Bakhshali manuscript. Bodleian Libraries, University of Oxford

As well as giving us the concept of zero, Indian mathematicians made seminal contributions to the study of trigonometry, algebra, arithmetic and negative numbers among other areas. Perhaps most significantly, the decimal system that we still employ worldwide today was first seen in India.
 

The number system

As far back as 1200 BC, mathematical knowledge was being written down as part of a large body of knowledge known as the Vedas. In these texts, numbers were commonly expressed as combinations of powers of ten. For example, 365 might be expressed as three hundreds (3×10²), six tens (6×10¹) and five units (5×10⁰), though each power of ten was represented with a name rather than a set of symbols. It is reasonable to believe that this representation using powers of ten played a crucial role in the development of the decimal-place value system in India.

Brahmi numerals. Wikimedia

From the third century BC, we also have written evidence of the Brahmi numerals, the precursors to the modern, Indian or Hindu-Arabic numeral system that most of the world uses today. Once zero was introduced, almost all of the mathematical mechanics would be in place to enable ancient Indians to study higher mathematics.
 

The concept of zero

Zero itself has a much longer history. The recently dated first recorded zeros, in what is known as the Bakhshali manuscript, were simple placeholders – a tool to distinguish 100 from 10. Similar marks had already been seen in the Babylonian and Mayan cultures in the early centuries AD and arguably in Sumerian mathematics as early as 3000-2000 BC.

But only in India did the placeholder symbol for nothing progress to become a number in its own right. The advent of the concept of zero allowed numbers to be written efficiently and reliably. In turn, this allowed for effective record-keeping that meant important financial calculations could be checked retroactively, ensuring the honest actions of all involved. Zero was a significant step on the route to the democratisation of mathematics.

No abacus needed. Shutterstock

These accessible mechanical tools for working with mathematical concepts, in combination with a strong and open scholastic and scientific culture, meant that, by around 600AD, all the ingredients were in place for an explosion of mathematical discoveries in India. In comparison, these sorts of tools were not popularised in the West until the early 13th century, though Fibonnacci’s book liber abaci.
 

Solutions of quadratic equations

In the seventh century, the first written evidence of the rules for working with zero were formalised in the Brahmasputha Siddhanta. In his seminal text, the astronomer Brahmagupta introduced rules for solving quadratic equations (so beloved of secondary school mathematics students) and for computing square roots.
 

Rules for negative numbers

Brahmagupta also demonstrated rules for working with negative numbers. He referred to positive numbers as fortunes and negative numbers as debts. He wrote down rules that have been interpreted by translators as: “A fortune subtracted from zero is a debt,” and “a debt subtracted from zero is a fortune”.

This latter statement is the same as the rule we learn in school, that if you subtract a negative number, it is the same as adding a positive number. Brahmagupta also knew that “The product of a debt and a fortune is a debt” – a positive number multiplied by a negative is a negative.

Negative cows. Shutterstock

For the large part, European mathematicians were reluctant to accept negative numbers as meaningful. Many took the view that negative numbers were absurd. They reasoned that numbers were developed for counting and questioned what you could count with negative numbers. Indian and Chinese mathematicians recognised early on that one answer to this question was debts.

For example, in a primitive farming context, if one farmer owes another farmer 7 cows, then effectively the first farmer has -7 cows. If the first farmer goes out to buy some animals to repay his debt, he has to buy 7 cows and give them to the second farmer in order to bring his cow tally back to 0. From then on, every cow he buys goes to his positive total.
 

Basis for calculus

This reluctance to adopt negative numbers, and indeed zero, held European mathematics back for many years. Gottfried Wilhelm Leibniz was one of the first Europeans to use zero and the negatives in a systematic way in his development of calculus in the late 17th century. Calculus is used to measure rates of changes and is important in almost every branch of science, notably underpinning many key discoveries in modern physics.

Leibniz: Beaten to it by 500 years.

But Indian mathematician Bhāskara had already discovered many of Leibniz’s ideas over 500 years earlier. Bhāskara, also made major contributions to algebra, arithmetic, geometry and trigonometry. He provided many results, for example on the solutions of certain “Doiphantine” equations, that would not be rediscovered in Europe for centuries.

The Kerala school of astronomy and mathematics, founded by Madhava of Sangamagrama in the 1300s, was responsible for many firsts in mathematics, including the use of mathematical induction and some early calculus-related results. Although no systematic rules for calculus were developed by the Kerala school, its proponents first conceived of many of the results that would later be repeated in Europe including Taylor series expansions, infinitessimals and differentiation.

The leap, made in India, that transformed zero from a simple placeholder to a number in its own right indicates the mathematically enlightened culture that was flourishing on the subcontinent at a time when Europe was stuck in the dark ages. Although its reputation suffers from the Eurocentric bias, the subcontinent has a strong mathematical heritage, which it continues into the 21st century by providing key players at the forefront of every branch of mathematics.

Christian Yates, Senior Lecturer in Mathematical Biology, University of Bath
 

This article was originally published on The Conversation. Read the original article.

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Professor D. N. Jha speaks on the myth of cow in India. In this part, he talks about the consumption of meat in the old Vedic times. He also tells us how the misconception of cow not being used as food in the ancient India, is being spread by the cow-vigilantes. Watch the second part of the talk.

