Matematik India

Matematik India — yang dimaksudkan di sini ialah matematik yang muncul Asia Selatan sejak zaman silam hingga akhir kurun ke-18 — bermula dalam tamadun Lembah Indus pada Zaman Gangsa (2600-1900SM) dan kebudayaan Veda pada Zaman Besi (1500-500SM). Semasa tempoh matematik India klasik (400M hingga 1200M), sumbangan-sumbangan penting telah dibuat oleh sarjana-sarjana seperti Aryabhatta, Brahmagupta, dan Bhaskara II. Ahli matematik India telah membuat sumbangan-sumbangan terawal terhadap pengkajian sistem nombor desimal,^[1] sifar,^[2] nombor negatif,^[3] aritmetik, dan algebra.^[4]

Di samping itu, trigonometri yang berkembang di dunia Hellenistik dan telah diperkenalkan ke India kuno melalui terjemahan buku Yunani, ^[5] berkembang lanjut di India, dan khususnya, takrifan-takrifan moden bagi sinus dan kosinus dimajukan di sana. ^[6] Konsep-konsep matematik ini dipindahkan ke Timur Tengah, China dan Eropah^[4] dan membawa kepada kemajuan lebih lanjut yang kini membentuk asas-asas bagi banyak bidang matematik.

Karya-karya matematik India kuno dan Zaman Pertengahan, semuanya ditulis dalam bahasa Sanskrit, selalunya mengandungi satu seksyen sutra yang merupakan satu set peraturan-peraturan atau masalah-masalah yang dinyatakan dengan cukup cermat dalam ayat supaya dapat membantu pelajar untuk menghafaznya. Ini diikuti dengan seksyen kedua yang mengandungi ulasan prosa (kadang-kala pelbagai ulasan oleh pelbagai sarjana) yang menjelaskan masalah itu dengan lebih terperinci dan menyediakan hujah bagi penyelesaian. Di seksyen prosa, bentuk itu tidak dianggap sebagai penting sepertimana idea terbabit. ^[7][8] Semua karya-karya matematik dipindahkan secara lisan sehinggalah sekitar tahun 500SM; selepas itu, semua karya itu dipindahkan secara lisan dan juga dalam bentuk manuskrip. Dokumen matematik tertua yang dihasilkan di India yang masih wujud ialah Manuskrip Bakhshali kulit kayu birch yang dijumpai pada tahun 1881 di kampung Bakhshali, berhampiran Peshawar, Pakistan; manuskrip itu kelihatan berasal dari tahun 200SM hingga 200M ^[9]. Sarjana-sarjana terdahulu telah berhujah bahawa ia mungkin berasal dari tahun 700M.^[10][11]

Satu mercu tanda terkemudian dalam matematik India adalah perkembangan pengembangan siri bagi fungsi trigonometri (sinus, kosinus dan kotangen) oleh ahli-ahli matematik aliran Kerala pada kurun ke-15. Karya mereka yang luar biasa yang disempurnakan dua kurun sebelum rekaan kalkulus di Eropah, menyediakan apa yang kini dianggap sebagai contoh pertama bagi siri kuasa (selain siri geometri).^[12] Bagaimana pun, mereka tidak merumuskan satu teori yang sistematik bagi pembezaan dan pengamiran, juga tiada bukti langsung bagi keputusan-keputusan mereka dipindahkan ke luar Kerala.^[13]

Bidang-bidang matematik India

Beberapa bidang matematik yang dikaji di India kuno dan Zaman Pertengahan termasuklah:

• Aritmetik: Sistem perpuluhan, Nombor negatif (lihat Brahmagupta), Sifar (lihat Sistem Angka Hindu-Arab), sistem nombor notasi kedudukan moden, nombor-nombor titik apungan (lihat aliran Kerala), teori nombor, Infiniti (lihat Yajur Veda), nombor transfinit, nombor tak nisbah (lihat Sulba Sutra)

