Matematik Babylon

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Batu bersurat tanah liat Babylon YBC 7289 dengan anotasi. Pepenjuru menggambarkan anggaran punca kuasa dua 2 dalam empat angka perenam-puluhan, yang sekitar enam angka perpuluhan.
1 + 24/60 + 51/602 + 10/603 = 1.41421296...

Matematik Babylon merujuk kepada matematik orang Mesopotamia (Iraq silam), dari zaman awal Sumeria hingga ke kejatuhan Babylon pada 539 SM. Berbeza dengan kekurangan sumber untuk Matematik Mesir, ilmu matematik Babylon boleh dirujuk dari 400 batu bersurat tanah liat yang ditemui sejak 1850-an. Ditulis dalam tulisan pepaku, batu bersurat tersebut ditulis sementara tanah liat masih lembab, dan dibakar keras dalam sebuah ketuhar atau oleh kepanasan matahari. Kebanyakan batu bersurat tersebut bertarikh dari 1800 hingga ke 1600 SM, dan meliputi topik yang termasuk pecahan, algebra, kuadratik dan kuasa tiga, teorem Pythagoras, dan pengiraan tigaan Pythagoras dan mungkin juga fungsi trigonometri (sila lihat Plimpton 322). Batu bersurat Babylon YBC 7289 memberikan suatu penganggaran \sqrt{2} tepat kepada hampir enam tempat perpuluhan.

Bilangan Babylon[sunting | sunting sumber]

Rencana utama: Bilangan Babylon

Sistem matematik Babylon adalah sexagesimal (asas-60) sistem bilangan. Dari ini kita melihat kegunaan hari moden 60 saat dalam satu minit, 60 minit dalam satu jam, dan 360 (60×6) darjah dalam sebuah bulatan. Orang Babylon dapat membuat kemajuan yang hebat dalam matematik berdasarkan dua alasan. Pertama, nombor 60 adalah suatua Bilangan Highly composite, mempunyai pembahagian 2, 3, 4, 5, 6, 10, 12, 15, 20, dan 30, pengiraan mudah dengan pecahan. Tambahan lagi, tidak seperti orang Mesir dan Rom, orang Babylon dan India mempunyai suatu sistem letak-nilai yang bear, di mana digit ditulis di column kiri mewakili nilai-nilai yang lebih besar (seperti dalam sistem asas sepuluh kita: 734 = 7×100 + 3×10 + 4×1).

Matematik Sumer (3000-2300 SM)[sunting | sunting sumber]

Bukti terawal matematik tulisan melatar belakang ke Sumer yang silam, yang membinakan peradaban di Mesopotamia. mereka membangunkan sistem kompleks metrologi dari 3000 SM. Dari 2600 SM selanjutnya, orang Sumer menulis jadual perdaraban pada batu bersurat tanah liat dan menguruskan dengan latihan geometri dan masalah pembahagian. Kesan-kesan terawal bilangan Babylon juga melatar belakang ke jangka ini.[1]

Matematik Babylon Lama (2000-1600 SM)[sunting | sunting sumber]

Zaman Babylon Lama adalah tempoh yang mana kebanyakan batu bersurat tanah liat mengenai asalnya matematik Babylon, dan oleh kerana itulah matematik Mesopotamia umumnya digelar matematik Babylon. Sesetengah batu bersurat tanah liat mengandungi senarai dan jadual, yang lain mengandungi dan jawapan yang dikerjakan.

Aritmetik[sunting | sunting sumber]

Orang Babylon menggunakan kegunaan lebsar pada jadual pra-kiraan untuk membantu dengan aritmetik. Contohnya, dua batu bersurat didapati di Senkerah di Euphrates pada 1854, bermula dari 2000 SM, memberikan senarai-senarai persegi bilangan ke atas 59 dan cubes bilangan ke atas 32. Orang Babylon menggunakan senarai-senarai persegi bersamaan dengan persamaan

ab = \frac{(a + b)^2 - a^2 - b^2}{2}
ab = \frac{(a + b)^2 - (a - b)^2}{4}

untuk memudahkan perdaraban.