Courtesy: Newsclick.in

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In this context, it would be educative to look closely at the prescribed textbooks for history and social studies teaching in Gujarat, with virtually no alterations since 1991, many books prescribed by the ICSE national board among others, and even some college level texts that contain these problematic formulations. 

One of the recurring myths about Indian culture, perpetuated ad noseum is that it is one of the most non-violent, peace-loving and tolerant

The state syllabus detailed in the texts being currently used by the Gujarat state board, outlines clearly for the teacher and student of history that when the author(s) of the text-book write about India they use the term for the modern nation as synonymous with “Hindu”. The student is instructed that the idea of studying social studies is to develop a true understanding of ancient India. The political implications of this assumption are significant and dangerous, because, immediately for the history learner paradigms have been drawn. It is only within these that adjustments are subsequently made for ‘synthesis’ or ‘syncretism.’

The syllabus for the standard five social studies text printed by the Gujarat State board, outlines the objective of the syllabus that has been laid out for the ten-year-old child:

‘Towards understanding the Indian Cultural heritage in a proper perspective’. 
This ‘perspective’, as described below in detail, outlines erroneously that the ancient age begins with Vedic times.It becomes clear from this introductory social studies text for the fifth standard child that no perspective of world ancient civilisations is given through the syllabus; that the desire is not just to begin and end with India, but ancient India has been made synonymous with the Vedic; and that values like ‘respectable status of women in Indian culture’ are rooted in the characters depicted through stories taken from the Vedas. There is no attempt to develop any sense of historical enquiry that could lead to a student understanding the quality of life and civilisations that existed pre-Vedas; the exchanges that took place between ancient peoples through river and sea routes etc. 

Not only is this kind of social studies self-limiting and restrictive, it is an approach that is set to stifle free thinking and enquiry. Here is how the objective of the syllabus is outlined: 

Ø  Ancient Age (From Vedic times to  Harshavardhan)

Ø  Is introduced to Vedic literature which is an expression of Indian Culture.

Ø    Knows about the respectable status of women in Indian culture. 

Ø Gets acquainted with the basic truths of life against a backdrop of Indian Culture.

Ø  Learns for himself the truth; that in the context of Indian culture a person acquires a high status not by right of birth but by merit.

Ø   Knows about how in the Indian Cultural context the rules were oriented towards the subjects. 
Ø   Imbibes the basic values of Indian Culture expressed by the narratives of the epics, Ramayana, Mahabharat, and by the main characters in it. 

Ø   For instance, the importance of 1) The purity of domestic life 2) Steadfastness in adhering to truth even at the cost of suffering.

Ø   Moulds the character which makes one abide by ones duty when there is a conflict between personal relationship and a sense of duty. 
(Social Studies text, Gujarat state board, Std. V)

Apart from the stated objective of portraying ancient Indian culture as synonymous with the Vedas, the Gujarat board texts also proceed to depict Indian culture as inherently superior to any other.

In the chapter titled, ‘The Cultural Heritage of Ancient India’, the child is told: “Ancient Indian history covers a period of about four thousand years. It can be divided into the following periods: The Indus valley civilisation period, the Vedic period, the post-Vedic period, the Epic period, the Age of Buddha and Mahavir, the Maurya and the Post-Gupta periods and the Early Muslim period.” The same text goes on to assert that from the beginning of the Indus valley period to the ‘end of Hindu supremacy’ the contribution of Indian civilisation was unique, implying that, thereafter, with the ‘Muslim period’ the contribution could not be measured in a similar fashion. 

“Right from the coming of the Aryans to India (around 2000 B.C.) to the end of the Hindu supremacy (around 1200 A.D.). The Indian civilisation made a unique contribution in many different fields of life, a contribution which includes certain high moral values. It is because of this reason that the ancient civilisation of India has survived today in the form of Indian culture while other ancient civilisation like those of Egypt, Mesopotamia (Iraq) and China have disappeared from the world. These countries do not have the continuity of culture, which is found in the Indian culture.”

The same, Std. IX text, that selectively excludes historical details like the Shaivite-Buddhist conflicts, oppression of women and the shudras, the state of Dalits even today, is however emphatic that “the inherent peace and tolerance of Indian culture” is one of its characteristics. This is one of the recurring myths that have been repeated ad nauseum about India and her ancient culture, the fact that it is “the most non-violent, peace-loving and tolerant”, a myth that is essential if the ‘Hindu’ is to be pitted as the quintessential Indian, a myth that sits well with the ‘others’  being labelled both ‘invaders’ and ‘foreigners’. It is also a myth that seeks to justify present-day violence against the country’s minorities, seeking justification for this in ‘the wrongs of yore.’

In a section titled ‘Tolerance and urge for peace’, the fourteen- year- old is told: “Tolerance and a strong desire for peace are two distinct features of Indian culture. Brahminism with its two main functions namely Shaivism and Vishnavism. Buddhism and Jainism were the main faith followed in ancient India. These faiths adopted a policy of tolerance towards one another. For examples the Satwahanas and the Guptas were followers of Brahminism. But they showed tolerance towards Buddhism and Jainism and gave financial grants to their places of worship.          

Archived from Communalism Combat, October 1999, Anniversary Issue (6th) Year 7  No. 52, Cover Story 6

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