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Babylonian mathematics
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{{see alsoBabylonian numerals}}File:Ybc7289bw.jpgthumb250pxrightBabylonian clay tablet YBC 7289 with annotations. The diagonal displays an approximation of the square root of 2 in four sexagesimal figures, 1 24 51 10, which is good to about six decimaldecimalBabylonian mathematics (also known as AssyroBabylonian mathematicsJOURNAL, Lewy, H., 1949, Studies in AssyroBabylonian mathematics and metrology, Orientalia, NS, 18, 40â€“67; 137â€“170, JOURNAL, Lewy, H., 1951, Studies in AssyroBabylonian mathematics and metrology, Orientalia, NS, 20, 1â€“12, JOURNAL, Bruins, E. M., 1953, La classification des nombres dans les mathÃ©matiques babyloniennes, Revue d'Assyriologie, 47, 4, 185â€“188, 23295221, JOURNAL, Cazalas, 1932, Le calcul de la table mathÃ©matique AO 6456, Revue d'Assyriologie, 29, 4, 183â€“188, 23284034, JOURNAL, Langdon, S., 1918, Assyriological notes: Mathematical observations on the ScheilEsagila tablet, Revue d'Assyriologie, 15, 3, 110â€“112, 23284735, BOOK, Robson, E., 2002, Guaranteed genuine originals: The Plimpton Collection and the early history of mathematical Assyriology, Mining the Archives: Festschrift for Christopher Walker on the occasion of his 60th birthday, C., Wunsch, ISLET, Dresden, 245â€“292, 3980846601, ) 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.BOOK, Aaboe, Asger, The culture of Babylonia: Babylonian mathematics, astrology, and astronomy, The Assyrian and Babylonian Empires and other States of the Near East, from the Eighth to the Sixth Centuries B.C., John, Boardman, I. E. S., Edwards, N. G. L., Hammond, E., Sollberger, C. B. F., Walker, Cambridge University Press, 1991, 0521227178, In respect of time they fall in two distinct groups: one from the Old Babylonian period (1830â€“1531 BC), the other mainly Seleucid from the last three or four centuries BC. In respect of content, there is scarcely any difference between the two groups of texts. Babylonian mathematics remained constant, in character and content, for nearly two millennia.In contrast to the scarcity of sources in Egyptian mathematics, 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, and baked hard in an oven or by the heat of the sun. The majority of recovered clay tablets date from 1800 to 1600 BC, and cover topics that include fractions, algebra, quadratic and cubic equations and the Pythagorean theorem. The Babylonian tablet YBC 7289 gives an approximation to sqrt{2} accurate to three significant sexagesimal digits (about six significant decimal digits). the content below is remote from Wikipedia
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Origins of Babylonian mathematics
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 early second millennium BC due to the wealth of data available. There has been debate over the earliest appearance of Babylonian mathematics, with historians suggesting a range of dates between the 5th and 3rd millennia BC.BOOK, An Aramaic Wisdom Text From Qumran: A New Interpretation Of The Levi Document, 86, Supplements to the Journal for the Study of Judaism, Henryk Drawnel, illustrated, BRILL, 2004, 9789004137530, 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, in the 5th millennium BC.BOOK, Ancient Mesopotamia: New Perspectives, Understanding ancient civilizations, Jane McIntosh, illustrated, ABCCLIO, 2005, 9781576079652, 265,Babylonian numerals
The Babylonian system of mathematics was a sexagesimal (base 60) numeral system. From this we derive the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 degrees in a circle.Michael A. Lombardi, "Why is a minute divided into 60 seconds, an hour into 60 minutes, yet there are only 24 hours in a day?", "Scientific American" March 5, 2007 The Babylonians were able to make great advances in mathematics for two reasons. Firstly, the number 60 is a superior highly composite number, having factors of 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 (including those that are themselves composite), facilitating calculations with fractions. Additionally, unlike the Egyptians and Romans, the Babylonians had a true placevalue system, where digits written in the left column represented larger values (much as, in our base ten system, 734 = 7Ã—100 + 3Ã—10 + 4Ã—1).BOOK, The Historical Roots of Elementary Mathematics, Lucas N. H. Bunt, Phillip S. Jones, Jack D. Bedient, reprint, Courier Corporation, 2001, 9780486139685, 44,Sumerian mathematics