Orang Babylon tidak mempunyai suatu algoritma untuk bahagi panjang. Daripada itu mereka berasaskan kaedah mereka ternyatanya bahawa

\frac{a}{b} = a \times \frac{1}{b}

bersama dengan sebuah jadual reciprocals. Bilangan yang hanyalah faktor perdana adalah 2, 3 atau 5 (digelar sebagai 5-smooth atau bilangan sering) mempunyai finite reciprocals dalam notasi sexagesimal, dan jadual-jadual dengan senarai-senarai extensive pada reciprocals ini telah ditemukan.

Resiprokal seperti 1/7, 1/11, 1/13, dll. tidak mempunyai pewakilan finite pada notasi sexagesimal. Untuk mengira 1/13 atau untuk membahagikan sebuah nombor dengan 13 orang Babylon akan menggunakan suatu anggaran seperti

\frac{1}{13} = \frac{7}{91} = 7 \times \frac {1}{91} \approx 7 \times \frac{1}{90}=7 \times \frac{40}{3600}.

Algebra[sunting | sunting sumber]

Dan juga pengiraan aritmetik, ahli matematik Babylon juga mengembang kaedah algebra pada penyelesaian persamaa. Sekali lagi, ini berasaskan jadual pra-kiraan.

Untuk menyelesai suatu persamaan kuadratik orang Babylon essentially menggunakan rumusan kuadratik piawai. Mereka menganggapkan persamaan kuadratik pada bentuk

\ x^2 + bx = c

di mana sini b dan c tidak seharusnya integer, tetapi c adalah sentiasa positif. Mereka mengetahui bahawa suatu jawapan ke bentuk persamaan ini adalah

x = - \frac{b}{2} + \sqrt{ \left ( \frac{b}{2} \right )^2 + c}

dan mereka akan menggunakan meja segi empat mereka dengan cara terbalik untuk mencari akar persegi. Mereka sentiasa menggunakan akar positif kerana ini masuk akal ketika menyelesaikan masalah "benar". Masalah-masalah jenis ini termasuk mencari dimensi sebuah segi empat tepat diberikan ruang ini dan jumlah yang mana panjangnya melebihi lebarnya.

Jadual nilai n3+n2 telah digunakan untuk menyelesai sesetengah persamaan kubik. Contohnya, anggapkan persamaan

\ ax^3 + bx^2 = c.

'''Pendaraban persamaan oleh''' a2 dan dibahagikan dengan b3 memberikan

\left ( \frac{ax}{b} \right )^3 + \left ( \frac {ax}{b} \right )^2 = \frac {ca^2}{b^3}.

Menggantikan y = ax/b memberikan

y^3 + y^2 = \frac {ca^2}{b^3}

yang dapat sekarang diselesai dengan melihat n3+n2 table untuk mendapatkan nilai terdekat ke sudut tangan kanan. Orang Babylon accomplished ini tanpa notasi algebra, menunjukkan suatu pendalaman remarkable pada kefahaman. Meskipun, mereka tidak mempunyai suatu kaedah untuk menyelesaikan persamaan kubik.

Geometri[sunting | sunting sumber]

The Babylonians may have known the general rules for measuring areas and volumes. They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if π is estimated as 3. 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. Also, there was a recent discovery in which a tablet used π as 3 and 1/8. The Babylonians are also known for the Babylonian mile, which was a measure of distance equal to about seven miles today. This measurement for distances eventually was converted to a time-mile used for measuring the travel of the Sun, therefore, representing time.[2]

Trigonometri[sunting | sunting sumber]

The ancient Babylonians had known of theorems on 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 were studied instead.[3]

The Babylonian astronomers kept detailed records on 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.[4]

There is also evidence that the Babylonians first used trigonometric functions, based on a table of numbers written on the Babylonian cuneiform tablet, Plimpton 322 (circa 1900 BC), which can be interpreted as a trigonometric table of secants.[5]

Plimpton 322[sunting | sunting sumber]

Rencana utama: Plimpton 322

In each row of the Plimpton 322 tablet, the number in the second column can be interpreted as the shortest side s of a right triangle, and the number in the third column can be interpreted as the hypotenuse d of the triangle. The number in the first column is either the fraction \tfrac{s^2}{l^2} or \tfrac{d^2}{l^2}, where l denotes the longest side of the same right triangle. However, scholars differ on how these numbers were generated and why the Babylonians would have been interested in such tables.

