Marshall (1961) ''The Science of Mechanics in the Middle Ages,'' (Madison: University of Wisconsin Press), pp. 332–45, 382–91.</ref> In a later mathematical commentary on Euclid's ''Elements'', Oresme made a more detailed general analysis in which he demonstrated that a body will acquire in each successive increment of time an increment of any quality that increases as the odd numbers. Since Euclid had demonstrated the sum of the odd numbers are the square numbers, the total quality acquired by the body increases as the square of the time.<ref>Nicole Oresme, "Questions on the ''Geometry'' of Euclid" Q. 14, pp. 560–65, in Marshall Clagett, ed., ''Nicole Oresme and the Medieval Geometry of Qualities and Motions,'' (Madison: University of Wisconsin Press, 1968).</ref>
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== Matematika Eropa modern dini ==
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[[Image:Pacioli.jpg|thumb|right|250px|Pacioli's portrait, a painting by [[Jacopo de' Barbari]], 1495, ([[Museo di Capodimonte]]).The open book to which he is pointing may be his ''Summa de Arithmetica, Geometria, Proportioni et Proportionalità''.<ref>Lauwers, Luc & Willekens, Marleen: "Five Hundred Years of Bookkeeping: A Portrait of Luca Pacioli" (Tijdschrift voor Economie en Management, [[Katholieke Universiteit Leuven]], 1994, vol:XXXIX issue:3 pages:289–304)[https://lirias.kuleuven.be/bitstream/123456789/119065/1/TEM1994-3_289-304p.pdf]</ref>]]
 
The development of [[mathematics]] and [[accounting]] was intertwined during the [[Renaissance]].<ref>Alan Sangster, Greg Stoner & Patricia McCarthy: "The market for Luca Pacioli’s Summa Arithmetica" (Accounting, Business & Financial History Conference, Cardiff, September 2007) p. 1–2</ref> While there is no direct relationship between algebra and accounting, the teaching of the subjects and the books published often intended for the children of merchants who were sent to reckoning schools (in [[Flanders]] and [[Germany]]) or [[abacus school]]s (known as ''abbaco'' in Italy), where they learned the skills useful for trade and commerce. There is probably no need for algebra in performing [[bookkeeping]] operations, but for complex bartering operations or the calculation of [[compound interest]], a basic knowledge of arithmetic was mandatory and knowledge of algebra was very useful.<ref>Heeffer, Albrecht: ''On the curious historical coincidence of algebra and double-entry bookkeeping'', Foundations of the Formal Sciences, [[Ghent University]], November 2009, p.7 [http://logica.ugent.be/albrecht/thesis/FOTFS2008-Heeffer.pdf]</ref>
 
[[Luca Pacioli]]'s ''"Summa de Arithmetica, Geometria, Proportioni et Proportionalità"'' (Italian: "Review of [[Arithmetic]], [[Geometry]], [[Ratio]] and [[Proportion]]") was first printed and published in [[Venice]] in 1494. It included a 27-page [[treatise]] on [[bookkeeping]], ''"Particularis de Computis et Scripturis"'' (Italian: "Details of Calculation and Recording"). It was written primarily for, and sold mainly to, merchants who used the book as a reference text, as a source of pleasure from the [[mathematical puzzles]] it contained, and to aid the education of their sons. In ''Summa Arithmetica'', Pacioli introduced symbols for [[plus and minus]] for the first time in a printed book, symbols that became standard notation in Italian Renaissance mathematics. ''Summa Arithmetica'' was also the first known book printed in Italy to contain [[algebra]].<ref>Alan Sangster, Greg Stoner & Patricia McCarthy: "The market for Luca Pacioli’s Summa Arithmetica" (Accounting, Business & Financial History Conference, Cardiff, September 2007) p.1–2 [http://www.cardiff.ac.uk/carbs/conferences/abfh07/summa.pdf]</ref>
 
