{
"type": "Article",
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"type": "Person",
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],
"givenNames": [
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"and",
"ArniS.R.Srinivasa"
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"description": [
{
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"content": [
"The\nIndian tradition in mathematics is long and\nglorious. It dates to the earliest times, and indeed many of the Indian\ndiscoveries from a period starting 5000 years ago correspond rather\nnaturally to modern mathematical results. Celebration of Indian\nmathematics needs to consider the personalities among ancient\nmathematicians who laid a solid foundation for modern thinking. Our main\npurpose here is, by presenting very briefly some of the main\ncontributions of ancient Indian mathematicians and astronomers, to argue\nand convince the reader that before the great Ramanujan, there have been\nthousands of years of rich mathematical discoveries in India and those\npersonalities’ work also needs to be honored on Indian Mathematics Day."
]
}
],
"identifiers": [],
"references": [
{
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"id": "bib-bib1",
"authors": [],
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"url": "https://dx.doi.org/10.1007/978-81-322-1053-5_18"
}
],
"title": "Ancient Indian mathematics needs an honorific place in modern mathematics celebration",
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"\nSanskrit and English numerals ",
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"content": [
"The\ngovernment of India announced in 2012 that every year, Indian\nmathematician Srinivasa Ramanujan’s birthday, December 22, will be\ncelebrated as national mathematics\nday.",
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"content": [
"Prime Minister’s speech at the 125th Birth Anniversary Celebrations of\nRamanujan at Chennai: ",
{
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"target": "https://archive.is/20120729041631/http://pmindia.nic.in/speech-details.php?nodeid=1117",
"content": [
"https://archive.is/20120729041631/http://pmindia.nic.in/speech-details.php?nodeid=1117"
]
},
"\n(accessed on April 15, 2023)."
]
}
]
},
" The year 2012 was the great\nRamanujan’s 125th birth anniversary. The government of India released a\ncommemorative stamp on that occasion as well. Brilliant contributions in\nnumber theory and combinatorics by Ramanujan are well known [",
{
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"2"
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},
", ",
{
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"target": "bib-bib1",
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"1"
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", ",
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"13"
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", ",
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"]. However, deep\nastronomical and mathematical developments in India are several thousand\nyears older than Ramanujan. In this comment, we try to recollect a few\ngems of the ancient Indian mathematics and its mathematicians who did\nfundamental work in number systems, mathematics of astronomy, calculus,\netc., over more than 5000 years."
]
},
{
"type": "Paragraph",
"id": "p2",
"content": [
"Our main purpose for writing this article is to\nargue and convince that, while giving Ramanujan’s brilliant achievements\nduring the past 125 years their due place, reducing the Mathematics Day\nin India to the celebration of Ramanujan’s birthday (who was born in the\n19th century) is somewhat short-sighted. Our goal is to make sure Indian\nMathematics Day is seen as a celebration of thousands of years of\ndeep-rooted mathematical thought processes and discoveries\nsince the\ntimes of ",
{
"type": "Emphasis",
"content": [
"Shulba sutra"
]
},
". Moreover, it should also be devoted to\ncelebrating many very strong mathematicians, such as, say, Harish\nChandra, who have come since Ramanujan’s time."