Neugebauer (1951) argued for a number-theoretic interpretation, pointing out that this table provides a list of (pairs of numbers from) Pythagorean triples. For instance, line 11 of the table can be interpreted as describing a triangle with short side 3/4 and hypotenuse 5/4, forming the side:hypotenuse ratio of the familiar (3,4,5) right triangle. If p and q are two coprime numbers, then ( p^2 - q^2,\, 2pq,\, p^2 + q^2 ) form a Pythagorean triple, and all Pythagorean triples can be formed in this way. For instance, line 11 can be generated by this formula with p = 1 and q = 1/2. As Neugebauer argues, each line of the tablet can be generated by a pair (p,q) that are both regular numbers, integer divisors of a power of 60. This property of p and q being regular leads to a denominator that is regular, and therefore to a finite sexagesimal representation for the fraction in the first column. Neugebauer's explanation is the one followed e.g. by Conway and Guy (1996). However, as Robson points out, Neugebauer's theory fails to explain how the values of p and q were chosen: there are 92 pairs of coprime regular numbers up to 60, and only 15 entries in the table. In addition, it does not explain why the table entries are in the order they are listed in, nor what the numbers in the first column were used for.

Joyce (1995) provides a trigonometric explanation: the values of the first column can be interpreted as the squared cosine or tangent (depending on the missing digit) of the angle opposite the short side of the right triangle described by each row, and the rows are sorted by these angles in roughly one-degree increments. However, Robson argues on linguistic grounds that this theory is "conceptually anachronistic": it depends on too many other ideas not present in the record of Babylonian mathematics from that time.

Robson (2001,2002), based on prior work by Bruins (1949,1955) and others, instead takes an approach that in modern terms would be characterized as algebraic, though she describes it in concrete geometric terms and argues that the Babylonians would also have interpreted this approach geometrically. Robson bases her interpretation on another tablet, YBC 6967, from roughly the same time and place.[6] This tablet describes a method for solving what we would nowadays describe as quadratic equations of the form x-\tfrac1x=c, by steps (described in geometric terms) in which the solver calculates a sequence of intermediate values v1 = c/2, v2 = v12, v3 = 1 + v2, and v4 = v31/2, from which one can calculate x = v4 + v1 and 1/x = v4 - v1. Robson argues that the columns of Plimpton 322 can be interpreted as the following values, for regular number values of x and 1/x in numerical order: v3 in the first column, v1 = (x - 1/x)/2 in the second column, and v4 = (x + 1/x)/2 in the third column. In this interpretation, x and 1/x would have appeared on the tablet in the broken-off portion to the left of the first column. For instance, row 11 of Plimpton 322 can be generated in this way for x = 2. Thus, the tablet can be interpreted as giving a sequence of worked-out exercises of the type solved by the method from tablet YBC 6967. It could, Robson suggests, have been used by a teacher as a problem set to assign to students.


Since the rediscovery of the Babylonian civilization, it has become apparent that Greek and Hellenistic mathematicians and astronomers, and in particular Hipparchus, borrowed a lot from the Chaldeans.

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):

  • 223 (synodic) months = 239 returns in anomaly (anomalistic month) = 242 returns in latitude (draconic month). This is now known as the saros period which is very useful for predicting eclipses.
  • 251 (synodic) months = 269 returns in anomaly
  • 5458 (synodic) months = 5923 returns in latitude
  • 1 synodic month = 29;31:50:08:20 days (sexagesimal; 29.53059413… days in decimals = 29 days 12 hours 44 min 3⅓ s)

The Babylonians expressed all periods in synodic months, probably because they used a lunisolar calendar. Various relations with yearly phenomena led to different values for the length of the year.

Similarly various relations between the periods of the planets were known. The relations that Ptolemy attributes to Hipparchus in Almagest IX.3 had all already been used in predictions found on Babylonian clay tablets.