Catatan

1. (Ifrah 2000, p. 346): "The measure of the genius of Indian civilisation, to which we owe our modern (number) system, is all the greater in that it was the only one in all history to have achieved this triumph. Some cultures succeeded, earlier than the Indian, in discovering one or at best two of the characteristics of this intellectual feat. But none of them managed to bring together into a complete and coherent system the necessary and sufficient conditions for a number-system with the same potential as our own."
2. (Bourbaki 1998, p. 46): "...our decimal system, which (by the agency of the Arabs) is derived from Hindu mathematics, where its use is attested already from the first centuries of our era. It must be noted moreover that the conception of zero as a number and not as a simple symbol of separation) and its introduction into calculations, also count amongst the original contribution of the Hindus."
3. (Bourbaki 1998, p. 49): "On this point, the Hindus are already conscious of the interpretation that negative numbers must have in certain cases (a debt in a commercial problem, for instance). In the following centuries, as there is a diffusion into the West (by intermediary of the Arabs) of the methods and results of Greek and Hindu mathematics, one becomes more used to the handling of these numbers, and one begins to have other "representation" for them which are geometric or dynamic."
4. "algebra" 2007. Britannica Concise Encyclopedia. Encyclopædia Britannica Online. 16 May 2007. Quote: "A full-fledged decimal, positional system certainly existed in India by the 9th century (AD), yet many of its central ideas had been transmitted well before that time to China and the Islamic world. Indian arithmetic, moreover, developed consistent and correct rules for operating with positive and negative numbers and for treating zero like any other number, even in problematic contexts such as division. Several hundred years passed before European mathematicians fully integrated such ideas into the developing discipline of algebra."
5. (Pingree 2003, p. 45) Quote: "Geometry, and its branch trigonometry, was the mathematics Indian astronomers used most frequently. In fact, the Indian astronomers in the third or fourth century, using a pre-Ptolemaic Greek table of chords, produced tables of sines and versines, from which it was trivial to derive cosines. This new system of trigonometry, produced in India, was transmitted to the Arabs in the late eighth century and by them, in an expanded form, to the Latin West and the Byzantine East in the twelfth century."
6. (Bourbaki 1998, p. 126): "As for trigonometry, it is disdained by geometers and abandoned to surveyors and astronomers; it is these latter (Aristarchus, Hipparchus, Ptolemy) who establish the fundamental relations between the sides and angles of a right angled triangle (plane or spherical) and draw up the first tables (they consist of tables giving the chord of the arc cut out by an angle ${\displaystyle \theta <\pi }$ on a circle of radius r, in other words the number ${\displaystyle 2r\sin \left(\theta /2\right)}$; the introduction of the sine, more easily handled, is due to Hindu mathematicians of the Middle Ages)."
7. (Filliozat 2004, pp. 140-143)
8. (Encyclopaedia Britannica (Kim Plofker) 2007, p. 1)
9. Ian Pearce (May 2002). "The Bakhshali manuscript". The MacTutor History of Mathematics archive. Dicapai pada 2007-07-24.
10. (Hayashi 1995)
11. (Encyclopaedia Britannica (Kim Plofker) 2007, p. 6)
12. (Stillwell 2004, p. 173)
13. (Bressoud 2002, p. 12)

Buku-buku sumber dalam bahasa Sanskrit

• Keller, Agathe (2006), written at Basel, Boston, and Berlin, Expounding the Mathematical Seed. Vol. 1: The Translation: A Translation of Bhaskara I on the Mathematical Chapter of the Aryabhatiya, Birkhäuser Verlag, 172 pages, ISBN 3764372915.
• Keller, Agathe (2006), written at Basel, Boston, and Berlin, Expounding the Mathematical Seed. Vol. 2: The Supplements: A Translation of Bhaskara I on the Mathematical Chapter of the Aryabhatiya, Birkhäuser Verlag, 206 pages, ISBN 3764372923.
• Neugebauer, Otto & David Pingree (eds.) (1970), written at New edition with translation and commentary, (2 Vols.), Copenhagen, The Pañcasiddhāntikā of Varāhamihira.
• Pingree, David (ed) (1978), written at edited, translated and commented by D. Pingree, Cambridge, MA, The Yavanajātaka of Sphujidhvaja, Harvard Oriental Series 48 (2 vols.).
• Sarma, K. V. (ed) (1976), written at critically edited with Introduction and Appendices, New Delhi, Āryabhaṭīya of Āryabhaṭa with the commentary of Sūryadeva Yajvan, Indian National Science Academy.
• Sen, S. N. & A. K. Bag (eds.) (1983), written at with Text, English Translation and Commentary, New Delhi, The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava, Indian National Science Academy.
• Shukla, K. S. (ed) (1976), written at critically edited with Introduction, English Translation, Notes, Comments and Indexes, New Delhi, Āryabhaṭīya of Āryabhaṭa with the commentary of Bhāskara I and Someśvara, Indian National Science Academy.
• Shukla, K. S. (ed) (1988), written at critically edited with Introduction, English Translation, Notes, Comments and Indexes, in collaboration with K.V. Sarma, New Delhi, Āryabhaṭīya of Āryabhaṭa, Indian National Science Academy.

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Pautan luar

• An overview of Indian mathematics, MacTutor History of Mathematics Archive, St Andrews University, 2000.
• 'Index of Ancient Indian mathematics', MacTutor History of Mathematics Archive, St Andrews University, 2004.
• Indian Mathematics: Redressing the balance, Student Projects in the History of Mathematics. Ian Pearce. MacTutor History of Mathematics Archive, St Andrews University, 2002.
• Online course material for InSIGHT Diarkibkan 2009-08-22 di Wayback Machine, a workshop on traditional Indian sciences for school children conducted by the Computer Science department of Anna University, Chennai, India.

Templat:Indian mathematics