]
},
{
"type": "Paragraph",
"id": "p3",
"content": [
"The origins of the mathematics that emerged in the Indian subcontinent\ncan be seen around the Shulba sutra period, around 1200 BCE to 500 BCE.\nDuring this period the numbers up to ",
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" were counted (in Vedic\nSanskrit this number was referred to as ",
{
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"Paradham"
]
},
"). The Vedic\nperiod mathematics was confined to the geometry\nof fire-altars and\nastronomy, and these concepts were used to perform rituals by the\npriests. Some of the famous names from that era are Baudhayana,\nApastamba, and Katyayana. In Table ",
{
"type": "Cite",
"target": "S0-T1",
"content": [
"1"
]
},
" we describe\nSanskrit sounds and their corresponding English numerals. Indian\nmathematics also introduced the decimal number system that is in use\ntoday and the concept of zero as a number. The concepts of sine (written\nas ",
{
"type": "Emphasis",
"content": [
"jaya"
]
},
" in Sanskrit) and cosine (",
{
"type": "Emphasis",
"content": [
"cojaya"
]
},
"), negative\nnumbers, arithmetic, and algebra were found in ancient Indian\nmathematics [",
{
"type": "Cite",
"target": "bib-bib6",
"content": [
"6"
]
},
"].\nThe mathematics developed in India\nwas later translated and transmitted to China, East Asia, West Asia,\nEurope, and Saudi Arabia. The classical period of Indian mathematics was\noften attributed to the interval from\n200 CE to 1400 CE, during which\nworks of several well-known mathematicians, like Varahamihira,\nAryabhata, Brahmagupta, Bhaskara, and Madhava have been translated into\nother languages and transmitted outside the sub-continent."
]
},
{
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"content": [
"The number systems present since the Vedic days, especially since the\nSukla Yajurveda and their Sanskrit sounds, were as follows: ",
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"),\n",
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},
"), ",
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"),\n",
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"), ",
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{
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"content": [
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{
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"content": [
"“Vishnu Sahasra Nama Stotra,”"
]
},
" which dates\nback to the Mahabharata epic. In this, there is a verse that sounds like\nSahasra ",
{
"type": "Emphasis",
"content": [
"“Koti Yugadharine Namah.”"
]
},
" If we translate this verse,\nthen, as we saw above, ",
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" means ",
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"\nmeans ",
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"\ncould mean ",
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". The entire phrase has been interpreted in\ndifferent ways. We do not list here all possible interpretations and\nconfine ourselves to number systems."
]
},
{
"type": "Paragraph",
"id": "p6",
"content": [
"The deep investigations in astronomy and the solar system, geometry, and\nground-breaking mathematical calculations by ancient and medieval great\nscholars in India, for example,\nBaudhayana, Varahamihira, Aryabhata,\nBhaskara I & II, Pingala,\nMadhava, and many more, are\nwell known (see [",
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"10"
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", ",
{
"type": "Cite",
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"] and [",
{
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"17"
]
},
", p. 423]). "
]
},
{
"type": "Paragraph",
"id": "p7",
"content": [
"It seems\nthat celebrating national mathematics day in India only as part of\nRamanujan’s birthday is confining the glory and celebration of Indian\nmathematics to a little over 100 years of the past. Schools and colleges\nacross India have celebrated Ramanujan’s birthday for many decades, but\nthat is different from exclusively limiting national day only to the\ngreat\nRamanujan."
]
},
{
"type": "Paragraph",
"id": "p8",
"content": [
"A good deal can be written on ancient scholar’s work from India;\nmaterial on this can be found, for example in [",
{
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"content": [
"8"
]
},
", ",
{
"type": "Cite",
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"content": [
"12"
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},
", ",
{
"type": "Cite",
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"content": [
"15"
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},
", ",
{
"type": "Cite",
"target": "bib-bib6",
"content": [
"6"
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", ",
{
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"]. In this opinion piece, we\nhighlight only a few of them."
]
},
{
"type": "Paragraph",
"content": [
"Shulba sutras were believed to have started in India around 2000 BCE through verbal usage. Their compilation in Sanskrit started\nperhaps 1000 years later by Baudhayana then\nby Manava, Apastamba, Katyayana and consisted of geometric-shaped fire-altars for\nperforming ancient Indian rituals [",
{
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"target": "bib-bib3",
"content": [
"3"
]
},
", ",
{
"type": "Cite",
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"12"
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},
"]. Some of these sutras also contain the\nstatements of Pythagorean theorems and triples. For example, Apastamba\nprovided the following triples:"
]
},
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"for constructing fire-altars [",
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"], using the expression"
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"content": [
"Magic squares (",
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") for the Sun (top) and for the other eight planets (bottom) taken from ancient Indian literature."
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{
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") were used to please and worship nine planets of the solar\nsystem [",
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"9"
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"]. Figure ",
{
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" is about ancient magic\nsquares for the Sun and the other eight planets in our solar system."
