# Easy math from Ancient India

C K Raju5 Jun 2012 3:20 AM GMT

Millions of people drop mathematics in school because they find it difficult. They hence regard themselves as inferior in some way. Is there any hope for them in the national year of mathematics?

Indeed there is. Most students can manage arithmetic. Their difficulties begin with the calculus. The calculus is normally taught over several years in school and undergraduate courses. Current calculus texts typically run into 1400 large pages in double column and fine print, and weigh a few kilos. But there is an easy way to teach calculus. Last week, in Tehran I taught calculus in just five days to students of humanities and engineering. The week before that I taught a similar course to students of social sciences in the Ambedkar University, Delhi. This experiment was earlier performed with five groups of university students, in Sarnath and Penang, and the outcome reported in scholarly journals. The students were happy to learn calculus quickly. They overcame their fear of mathematics and their consequent feeling of inferiority.

So, what is the secret? Let us try to understand it. In many a Bollywood film, the hero loses his memory, but regains it later. Loss or gain of memory completely changes our hero’s actions. Likewise, changing perceived history changes our behaviour. This applies also to the history of mathematics. So long as we believed the false story that the calculus began with Newton and Leibniz, it seemed natural to imitate its Western understanding. Today we know that the calculus originated in India, and that shows us another way of doing it – a very easy way. A key difference between the two approaches is this: Indians did mathematics for its practical use, while the West struggled to reconcile the practical use of the calculus with their religious view of mathematics.

The practical uses of the calculus in India related to the two traditional sources of wealth: agriculture and overseas trade. Indian agriculture is monsoon-driven. It requires a good calendar which can tell the rainy season, and that requires accurate astronomical models. Overseas trade required reliable techniques of navigation. Accurate trigonometric values were needed for both purposes. In the 5th century, Aryabhata calculated trigonometric values accurately to five decimal places by a radical innovation: he solved differential equations. His method of solution is very easy. He used just the elementary rule of three which one learns in primary or middle school. (The 'rule of three' refers to arithmetic problems of the following sort: 'if five men can do a given work in 10 days, in how many days can 10 men do it'.)

By the 15th century, Aryabhata’s followers in Kerala had increased the accuracy of trigonometric values to nine decimal places. They explicitly used infinite series, and had a sophisticated way to sum them. I have called this zeroism, because it is similar to Buddhist sunyavada. From a practical point of view, infinite series present no problem. For example, the number pi which is the ratio of the circumference of a circle to its diameter is written as 3.1415... This corresponds to the infinite series 3 + (1/10) + (4/100)+... In practice, one simply uses as many decimal places of precision as one needs, for example, 3.14 or 3.14159 and so on. Even rocket science rarely requires more than 16 decimal places of precision. Zeroism rigorously justifies this practical procedure.

Now, the practical importance of calculus for navigation was, of course, clear to the Jesuits who transmitted calculus from India to Europe. However, the infinite series created great theoretical difficulties for Western mathematicians because of their religiosity about mathematics. If this seems surprising, consider that the very word 'mathematics' derives from 'mathesis', which means learning. Plato explained that 'all learning is recollection' of eternal ideas in the soul. He prescribed the teaching of mathematics for the good of the soul, not for its practical value. In Plato’s dialogue called Meno, Socrates demonstrates Plato’s theory by questioning a slave boy about mathematics. Having established the slave boy’s innate ideas about mathematics, Socrates claims to have proved the existence of the soul. A 5th century commentator, Proclus, explained that Socrates used mathematics for this demonstration and not any other subject, since mathematics incorporates eternal truths, which most readily aroused the eternal soul. This is the principle that 'like arouses like', which we would today call sympathetic magic.