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 and division problems. The earliest traces of the Babylonian numerals also date back to this period.Duncan J. Melville (2003). Third Millennium Chronology, Third Millennium Mathematics. St. Lawrence University.Old Babylonian mathematics (2000â€“1600 BC)
Most clay tablets that describe Babylonian mathematics belong to the Old Babylonian, which is why the mathematics of Mesopotamia is commonly known as Babylonian mathematics. Some clay tablets contain mathematical lists and tables, others contain problems and worked solutions.Arithmetic
The Babylonians used precalculated 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:
ab = frac{(a + b)^2  a^2  b^2}{2}
ab = frac{(a + b)^2  (a  b)^2}{4}
to simplify multiplication.The Babylonians did not have an algorithm for long division.ARTICLE,weblink An Overview of Babylonian mathematics, Instead they based their method on the fact that:
frac{a}{b} = a times frac{1}{b}
together with a table of reciprocals. Numbers whose only prime factors are 2, 3 or 5 (known as 5smooth or regular numbers) have finite reciprocals in sexagesimal notation, and tables with extensive lists of these reciprocals have been found.Reciprocals such as 1/7, 1/11, 1/13, etc. do not have finite representations in sexagesimal notation. To compute 1/13 or to divide a number by 13 the Babylonians would use an approximation such as:
frac{1}{13} = frac{7}{91} = 7 times frac {1}{91} approx 7 times frac{1}{90}=7 times frac{40}{3600} = frac{280}{3600} = frac{4}{60} + frac{40}{3600}.
Algebra
{{See alsoSquare root of 2#History}}The Babylonian clay tablet YBC 7289 (c. 1800â€“1600 BC) gives an approximation of {{math{{sqrt2}}}} in four sexagesimal figures, {{nowrap1 24 51 10}}, which is accurate to about six decimal digits,Fowler and Robson, p. 368.Photograph, illustration, and description of the root(2) tablet from the Yale Babylonian CollectionHigh resolution photographs, descriptions, and analysis of the root(2) tablet (YBC 7289) from the Yale Babylonian Collection and is the closest possible threeplace sexagesimal representation of {{math{{sqrt2}}}}:
1 + frac{24}{60} + frac{51}{60^2} + frac{10}{60^3} = frac{30547}{21600} = 1.41421overline{296}.
As well as arithmetical calculations, Babylonian mathematicians also developed algebraic methods of solving equations. Once again, these were based on precalculated tables.To solve a quadratic equation, the Babylonians essentially used the standard quadratic formula. They considered quadratic equations of the form:
x^2 + bx = c
where b and c were not necessarily integers, but c was always positive. They knew that a solution to this form of equation is:{{Citation neededdate=December 2011}}
x =  frac{b}{2} + sqrt{ left ( frac{b}{2} right )^2 + c}
and they found square roots efficiently using division and averaging.JOURNAL, Allen, Arnold, The American Mathematical Monthly, January 1999, Reviews: Mathematics: From the Birth of Numbers. By Jan Gullberg, 106, 1, 77â€“85, 2589607, 10.2307/2589607, They always used the positive root because this made sense when solving "real" problems. Problems of this type included finding the dimensions of a rectangle given its area and the amount by which the length exceeds the width.Tables of values of n3 + n2 were used to solve certain cubic equations. For example, consider the equation:
ax^3 + bx^2 = c.
Multiplying the equation by a2 and dividing by b3 gives:
left ( frac{ax}{b} right )^3 + left ( frac {ax}{b} right )^2 = frac {ca^2}{b^3}.