Driven by the demands of navigation and the growing need for accurate maps of large areas, [[trigonometry]] grew to be a major branch of mathematics. [[Bartholomaeus Pitiscus]] was the first to use the word, publishing his ''Trigonometria'' in 1595. Regiomontanus's table of sines and cosines was published in 1533.<ref>{{cite book | last = Grattan-Guinness | first = Ivor | year = 1997 | title = The Rainbow of Mathematics: A History of the Mathematical Sciences | publisher = W.W. Norton | isbn = 0-393-32030-8}}</ref>
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== Abad ke-17 ==
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The 17th century saw an unprecedented explosion of mathematical and scientific ideas across Europe. [[Galileo]], an Italian, observed the moons of Jupiter in orbit about that planet, using a telescope based on a toy imported from Holland. [[Tycho Brahe]], a Dane, had gathered an enormous quantity of mathematical data describing the positions of the planets in the sky. His student, [[Johannes Kepler]], a German, began to work with this data. In part because he wanted to help Kepler in his calculations, [[John Napier]], in Scotland, was the first to investigate [[natural logarithm]]s. Kepler succeeded in formulating mathematical laws of planetary motion. The [[analytic geometry]] developed by [[René Descartes]] (1596–1650), a French mathematician and philosopher, allowed those orbits to be plotted on a graph, in [[Cartesian coordinates]].
 
Building on earlier work by many predecessors, [[Isaac Newton]], an Englishman, discovered the laws of physics explaining [[Kepler's Laws]], and brought together the concepts now known as [[infinitesimal calculus]]. Independently, [[Gottfried Wilhelm Leibniz]], in Germany, developed calculus and much of the calculus notation still in use today. Science and mathematics had become an international endeavor, which would soon spread over the entire world.<ref>Eves, Howard, An Introduction to the History of Mathematics, Saunders, 1990, ISBN 0-03-029558-0, p. 379, "...the concepts of calculus...(are) so far reaching and have exercised such an impact on the modern world that it is perhaps correct to say that without some knowledge of them a person today can scarcely claim to be well educated."</ref>
 
In addition to the application of mathematics to the studies of the heavens, applied mathematics began to expand into new areas, with the correspondence of [[Pierre de Fermat]] and [[Blaise Pascal]]. Pascal and Fermat set the groundwork for the investigations of [[probability theory]] and the corresponding rules of [[combinatorics]] in their discussions over a game of [[gambling]]. Pascal, with his [[Pascal's Wager|wager]], attempted to use the newly developing probability theory to argue for a life devoted to religion, on the grounds that even if the probability of success was small, the rewards were infinite. In some sense, this foreshadowed the development of [[utility theory]] in the 18th–19th century.
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== Abad ke-18 ==
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[[Image:Leonhard Euler.jpg|left|thumb|[[Leonhard Euler]] by [[Emanuel Handmann]].]]
 
The most influential mathematician of the 1700s was arguably [[Leonhard Euler]]. His contributions range from founding the study of [[graph theory]] with the [[Seven Bridges of Königsberg]] problem to standardizing many modern mathematical terms and notations. For example, he named the square root of minus 1 with the symbol <font face="times new Roman">[[Imaginary unit|''i'']]</font>, and he popularized the use of the Greek letter <math>\pi</math> to stand for the ratio of a circle's circumference to its diameter. He made numerous contributions to the study of topology, graph theory, calculus, combinatorics, and complex analysis, as evidenced by the multitude of theorems and notations named for him.
 
Other important European mathematicians of the 18th century included [[Joseph Louis Lagrange]], who did pioneering work in number theory, algebra, differential calculus, and the calculus of variations, and [[Laplace]] who, in the age of [[Napoleon]] did important work on the foundations of [[celestial mechanics]] and on [[statistics]].
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== Abad ke-19 ==
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[[Image:noneuclid.svg|right|thumb|400px|Behavior of lines with a common perpendicular in each of the three types of geometry]]
Throughout the 19th century mathematics became increasingly abstract. In the 19th century lived [[Carl Friedrich Gauss]] (1777–1855). Leaving aside his many contributions to science, in pure mathematics he did revolutionary work on [[function (mathematics)|function]]s of [[complex variable]]s, in [[geometry]], and on the convergence of [[series (mathematics)|series]]. He gave the first satisfactory proofs of the [[fundamental theorem of algebra]] and of the [[quadratic reciprocity law]].
 