]
},
{
"type": "Paragraph",
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"content": [
"In\nthe 5th century CE, Aryabhata calculated, among many other things,\nthat the moon orbit takes 27.396 days, the value of ",
{
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"meta": {
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", etc.\nHe is believed to have started the study of properties of sine and\ncosine in trigonometry."
]
},
{
"type": "Paragraph",
"content": [
"According to [",
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"], the Leibniz infinite series"
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},
{
"type": "Paragraph",
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"was known in the works of Indian mathematician Madhava, who lived three\ncenturies before Leibniz."
]
},
{
"type": "Paragraph",
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"content": [
"In\nthe 12th century CE, Bhaskara described in his famous book\n",
{
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" the rules of algebraic operations on positive,\nnegative signs, rules of zero (",
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},
"), and infinity\n(",
{
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"). His book also shows how to obtain solutions to\nintermediate equations of the first degree [",
{
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"content": [
"14"
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},
"].\nBhaskara’s book titled ",
{
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"Siddhanta siromani"
]
},
"\nprovided a detailed account of Indian astronomy and its development.\nComputations of the planetary movements, shapes of planets, rotation\naxis, lunar month days, etc., were explained in detail. See Figure ",
{
"type": "Cite",
"target": "footnote2",
"content": [
"2"
]
},
".\nPavuluri Mallana translated Mahavira’s ",
{
"type": "Emphasis",
"content": [
"Ganitasarasamgraha"
]
},
" from ",
{
"type": "Emphasis",
"content": [
"Sanskrit"
]
},
" in the 11th century to another ancient indian\nlanguage, ",
{
"type": "Emphasis",
"content": [
"Telugu"
]
},
"; Joseph [",
{
"type": "Cite",
"target": "bib-bib11",
"content": [
"11"
]
},
"] thinks that this stood as a role model for other subsequent translations. Bhaskara’s\n",
{
"type": "Emphasis",
"content": [
"Lilavati Ganitam"
]
},
" was for the first time translated from ",
{
"type": "Emphasis",
"content": [
"Sanskrit"
]
},
" to ",
{
"type": "Emphasis",
"content": [
"Telugu"
]
},
" in the 12th century by Eluganti Peddana [",
{
"type": "Cite",
"target": "bib-bib4",
"content": [
"4"
]
},
"],\nand into English first in 1816 by John Taylor, then in 1817 by Henry Thomas Holbrooke, who was considered as the\nfirst European Sanskrit scholar."
]
},
{
"type": "Paragraph",
"id": "p15",
"content": [
"What we advocate in this piece is for an exposition of deep-rooted\nmathematical knowledge in India, and not an exhaustive account of all\npossible results and conclusions. Several of the ancient texts in the\nlanguage ",
{
"type": "Emphasis",
"content": [
"Sanskrit"
]
},
" are either lost or preserved in museums."
]
},
{
"type": "Figure",
"id": "S0-F2",
"caption": [
{
"type": "Paragraph",
"content": [
" Screenshots from the book “Lilavati Ganitamu”\nwritten by Bhaskara (also known as Bhaskaracharya) in the year 1114 CE.\nTop: Bhaskara wrote the book in Sanskrit. These screenshots were taken\nfrom its translated version into another ancient Indian language,\nTelugu, by Pidaparti Krishnamurti Sastri in 1936, published by Maharaja\nCollege, Vizianagaram, Andhra Pradesh, India.",
{
"type": "Note",
"id": "idm399",
"noteType": "Footnote",
"content": [
{
"type": "Paragraph",
"id": "footnote2",
"content": [
"This book is available for free at\n",
{
"type": "Link",
"target": "https://upload.wikimedia.org/wikipedia/commons/7/75/Lilavatiganitamu00bhassher.pdf",
"content": [
"https://upload.wikimedia.org/wikipedia/commons/7/75/Lilavatiganitamu00bhassher.pdf"
]
},
".\n© Sundarayya Vignana Kendram, Bagh\nLingampally, Hyderabad, India"
]
}
]
},
" Bottom: Here the contents of the book are\nmentioned in Telugu, as well as in English.\n"
]
}
],
"licenses": [
{
"type": "CreativeWork",
"url": "https://creativecommons.org/licenses/by-sa/4.0/legalcode",
"content": [
{
"type": "Paragraph",
"content": [
"Bhaskaracarya / Wikimedia Commons / CC BY-SA 4.0"
]
}
]
}
],
"content": [
{
"type": "ImageObject",
"contentUrl": "image3.png",
"mediaType": "image/png",
"meta": {
"inline": false
}
},
{
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"content": []
},
{
"type": "ImageObject",
"contentUrl": "image4.png",
"mediaType": "image/png",
"meta": {
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]
},
{
"type": "Paragraph",
"id": "p16",
"content": [
"Srinivasa Ramanujan’s work undoubtedly shines as part of modern Indian\nmathematics but thousands of years ancient mathematical discoveries, the\nintroduction of various branches of pure and applied mathematics needs a\nproper representation in any celebration of India’s contribution to\nworld mathematics."