Substituting y = ax/b gives:
y^3 + y^2 = frac {ca^2}{b^3}
which could now be solved by looking up the n3 + n2 table to find the value closest to the right hand side. The Babylonians accomplished this without algebraic notation, showing a remarkable depth of understanding. However, they did not have a method for solving the general cubic equation.Growth
Babylonians modeled exponential growth, constrained growth (via a form of sigmoid functions), and doubling time, the latter in the context of interest on loans.Clay tablets from c. 2000 BCE include the exercise "Given an interest rate of 1/60 per month (no compounding), compute the doubling time." This yields an annual interest rate of 12/60 = 20%, and hence a doubling time of 100% growth/20% growth per year = 5 years.Why the "Miracle of Compound Interest" leads to Financial Crises, by Michael HudsonHave we caught your interest? by John H. WebbPlimpton 322
The Plimpton 322 tablet contains a list of "Pythagorean triples", i.e., integers (a,b,c) such that a^2+b^2=c^2.The triples are too many and too large to have been obtained by brute force.Much has been written on the subject, including some speculation (perhaps anachronistic) as to whether the tablet could have served as an early trigonometrical table. Care must be exercised to see the tablet in terms of methods familiar or accessible to scribes at the time.[...] the question "how was the tablet calculated?" does not have to have thesame answer as the question "what problems does the tablet set?" The first can be answeredmost satisfactorily by reciprocal pairs, as first suggested half a century ago, and the secondby some sort of righttriangle problems.(E. Robson, "Neither Sherlock Holmes nor Babylon: a reassessment of Plimpton 322", Historia Math. 28 (3), p. 202).Geometry
Babylonians knew the common rules for measuring volumes and areas. They measured the circumference of a circle as three times the diameter and the area as onetwelfth the square of the circumference, which would be correct if {{pi}} is estimated as 3. They were aware that this was an approximation, and one Old Babylonian mathematical tablet excavated near Susa in 1936 (dated to between the 19th and 17th centuries BCE) gives a better approximation of {{pi}} as 25/8 = 3.125, about 0.5 percent below the exact value.David Gilman Romano, Athletics and Mathematics in Archaic Corinth: The Origins of the Greek Stadion, American Philosophical Society, 1993, p. 78."A group of mathematical clay tablets from the Old Babylonian Period, excavated at Susa in 1936, and published by E.M. Bruins in 1950, provide the information that the Babylonian approximation of {{pi}} was 3â…› or 3.125."E. M. Bruins, Quelques textes mathÃ©matiques de la Mission de Suse, 1950.E. M. Bruins and M. Rutten, Textes mathÃ©matiques de Suse, MÃ©moires de la Mission archÃ©ologique en Iran vol. XXXIV (1961).See also {{citationfirst=Petrlast=Beckmanntitle=A History of Pipublisher=St. Martin's Pressplace=New Yorkyear=1971pages=12, 21â€“22}}"in 1936, a tablet was excavated some 200 miles from Babylon. [...] The mentioned tablet, whose translation was partially published only in 1950, [...] states that the ratio of the perimeter of a regular hexagon to the circumference of the circumscribed circle equals a number which in modern notation is given by 57/60 + 36/(60)2 [i.e. {{pi}} = 3/0.96 = 25/8]".Jason Dyer, On the Ancient Babylonian Value for Pi, 3 December 2008.The volume of a cylinder was taken as the product of the base and the height, however, the volume of the frustum of a cone or a square pyramid was incorrectly taken as the product of the height and half the sum of the bases. The Pythagorean theorem was also known to the Babylonians.{{sfnNeugebauer1969page=36ps=. "In other words it was known during the whole duration of Babylonian mathematics that the sum of the squares on the lengths of the sides of a right triangle equals the square of the length of the hypotenuse."}}{{sfnHÃ¸yruppage=406ps=. "To judge from this evidence alone it is therefore likely that the Pythagorean rule was discovered within the lay surveyorsâ€™ environment, possibly as a spinoff from the problem treated in Db2146, somewhere between 2300 and 1825 BC." (Db2146 is an Old Babylonian clay tablet from Eshnunna concerning the computation of the sides of a rectangle given its area and diagonal.)}}{{sfnRobson2008page=109ps=. "Many Old Babylonian mathematical practitioners â€¦ knew that the square on the diagonal of a right triangle had the same area as the sum of the squares on the length and width: that relationship is used in the worked solutions to word problems on cutandpaste â€˜algebraâ€™ on seven different tablets, from EÅ¡nuna, Sippar, Susa, and an unknown location in southern Babylonia."