This century saw the development of the two forms of [[non-Euclidean geometry]], where the [[parallel postulate]] of [[Euclidean geometry]] no longer holds.
The Russian mathematician [[Nikolai Ivanovich Lobachevsky]] and his rival, the Hungarian mathematician [[Janos Bolyai]], independently defined and studied [[hyperbolic geometry]], where uniqueness of parallels no longer holds. In this geometry the sum of angles in a triangle add up to less than 180°. [[Elliptic geometry]] was developed later in the 19th century by the German mathematician [[Bernhard Riemann]]; here no parallel can be found and the angles in a triangle add up to more than 180°. Riemann also developed [[Riemannian geometry]], which unifies and vastly generalizes the three types of geometry, and he defined the concept of a [[manifold]], which generalize the ideas of [[curve]]s and [[surface]]s.
The 19th century saw the beginning of a great deal of [[abstract algebra]]. [[Hermann Grassmann]] in Germany gave a first version of [[vector space]]s, [[William Rowan Hamilton]] in Ireland developed [[noncommutative algebra]]. The British mathematician [[George Boole]] devised an algebra that soon evolved into what is now called [[Boolean logic|Boolean algebra]], in which the only numbers were 0 and 1 and in which, 1&nbsp;+&nbsp;1&nbsp;=&nbsp;1. Boolean algebra is the starting point of [[mathematical logic]] and has important applications in [[computer science]].
 
[[Augustin-Louis Cauchy]], [[Bernhard Riemann]], and [[Karl Weierstrass]] reformulated the calculus in a more rigorous fashion.
 
Also, for the first time, the limits of mathematics were explored. [[Niels Henrik Abel]], a Norwegian, and [[Évariste Galois]], a Frenchman, proved that there is no general algebraic method for solving polynomial equations of degree greater than four. Other 19th century mathematicians utilized this in their proofs that straightedge and compass alone are not sufficient to [[trisect an arbitrary angle]], to construct the side of a cube twice the volume of a given cube, nor to construct a square equal in area to a given circle. Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks.
 
Abel and Galois's investigations into the solutions of various polynomial equations laid the groundwork for further developments of [[group theory]], and the associated fields of [[abstract algebra]]. In the 20th century physicists and other scientists have seen group theory as the ideal way to study [[symmetry]].
 
In the later 19th century, [[Georg Cantor]] established the first foundations of [[set theory]], which enabled the rigorous treatment of the notion of infinity and has become the common language of nearly all mathematics. Cantor's set theory, and the rise of [[mathematical logic]] in the hands of [[Peano]], [[L. E. J. Brouwer]], [[David Hilbert]], [[Bertrand Russell]], and [[A.N. Whitehead]], initiated a long running debate on the [[foundations of mathematics]].
 
The 19th century saw the founding of a number of national mathematical societies: the [[London Mathematical Society]] in 1865, the [[Société Mathématique de France]] in 1872, the [[Circolo Mathematico di Palermo]] in 1884, the [[Edinburgh Mathematical Society]] in 1883, and the [[American Mathematical Society]] in 1888.
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[[Image:Four Colour Map Example.svg|thumb|A map illustrating the [[Four Color Theorem]]]]
 
The 20th century saw mathematics become a major profession. Every year, thousands of new Ph.D.s in mathematics are awarded, and jobs are available in both teaching and industry. In earlier centuries, there were few creative mathematicians in the world at any one time. For the most part, mathematicians were either born to wealth, like [[John Napier|Napier]], or supported by wealthy patrons, like [[Gauss]]. A few, like [[Joseph Fourier|Fourier]], derived meager livelihoods from teaching in universities. [[Niels Henrik Abel]], unable to obtain a position, died in poverty of malnutrition and tuberculosis at the age of twenty-six.
 
As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest [[Mathematics Subject Classification]] runs to 46 pages<ref>[http://www.ams.org/mathscinet/msc/pdfs/classification2010.pdf Mathematics Subject Classification 2010]</ref>. More and more [[mathematical journal]]s were published and, by the end of the century, the development of the [[world wide web]] led to online publishing.
 