]
},
{
"type": "Paragraph",
"id": "p17",
"content": [
"We\nhope that this short list of significant examples will convince\nthe readers as well as the decision makers of the need to incorporate\nand celebrate all the rich past and contemporary history of Indian\nmathematics during the Indian Mathematics Day."
]
},
{
"type": "Paragraph",
"id": "p18",
"content": [
{
"type": "Emphasis",
"content": [
"Acknowledgements. "
]
},
"\nWe are thankful to the reviewer for the\ncomments that helped us to make\nthe point in the article clearer. We are\nalso thankful to Prof. Ralf Krömer, editor of the\nEMS Magazine, and the editorial staff and copyeditor of EMS Press for their edits to\nour article."
]
},
{
"type": "Paragraph",
"id": "authorinfo",
"content": [
"\nSteven G. Krantz is a professor of mathematics at Washington University\nin St. Louis. He received a BA degree from the University of California\nat Santa Cruz in 1971 and the PhD from Princeton University in 1974.\nKrantz has taught at UCLA, Princeton University, Penn State, and\nWashington University in St. Louis. He was Chair of the latter\ndepartment for five years. Krantz has had 9 master’s students and 20 PhD\nstudents. He has written more than 135 books and more than 330 scholarly\npapers. He edits five journals and is managing editor of three. He is\nthe founding editor of the ",
{
"type": "Emphasis",
"content": [
"Journal of Geometric Analysis"
]
},
". He is the\ncreator, founder, and editor of the new journal ",
{
"type": "Emphasis",
"content": [
"Complex Analysis and its\nSynergies"
]
},
". He is an\nAMS Fellow.\n",
{
"type": "Link",
"target": "mailto:sk@math.wustl.edu",
"content": [
"sk@math.wustl.edu"
]
},
"Arni S. R. Srinivasa Rao is a professor and Director of the Laboratory\nfor Theory and Mathematical Modeling, at Medical College of Georgia,\nAugusta, USA. Until 2012, he held a permanent faculty position at Indian\nStatistical Institute, Kolkata. He conducted research and/or taught at\nseveral institutions, such as the Indian Statistical Institute, the\nIndian Institute of Science, and the University of Oxford. Dr. Rao’s\nmodels assisted in the national AIDS control planning in India.\nHe served on various committees and as consultant on mathematical modeling,\npublic health, and artificial intelligence (He developed the first\nAI-based approach for COVID-19 mobile apps). He taught\ncourses on real analysis, complex analysis, differential equations,\nmathematical biology, demography, and stochastic processes. Rao’s other\nnoted contributions include his Partition Theorem in Populations,\nmultilevel contours in complex bundles, a fundamental theorem in\nstationary population models (Rao–Carey Theorem), blockchain technology\nin healthcare, Exact Deep Learning Machines (EDLM).\n",
{
"type": "Link",
"target": "mailto:arrao@augusta.edu",
"content": [
"arrao@augusta.edu"
]
}
]
}
]
}