The church, however, cursed this notion of soul in the 6th century, and banished all mathematicians from Christendom. Many centuries later, during the Crusades, the church accepted back mathematics after divorcing it from mathesis and the underlying notion of soul. It reinterpreted mathematics as merely a tool to teach reasoning for persuasion, for the church was interested only in persuasion. Nevertheless, the idea of mathematics as eternal truth remained stuck in Western tradition which hence believed mathematics ought to be 'perfect'. Thus, Descartes reasoned that summing the entire infinite series for pi would take an infinite amount of time, while stopping at any point would not give us the exact sum, and hence would not be mathematics. He hence declared that the length of a curved line was 'beyond the human mind', though children in India were taught to measure curved lines with a string! The religious Western hangup with perfection and exactitude led to an enormous metaphysical overload which makes math difficult today. But all this metaphysics, from Newton’s fluxions, to Dedekind’s formal real numbers to Cantor’s set theory and its axiomatisation, is irrelevant to the practical applications of mathematics. On the contrary, it impedes applications: Newton’s physics failed just because he tried to make the calculus 'perfect' – but that is another story.

Even today, most applications of calculus involve differential equations (of some sort). Aryabhata’s technique of solution can still be used to solve all kinds of non-linear differential equations (though the technique can, of course, be improved). This numerical technique is well adapted to computers. Following this historical route, we can and should teach differential equations at the very beginning of the calculus (using only the rule of three). Because that makes calculus very easy, it enables school students to do mathematics that was previously considered 'very advanced'. This was first demonstrated by the school projects carried out by my own children.

Unfortunately, even 65 years after Independence, we have not bothered to re-examine our mathematics syllabus. The millions of students who drop out of math courses every year are stakeholders, as are their parents. They are interested in mathematics only for its usefulness, but they have no say in framing the mathematics syllabus. The same applies also to students of social science, engineering and science. Even the teachers just teach the received syllabus. That syllabus is decided, behind closed doors, by a handful of 'experts' – Western-trained mathematicians who have never explained publicly what practical value they ever contributed to the people of India. Their main qualification is that they have certificates of approval from the West. Changing the math syllabus would conflict with their self-interest. To make math easy, we perhaps need a mass movement for educational independence, which would liberate us from the terrible superstition that imitating the West is the only way forward.

Indeed there is. Most students can manage arithmetic. Their difficulties begin with the calculus. The calculus is normally taught over several years in school and undergraduate courses. Current calculus texts typically run into 1400 large pages in double column and fine print, and weigh a few kilos. But there is an easy way to teach calculus. Last week, in Tehran I taught calculus in just five days to students of humanities and engineering. The week before that I taught a similar course to students of social sciences in the Ambedkar University, Delhi. This experiment was earlier performed with five groups of university students, in Sarnath and Penang, and the outcome reported in scholarly journals. The students were happy to learn calculus quickly. They overcame their fear of mathematics and their consequent feeling of inferiority.

So, what is the secret? Let us try to understand it. In many a Bollywood film, the hero loses his memory, but regains it later. Loss or gain of memory completely changes our hero’s actions. Likewise, changing perceived history changes our behaviour. This applies also to the history of mathematics. So long as we believed the false story that the calculus began with Newton and Leibniz, it seemed natural to imitate its Western understanding. Today we know that the calculus originated in India, and that shows us another way of doing it – a very easy way. A key difference between the two approaches is this: Indians did mathematics for its practical use, while the West struggled to reconcile the practical use of the calculus with their religious view of mathematics.

The practical uses of the calculus in India related to the two traditional sources of wealth: agriculture and overseas trade. Indian agriculture is monsoon-driven. It requires a good calendar which can tell the rainy season, and that requires accurate astronomical models. Overseas trade required reliable techniques of navigation. Accurate trigonometric values were needed for both purposes. In the 5th century, Aryabhata calculated trigonometric values accurately to five decimal places by a radical innovation: he solved differential equations. His method of solution is very easy. He used just the elementary rule of three which one learns in primary or middle school. (The 'rule of three' refers to arithmetic problems of the following sort: 'if five men can do a given work in 10 days, in how many days can 10 men do it'.)

