印度數學在公元前 1200 年 於印度次大陸出現，到 18 世紀結束。在印度數學的古典時期（公元 400 年至 1200 年），阿耶波多、婆羅摩笈多、婆什迦羅第二和伐羅訶密希羅等學者做出了重要的貢獻。印度數學首先記錄了今天使用的十進位制 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 numbersystem with the same potential as our own。印度數學家早期的貢獻包括對 0 作為數字的概念的研究 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，負數， Modern arithmetic was known during medieval times as Modus Indorum or method of the Indians Leonardo of Pisa wrote that compared to method of the Indians all other methods is a mistake This method of the Indians is none other than our very simple arithmetic of addition subtraction multiplication and division Rules for these four simple procedures was first written down by Brahmagupta during 7th century AD 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 otherrepresentationfor them which are geometric or dynamic 算數，以及代數。另外，三角學 Quote Geometry and its branch trigonometry was the mathematics Indian astronomers used most frequently Greek mathematicians used the full chord and never imagined the half chord that we use today Half chord was first used by Aryabhata which made trigonometry much more simple In fact the Indian astronomers in the third or fourth century using a prePtolemaic 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 在印度更加先進，特別是發展出了正弦和餘弦的現代定義。 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 theta pi on a circle of radius r in other words the number 2rsinleft theta2right the introduction of the sine more easily handled is due to Hindu mathematicians of the Middle Ages 這些數學概念被傳播到中東，中國和歐洲 algebra 2007 Britannica Concise Encyclopedia Encyclopædia Britannica Online 16 May 2007 Quote A fullfledged 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，從而導致了進一步的發展，形成了現在許多數學領域的基礎。 古代和中世紀的印度數學作品，都是用梵語寫成，通常由稱作契經的一部分組成，在其中為了幫助學生記憶，用極少的字詞陳述了一些規則和問題。在這之後是第二部分，包括一篇散文評論（有時是來自不同學者的多篇評論），其更詳細地解釋了問題併為解決方案提供了更多的理由。在散文部分，形式（和記憶）比起其涉及的思想來說並不是很重要。在公元前 500 年之前，所有的數學作品都是由口頭傳播，之後同時以口頭和手稿的形式傳播。現存的印度次大陸上最古老的數學文獻是寫在樺樹皮上的巴赫沙利手稿，它在 1881 年於巴赫沙利村被發現，靠近白沙瓦（現位於巴基斯坦）並可能來自公元 7 世紀。 印度數學的後期里程碑是公元 15 世紀喀拉拉邦學派的數學家對三角函式（正弦，餘弦和反正切）的級數展開的發展。 他們的卓越工作，在歐洲發明微積分之前兩個世紀完成，提供了現在被稱為冪級數的第一個例子（除了等比數列） 。 然而，他們沒有制定出系統的微分和積分理論，也沒有任何直接證據證明他們的結果是在喀拉拉邦以外傳播的 Quote There is no evidence that the Indian work on series was known beyond India or even outside Kerala until the nineteenth century Gold and Pingree assert 4 that by the time these series were rediscovered in Europe they had for all practical purposes been lost to India The expansions of the sine cosine and arc tangent had been passed down through several generations of disciples but they remained sterile observations for which no one could find much use Quote It is not unusual to encounter in discussions of Indian mathematics such assertions as that the concept of differentiation was understood in India from the time of Manjula in the 10th century Joseph 1991 300 or thatwe may consider Madhava to have been the founder of mathematical analysisJoseph 1991 293 or that Bhaskara II may claim to bethe precursor of Newton and Leibniz in the discovery of the principle of the differential calculusBag 1979 294 The points of resemblance particularly between early European calculus and the Keralese work on power series have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world eg in Bag 1979 285 It should be borne in mind however that such an emphasis on the similarity of Sanskrit or Malayalam and Latin mathematics risks diminishing our ability fully to see and comprehend the former To speak of the Indiandiscovery of the principle of the differential calculussomewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa as in the examples we have seen remained within that specific trigonometric context The differentialprinciple was not generalised to arbitrary functionsin