}}The "Babylonian mile" was a measure of distance equal to about 11.3 km (or about seven modern miles).This measurement for distances eventually was converted to a "timemile" used for measuring the travel of the Sun, therefore, representing time.Eves, Chapter 2.The ancient Babylonians had known of theorems concerning the ratios of the sides of similar triangles for many centuries, but they lacked the concept of an angle measure and consequently, studied the sides of triangles instead.BOOK, Boyer, Carl Benjamin Boyer, A History of Mathematics, 1991, Greek Trigonometry and Mensuration, 158â€“159, The Babylonian astronomers kept detailed records of the rising and setting of stars, the motion of the planets, and the solar and lunar eclipses, all of which required familiarity with angular distances measured on the celestial sphere.BOOK, Trigonometric Delights, Eli, Maor, 1998, Princeton University Press, 0691095418, 20, They also used a form of Fourier analysis to compute ephemeris (tables of astronomical positions), which was discovered in the 1950s by Otto Neugebauer.BOOK, The evolution of applied harmonic analysis: models of the real world, Elena, Prestini,weblink BirkhÃ¤user, 2004, 9780817641252, , p. 62BOOK,weblink Indiscrete thoughts, GianCarlo, Rota, Fabrizio, Palombi, GianCarlo Rota, BirkhÃ¤user, 1997, 9780817638665, , p. 11{{sfnNeugebauer1969}}JOURNAL, physics/0310126, Analyzing shell structure from Babylonian and modern times, Lisauthor1link= Lis BrackBernsen, Matthias, Brack, 10.1142/S0218301304002028, 13, 2004, International Journal of Modern Physics E, 247â€“260, 2004IJMPE..13..247B, To make calculations of the movements of celestial bodies, the Babylonians used basic arithmetic and a coordinate system based on the ecliptic, the part of the heavens that the sun and planets travel through.Tablets found in the British Museum provide evidence that the Babylonians even went so far as to have a concept of objects in an abstract mathematical space. The tablets date from between 350 and 50 B.C.E., revealing that the Babylonians understood and used geometry even earlier than previously thought. The Babylonians used a method for estimating the area under a curve by drawing a trapezoid underneath, a technique previously believed to have originated in 14th century Europe. This method of estimation allowed them to, for example, find the distance Jupiter had traveled in a certain amount of time.WEB, Babylonians Were Using Geometry Centuries Earlier Than Thought,weblink Smithsonian, 20160201, Jesse, Emspak, Influence{{refimprove sectiondate=October 2017}}Since the rediscovery of the Babylonian civilization, it has become apparent that Greek and Hellenistic mathematicians and astronomers, and in particular Hipparchus, borrowed greatly from the Babylonians.Franz Xaver Kugler demonstrated in his book Die Babylonische Mondrechnung ("The Babylonian lunar computation", Freiburg im Breisgau, 1900) the following: Ptolemy had stated in his Almagest IV.2 that Hipparchus improved the values for the Moon's periods known to him from "even more ancient astronomers" by comparing eclipse observations made earlier by "the Chaldeans", and by himself. However, Kugler found that the periods that Ptolemy attributes to Hipparchus had already been used in Babylonian ephemerides, specifically the collection of texts nowadays called "System B" (sometimes attributed to Kidinnu). Apparently, Hipparchus only confirmed the validity of the periods he learned from the Chaldeans by his newer observations.It is clear that Hipparchus (and Ptolemy after him) had an essentially complete list of eclipse observations covering many centuries. Most likely these had been compiled from the "diary" tablets: these are clay tablets recording all relevant observations that the Chaldeans routinely made. Preserved examples date from 652 BC to AD 130, but probably the records went back as far as the reign of the Babylonian king Nabonassar: Ptolemy starts his chronology with the first day in the Egyptian calendar of the first year of Nabonassar, i.e., 26 February 747 BC.This raw material by itself must have been hard to use, and no doubt the Chaldeans themselves compiled extracts of e.g., all observed eclipses (some tablets with a list of all eclipses in a period of time covering a saros have been found). This allowed them to recognise periodic recurrences of events. Among others they used in System B (cf. Almagest IV.2):
See also
Notes{{reflist30em}}References
, Robson, Eleanor, Eleanor Robson , , Neither Sherlock Holmes nor Babylon: a reassessment of Plimpton 322 , 2001 , Historia Math. , 28 , 3 , 167â€“206 , 10.1006/hmat.2001.2317 , 1849797

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