By the 15th century, Aryabhata’s followers in Kerala had increased the accuracy of trigonometric values to nine decimal places. They explicitly used infinite series, and had a sophisticated way to sum them. I have called this zeroism, because it is similar to Buddhist sunyavada. From a practical point of view, infinite series present no problem. For example, the number pi which is the ratio of the circumference of a circle to its diameter is written as 3.1415... This corresponds to the infinite series 3 + (1/10) + (4/100)+... In practice, one simply uses as many decimal places of precision as one needs, for example, 3.14 or 3.14159 and so on. Even rocket science rarely requires more than 16 decimal places of precision. Zeroism rigorously justifies this practical procedure.

Now, the practical importance of calculus for navigation was, of course, clear to the Jesuits who transmitted calculus from India to Europe. However, the infinite series created great theoretical difficulties for Western mathematicians because of their religiosity about mathematics. If this seems surprising, consider that the very word 'mathematics' derives from 'mathesis', which means learning. Plato explained that 'all learning is recollection' of eternal ideas in the soul. He prescribed the teaching of mathematics for the good of the soul, not for its practical value. In Plato’s dialogue called Meno, Socrates demonstrates Plato’s theory by questioning a slave boy about mathematics. Having established the slave boy’s innate ideas about mathematics, Socrates claims to have proved the existence of the soul. A 5th century commentator, Proclus, explained that Socrates used mathematics for this demonstration and not any other subject, since mathematics incorporates eternal truths, which most readily aroused the eternal soul. This is the principle that 'like arouses like', which we would today call sympathetic magic.

The church, however, cursed this notion of soul in the 6th century, and banished all mathematicians from Christendom. Many centuries later, during the Crusades, the church accepted back mathematics after divorcing it from mathesis and the underlying notion of soul. It reinterpreted mathematics as merely a tool to teach reasoning for persuasion, for the church was interested only in persuasion. Nevertheless, the idea of mathematics as eternal truth remained stuck in Western tradition which hence believed mathematics ought to be 'perfect'. Thus, Descartes reasoned that summing the entire infinite series for pi would take an infinite amount of time, while stopping at any point would not give us the exact sum, and hence would not be mathematics. He hence declared that the length of a curved line was 'beyond the human mind', though children in India were taught to measure curved lines with a string! The religious Western hangup with perfection and exactitude led to an enormous metaphysical overload which makes math difficult today. But all this metaphysics, from Newton’s fluxions, to Dedekind’s formal real numbers to Cantor’s set theory and its axiomatisation, is irrelevant to the practical applications of mathematics. On the contrary, it impedes applications: Newton’s physics failed just because he tried to make the calculus 'perfect' – but that is another story.

Even today, most applications of calculus involve differential equations (of some sort). Aryabhata’s technique of solution can still be used to solve all kinds of non-linear differential equations (though the technique can, of course, be improved). This numerical technique is well adapted to computers. Following this historical route, we can and should teach differential equations at the very beginning of the calculus (using only the rule of three). Because that makes calculus very easy, it enables school students to do mathematics that was previously considered 'very advanced'. This was first demonstrated by the school projects carried out by my own children.

Unfortunately, even 65 years after Independence, we have not bothered to re-examine our mathematics syllabus. The millions of students who drop out of math courses every year are stakeholders, as are their parents. They are interested in mathematics only for its usefulness, but they have no say in framing the mathematics syllabus. The same applies also to students of social science, engineering and science. Even the teachers just teach the received syllabus. That syllabus is decided, behind closed doors, by a handful of 'experts' – Western-trained mathematicians who have never explained publicly what practical value they ever contributed to the people of India. Their main qualification is that they have certificates of approval from the West. Changing the math syllabus would conflict with their self-interest. To make math easy, we perhaps need a mass movement for educational independence, which would liberate us from the terrible superstition that imitating the West is the only way forward.

*C K Raju is professor of mathematics at the University of Science, Malaysia.*### C K Raju

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