All this knowledge was transferred to the Greeks probably shortly after the conquest by Alexander the Great (331 BC). According to the late classical philosopher Simplicius (early 6th century AD), Alexander ordered the translation of the historical astronomical records under supervision of his chronicler Callisthenes of Olynthus, who sent it to his uncle Aristotle. It is worth mentioning here that although Simplicius is a very late source, his account may be reliable. He spent some time in exile at the Sassanid (Persian) court, and may have accessed sources otherwise lost in the West. It is striking that he mentions the title tèresis (Greek: guard) which is an odd name for a historical work, but is in fact an adequate translation of the Babylonian title massartu meaning "guarding" but also "observing". Anyway, Aristotle's pupil Callippus of Cyzicus introduced his 76-year cycle, which improved upon the 19-year Metonic cycle, about that time. He had the first year of his first cycle start at the summer solstice of 28 June 330 BC (Julian proleptic date), but later he seems to have counted lunar months from the first month after Alexander's decisive battle at Gaugamela in fall 331 BC. So Callippus may have obtained his data from Babylonian sources and his calendar may have been anticipated by Kidinnu. Also it is known that the Babylonian priest known as Berossus wrote around 281 BC a book in Greek on the (rather mythological) history of Babylonia, the Babyloniaca, for the new ruler Antiochus I; it is said that later he founded a school of astrology on the Greek island of Kos. Another candidate for teaching the Greeks about Babylonian astronomy/astrology was Sudines who was at the court of Attalus I Soter late in the 3rd century BC.

In any case, the translation of the astronomical records required profound knowledge of the cuneiform script, the language, and the procedures, so it seems likely that it was done by some unidentified Chaldeans. Now, the Babylonians dated their observations in their lunisolar calendar, in which months and years have varying lengths (29 or 30 days; 12 or 13 months respectively). At the time they did not use a regular calendar (such as based on the Metonic cycle like they did later), but started a new month based on observations of the New Moon. This made it very tedious to compute the time interval between events.

What Hipparchus may have done is transform these records to the Egyptian calendar, which uses a fixed year of always 365 days (consisting of 12 months of 30 days and 5 extra days): this makes computing time intervals much easier. Ptolemy dated all observations in this calendar. He also writes that "All that he (=Hipparchus) did was to make a compilation of the planetary observations arranged in a more useful way" (Almagest IX.2). Pliny states (Naturalis Historia II.IX(53)) on eclipse predictions: "After their time (=Thales) the courses of both stars (=Sun and Moon) for 600 years were prophesied by Hipparchus, …". This seems to imply that Hipparchus predicted eclipses for a period of 600 years, but considering the enormous amount of computation required, this is very unlikely. Rather, Hipparchus would have made a list of all eclipses from Nabonasser's time to his own.

Other traces of Babylonian practice in Hipparchus' work are:

  • first Greek known to divide the circle in 360 degrees of 60 arc minutes.
  • first consistent use of the sexagesimal number system.
  • the use of the unit pechus ("cubit") of about 2° or 2½°.
  • use of a short period of 248 days = 9 anomalistic months.

Matematik Babylon di Alexandria[sunting | sunting sumber]

Sewaktu zaman Hellen, astronomi Babylon dan matematik exerted suatu pengaruh hebat pada ahli matematik Alexandria, di Mesir Ptolemy dan Mesir Rom. Ini adalah terutamanya apparent dalam astronomi dan karya matematik Hipparchus, Ptolemy, Hero dari Alexandria, dan Diophantus. Dalam kes Diophantus, pengaruh Babylon adalah sangat kuat dalam Arithmeticanya yang sesetengah sarjana telah argued bahawa beliau sendiri mungkin adalah seorang Babylon Hellen.[7] Pengaruh Babylon kuat pada karya Hero telah membawakan ke spekulasi bahwa beliau adalah seorang Phoenicia.[8]

Matematik Islam di Mesopotamia[sunting | sunting sumber]