In a 1900 speech to the [[International Congress of Mathematicians]], [[David Hilbert]] set out a list of [[Hilbert's problems|23 unsolved problems in mathematics]]. These problems, spanning many areas of mathematics, formed a central focus for much of 20th century mathematics. Today, 10 have been solved, 7 are partially solved, and 2 are still open. The remaining 4 are too loosely formulated to be stated as solved or not.
 
Notable historical conjectures were finally proved. In 1976, [[Wolfgang Haken]] and [[Kenneth Appel]] used a computer to prove the [[four color theorem]]. [[Andrew Wiles]], building on the work of others, proved [[Fermat's Last Theorem]] in 1995. [[Paul Cohen (mathematician)|Paul Cohen]] and [[Kurt Gödel]] proved that the [[continuum hypothesis]] is [[logical independence|independent]] of (could neither be proved nor disproved from) the [[ZFC|standard axioms of set theory]]. In 1998 [[Thomas Callister Hales]] proved the [[Kepler conjecture]].
 
Mathematical collaborations of unprecedented size and scope took place. An example is the [[classification of finite simple groups]] (also called the "enormous theorem"), whose proof between 1955 and 1983 required 500-odd journal articles by about 100 authors, and filling tens of thousands of pages. A group of French mathematicians, including [[Jean Dieudonné]] and [[André Weil]], publishing under the [[pseudonym]] "[[Nicolas Bourbaki]]," attempted to exposit all of known mathematics as a coherent rigorous whole. The resulting several dozen volumes has had a controversial influence on mathematical education.<ref>Maurice Mashaal, 2006. ''Bourbaki: A Secret Society of Mathematicians''. [[American Mathematical Society]]. ISBN 0-8218-3967-5, ISBN 978-0-8218-3967-6.</ref>
 
[[Differential geometry]] came into its own when [[Einstein]] used it in [[general relativity]]. Entire new areas of mathematics such as [[mathematical logic]], [[topology]], and [[John von Neumann]]'s [[game theory]] changed the kinds of questions that could be answered by mathematical methods. All kinds of [[Mathematical structure|structures]] were abstracted using axioms and given names like [[metric space]]s, [[topological space]]s etc. As mathematicians do, the concept of an abstract structure was itself abstracted and led to [[category theory]]. [[Grothendieck]] and [[Jean-Pierre Serre|Serre]] recast [[algebraic geometry]] using [[Sheaf (mathematics)|sheaf theory]]. Large advances were made in the qualitative study of [[dynamical systems theory|dynamical systems]] that [[Poincaré]] had began in the 1890s. [[Measure theory]] was developed in the late 19th and early 20th century. Applications of measures include the [[Lebesgue integral]], [[Kolmogorov]]'s axiomatisation of [[probability theory]], and [[ergodic theory]]. [[Knot theory]] greatly expanded. Other new areas include [[functional analysis]], [[Laurent Schwarz]]'s [[Distribution (mathematics)|distribution theory]], [[fixed point theory]], [[singularity theory]] and [[René Thom]]'s [[catastrophe theory]], [[model theory]], and [[Mandelbrot]]'s [[fractals]].
 
The development and continual improvement of [[computer]]s, at first mechanical analog machines and then digital electronic machines, allowed [[industry]] to deal with larger and larger amounts of data to facilitate mass production and distribution and communication, and new areas of mathematics were developed to deal with this: [[Alan Turing]]'s [[computability theory]]; [[Computational complexity theory|complexity theory]]; [[Claude Shannon]]'s [[information theory]]; [[signal processing]]; [[data analysis]]; [[optimization (mathematics)|optimization]] and other areas of [[operations research]]. In the preceding centuries much mathematical focus was on [[calculus]] and continuous functions, but the rise of computing and communication networks led to an increasing importance of [[discrete mathematics|discrete]] concepts and the expansion of [[combinatorics]] including [[graph theory]]. The speed and data processing abilities of computers also enabled the handling of mathematical problems that were too time-consuming to deal with by pencil and paper calculations, leading to areas such as [[numerical analysis]] and [[symbolic computation]].
 