fact the explicit notion of an arbitrary function not to mention that of its derivative or an algorithm for taking the derivative is irrelevant hereQuoteOne example I can give you relates to the Indian Mādhavas demonstration in about 1400 AD of the infinite power series of trigonometrical functions using geometrical and algebraic arguments When this was first described in English by Charles Matthew Whish in the 1830s it was heralded as the Indians discovery of the calculus This claim and Mādhavas achievements were ignored by Western historians presumably at first because they could not admit that an Indian discovered the calculus but later because no one read anymore the Transactions of the Royal Asiatic Society in which Whishs article was published The matter resurfaced in the 1950s and now we have the Sanskrit texts properly edited and we understand the clever way that Mādhava derived the series without the calculus but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found In this case the elegance and brilliance of Mādhavas mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solutionQuoteHow close did Islamic and Indian scholars come to inventing the calculus Islamic scholars nearly developed a general formula for finding integrals of polynomials by AD 1000and evidently could find such a formula for any polynomial in which they were interested But it appears they were not interested in any polynomial of degree higher than four at least in any of the material that has come down to us Indian scholars on the other hand were by 1600 able to use ibn alHaythams sum formula for arbitrary integral powers in calculating power series for the functions in which they were interested By the same time they also knew how to calculate the differentials of these functions So some of the basic ideas of calculus were known in Egypt and India many centuries before Newton It does not appear however that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus They were apparently only interested in specific cases in which these ideas were needed There is no danger therefore that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented calculus They were certainly the ones who were able to combine many differing ideas under the two unifying themes of the derivative and the integral show the connection between them and turn the calculus into the great problemsolving tool we have today。 史前時期 在摩亨佐 達羅、哈拉帕與其他幾個印度河流域文明的遺址中，都發現了使用 實用數學 的證據。他們使用 421 的比例製造磚頭，此作法被認為有利於磚頭結構的穩定性。他們使用了標準化的砝碼，以比例：120 110 15 12 1 2 5 10 20 50 100 200 與 500 進行製作，其中的一單位的重量約等於 28 克。他們大量生產有規則形狀的砝碼，有六面體、桶形、錐形與柱形，從而展示出對於基本幾何的理解。 印度文明的居民曾試圖對於長度量測進行高精準度的標準化。他們設計出了摩亨佐達羅尺，尺上的單位長度約為 34 公分，並且再分割成十等分。古代摩亨若達羅的磚頭尺寸通常是此單位長度的整數倍。 在洛沙爾 Lothal2200 BCE 與卓拉維拉 Dholavira 發現的貝殼製中空柱狀物體上面有 8 個裂縫，被認為可用來製作羅盤，證明瞭能在平面上測量角度的能力，並可藉此確定星星的位置進行導航。S R Rao 1992 Marine Archaeology Vol 3 pp 6162 Link httpdrsnioorgdrsbitstreamhandle22643082J_Mar_Archaeol_3_61pdfsequence2 吠陀時期 吠陀時期的宗教文字提供了使用大數的證據。在《夜柔吠陀》（公元前 1200900 年）文字中，包含的數字高達。 耆那學者（前 400 年 200 年） 耆那學者認為世界是永恆的，只有形式上的變化，與婆羅門教創造萬物的理論不同。耆那教由筏陀摩那在前 6 世紀創立，但耆那數學著作大部分是在前 6 世紀後撰寫的。耆那學者將數字分為三類。 耆那重要數學家包括 Bhadrabahu，卒於公元前 298 年），他是兩部天文著作的作者。 書寫形式 巴赫沙利手稿 古典時期（4001600 年） 參見 婆羅摩笈多 斯里尼瓦瑟拉馬努金 巴赫沙利手稿 數學史 引用 參考文獻 延伸閱讀 梵語文獻 外部連結 Science and Mathematics in India An overview of Indian mathematics MacTutor History of Mathematics archive St Andrew University 2000 Indian Mathematicians 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 InSIGHT 2009 a workshop on traditional Indian sciences for school children conducted by the Computer Science department of Anna University Chennai India Mathematics in ancient India by R Sridharan Combinatorial methods in ancient India Mathematics before S Ramanujan Category 印度歷史 Category 印度文化 Category 數學史