Selepas penaklukan Islam Mesopotamia Farsi, daerah Mesopotamia yang digelar "Iraq" dalam bahasa Arab. Di bawah khilafah Abbasid, ibu negara Empayar Arab adalah Baghdad, yang dibina di Iraq sewaktu abad ke-8. Dari abad ke-8 hingga ke-13, sering digelar "Zaman Keemasan Islam", Iraq/Mesopotamia sekali lagi menjadi pusat aktiviti matematik. Banyak para ahli matematik pada waktu itu adalah aktif di Iraq, termasuk Muḥammad ibn Mūsā al-Khwārizmī (Algoritmi), Al-Abbās ibn Said al-Jawharī, 'Abd al-Hamīd ibn Turk, Al-Kindi (Alkindus), Hunayn ibn Ishaq (Johannitius), adik-beradik lelaki Banū Mūsā, keluarga Thābit ibn Qurra, Muhammad ibn Jābir al-Harrānī al-Battānī (Albatenius), the Brethren of Purity, Al-Saghani, Abū Sahl al-Qūhī, Ibn Sahl, Abu Nasr Mansur ibn Iraq, Ibn al-Haytham (Alhazen), Ibn Tahir al-Baghdadi, dan Ibn Yahyā al-Maghribī al-Samaw'al. Altiviti matematik berakhir di Iraq/Mesopotamia selepas sack of Baghdad pada 1258.

Lihat juga[sunting | sunting sumber]

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Nota[sunting | sunting sumber]

  1. Duncan J. Melville (2003). Third Millennium Chronology, Third Millennium Mathematics. St. Lawrence University.
  2. Eves, Chapter 2.
  3. Boyer (1991). "Greek Trigonometry and Mensuration". ms. 158–159.  Missing or empty |title= (bantuan)
  4. Maor, Eli (1998), Trigonometric Delights, Princeton University Press, ms. 20, ISBN 0691095418 
  5. Joseph, pp. 383-4
  6. Neugebauer, O.; Sachs, A. J. (1945). Mathematical Cuneiform Texts. American Oriental Series, vol. 29. New Haven: American Oriental Society and the American Schools of Oriental Research. ms. text Ua. 
  7. D. M. Burton (1991, 1995), History of Mathematics, Dubuque, IA (Wm.C. Brown Publishers):
    "Diophantos was most likely a Hellenized Babylonian."
  8. Boyer (1968 [1991]). "Greek Trigonometry and Mensuration". A History of Mathematics. ms. 171–2. :
    Sekurang-kurangnya dari hari Iskandar Agung berhampiran dengan dunia klasik, tidak diserbasalahkan bahwa adanya antarahubungan di antara Greece dan Mesopotamia, dan ia kelihatan bahawa jelas bahawa geometri aritmetik dan algebra berlanjut untuk memberikan considerable pengaruh dalam dunia Hellen. Aspek ini pada matematik, contohnya, kelihatan sangat kuat pada Heron dari Alexandria (fl. ca. A.D. 100) bahawa Heron sekali dahulu dianggap seorang Mesir atau Phoenicia daripada orang Greece. Sekarang ia difikir bahawa Heron menggambarkan sejenis matematik telah lama berada di Greece tetapi tidak dapat mendapatkan seorang pewakilan di kalangan tokoh-tokoh hebat - kecuali mungkin digambarkan oleh Ptolemy di Tetrabiblos.

Rujukan[sunting | sunting sumber]

  • Berriman, A. E., The Babylonian quadratic equation (1956).
  • Boyer, C. B., A History of Mathematics, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, (1989) ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7).
  • Joseph, G. G., The Crest of the Peacock, Princeton University Press (October 15, 2000), ISBN 0-691-00659-8.
  • Joyce, David E. (1995). "Plimpton 322". 
  • Neugebauer, O., "Exact Sciences of Antiquity", Dover (1969).
  • O'Connor, J. J. and Robertson, E. F., "An overview of Babylonian mathematics", MacTutor History of Mathematics, (December 2000).
  • Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a reassessment of Plimpton 322". Historia Math. 28 (3): 167–206. doi:10.1006/hmat.2001.2317. Templat:MathSciNet. 
  • Eleanor Robson, Words and pictures: New light on Plimpton 322, The American Mathematical Monthly. Washington: Feb 2002. Vol. 109, Iss. 2; pg. 105
  • Toomer, G. J., Hipparchus and Babylonian Astronomy, (1981).

Pautan luar[sunting | sunting sumber]