At the same time, deep insights were made about the limitations to mathematics. In 1929 and 1930, it was proved the truth or falsity of all statements formulated about the [[natural number]]s plus one of addition and multiplication, was [[decidable]], i.e. could be determined by algorithm. In 1931, [[Kurt Gödel]] found that this was not the case for the natural numbers plus both addition and multiplication; this system, known as [[Peano arithmetic]], was in fact [[incompleteness theorem|incompletable]]. (Peano arithmetic is adequate for a good deal of [[number theory]], including the notion of [[prime number]].) A consequence of Gödel's two [[incompleteness theorem]]s is that in any mathematical system that includes Peano arithmetic (including all of [[mathematical analysis|analysis]] and [[geometry]]), truth necessarily outruns proof, i.e. there are true statements that [[Incompleteness theorem|cannot be proved]] within the system. Hence mathematics cannot be reduced to mathematical logic, and [[David Hilbert]]'s dream of making all of mathematics complete and consistent died.
 
One of the more colorful figures in 20th century mathematics was [[Srinivasa Aiyangar Ramanujan]] (1887–1920) who, despite being largely self-educated, conjectured or proved over 3000 theorems, including properties of [[highly composite number]]s, the [[partition function (number theory)|partition function]] and its [[asymptotics]], and [[Ramanujan theta function|mock theta functions]]. He also made major investigations in the areas of [[gamma function]]s, [[modular form]]s, [[divergent series]], [[hypergeometric series]] and [[prime number theory]].
 
[[Paul Erdős]] published more papers than any other mathematician in history, working with hundreds of collaborators. Mathematicians have a game equivalent to the [[Kevin Bacon Game]], which leads to the [[Erdős number]] of a mathematician. This describes the "collaborative distance" between a person and Paul Erdős, as measured by joint authorship of mathematical papers.
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In 2000, the [[Clay Mathematics Institute]] announced the [[Millennium Prize Problems]], and in 2003 the [[Poincaré conjecture]] was solved by [[Grigori Perelman]].
 
Most mathematical journals now have online versions as well as print versions, and many online-only journals are launched. There is an increasing drive towards [[Open access (publishing)|open access publishing]], first popularized by the [[arXiv]].
 
Beginning in the late 20th century, but particularly in the 21st, mathematical research has become a global endeavor, such that it doesn't make much sense to speak of "ethnic" schools (e.g. Greek, Indian, etc...) of mathematics anymore.
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== Lihat pula ==
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*[[List of important publications in mathematics]]
*[[History of algebra]]
*[[History of calculus]]
*[[History of combinatorics]]
*[[History of geometry]]
*[[History of logic]]
*[[History of mathematical notation]]
*[[History of statistics]]
*[[History of trigonometry]]
*[[History of writing numbers]]
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== Referensi ==
 
== Bacaan lanjutan ==
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== Pranala luar ==
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*[http://www-history.mcs.st-andrews.ac.uk/ MacTutor History of Mathematics archive] (John J. O'Connor and Edmund F. Robertson; University of St Andrews, Scotland). An award-winning website containing detailed biographies on many historical and contemporary mathematicians, as well as information on notable curves and various topics in the history of mathematics.
*[http://aleph0.clarku.edu/~djoyce/mathhist/ History of Mathematics Home Page] (David E. Joyce; Clark University). Articles on various topics in the history of mathematics with an extensive bibliography.
*[http://astech.library.cornell.edu/ast/math/find/Collected-Works-of-Mathematicians.cfm A Bibliography of Collected Works and Correspondence of Mathematicians] (Steven W. Rockey; Cornell University Library).
*[http://www.mathourism.com Mathourism - Places with a mathematical historic interest]
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=== Jurnal ===
 
=== Direktori ===
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*[http://www.dcs.warwick.ac.uk/bshm/resources.html Links to Web Sites on the History of Mathematics] (The British Society for the History of Mathematics)
*[http://archives.math.utk.edu/topics/history.html History of Mathematics] Math Archives (University of Tennessee, Knoxville)
*[http://www.abc.se/~m9847/matre/history.html Mathematical Resources: History of Mathematics] (Bruno Kevius)
*[http://www.dm.unipi.it/~tucci/index.html History of Mathematics] (Roberta Tucci)
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[[Kategori:Wikipediawan yang bergabung bulan Juni 2010]]