 Numbers. Human life cannot be imagined without numbers. How did the humans learn to count? And where did the numbers and their symbols come from? Numbers always, in the civilizational sense that we are talking about, came from everyday activities. How many people are there, what work they would do, if there is an exchange, how many chickens you will exchange, say, for how many boxes of grains. So the first simplest would have been just one number is all you need. And then one mark, two mark, three marks, this is how the number system would have started in different languages or alphabets that we have. Counting systems have been known from, first of all, because the Babylonians, Egyptians, another time from Mayan system, Mayans, Chinese and of course from India. Different civilizations came up with different ways of recording higher numbers. For instance, the Greek, Hebrew and Egyptian numerals were extensions of tally marks with new symbols added to represent larger magnitudes. If you can see, for instance, even today in what we call the Roman numerals, one, two, three are repetitions of just increasing the number of marks that we have. When you go to four, you start using actually five and one in the subtractive sense, if you will, by putting it on the left. So you start then six, again you start repeating. Then when it comes to nine, it's really ten minus one. That's how we would see the number. What we call the Roman numbers which are still used in some clocks and some things. So with those, it is not so easy to do calculations. This has to do with the fact that they are not using the way of writing for calculations. The calculations were actually done using the abacus, or abacus-like device. And it's only the recording that involved the representation. You want to compute, for instance, how much of grains have been produced. So you have to have a way to measure how much the land will produce and you have to have a measure of how much a total land area that has been cultivated by, say, a landowner. Why is that important? Because you already have a concept of taxation, the kings, the temples, levy taxes. And with the levy taxes, they did not really know how much you could produce. But they would do levy taxes based on what is the land you have and how much you have produced. Therefore the percentage of your product would then be levy tax. Using those symbols, adding, subtracting, still possible. When it comes to multiplying and dividing, which we will have to do if we are talking about taxes, we are talking about, for instance, areas, it becomes increasingly more and more difficult. To write and calculate larger numbers, some civilizations came up with a more useful system known today as the Positional Notation or the Place Value Notation. Conceptualizing, subtracting the hundreds, the tens and one units and then composing the number. This is what is called the Positional System. This is the Decimal Positional System. So let us understand our own Decimal System. Take, for example, the number 231. Here, the number 1 is in the unit's place. The number 3 is in the tens place and the number 2 is in the hundreds place. What does it mean by hundreds, tens and units? What it means is that we multiply 2 by 100. We add 3 multiplied by 10. Let me put in parenthesis and we add 1 multiplied by 1. This is the same as 2 into 10 square plus 3 into 10 to the power 1 plus 1 multiplied by 10 to the power 0 because 10 to the power 0 is 1. So this is the Decimal System. The decimal becomes important because here as you see there is all 10, 10 and 10. In certain other cultures, the system has been different. The Mayans had basically a system based on base 20. The Chinese, on the other hand, had a system which was based on 10 as the base. So something very similar to what we did, what we had. Except that there is no position plotted to 0. A similar thing happened also in the Babylonians. The Babylonians did not have the decimal number system. They had a sexagesimal number system. Just like in decimal number system, we have the 9 digits 1 through 9 plus 0. And so we write 11 as 11, that is 10 plus 1. They had 60 digits and that is the sexagesimal system. However, they had a very nice way, a very clever way instead of remembering 60 digits. They have a very clever way of using 12 as a base to create the other numbers in the first 16 digits. 60 gets divided by many more numbers. So arithmetic is relatively easier with a base 60. But the disadvantage is 60 is a large number. It is not a small number. It still is there for instance in minutes, hours, seconds, which is still on a sexagesimal basis. The remnant of the Babylonian system is still there in our use. But we do not use it as a number base anymore. And the number base, it becomes more convenient if you have something called 10 as a number base. Of course, it originates from our 5 plus 5, 10 fingers. In India, people talked of numbers already from the Vedic period in powers of 10. So they had names for 1, 10, 1, etc. Early times it went up to 10 to 12 and then it was 10 to 17. In the Buddhist tradition, there were even much larger numbers, 10 to 54 for that order. But the Jains actually went far beyond that because they were trying to reach, shall we say, infinity in some way. And trying to reach infinity, their method was to construct very large numbers and say actually the universe is bigger than even this. So that's where it originates from the quest is really a cosmological one. And for that cosmological quest for large numbers, they need to express or show how large numbers can be written. Now, in regards to the concrete, the decimal representation of numbers, there are records. The epigraphic record, that's in terms of writing on stone, comes only from the 9th century. But there are some earlier records, again there are some disputes about them. In the study of ancient Indian mathematics, there is considerable degree of uncertainty about various sources and so on. So in a way, not much is written on stone, as one would say. So the systems were evolving through different various cultures. But this came to be practiced actually both in India and in China. The decimal system was being used. And the Arabs, this went to Arabs. I should mention that, see, Brahmahupta was the one who systematically wrote about how this arithmetic works. And some of these manuscripts made their way to the Arab and Persian lands. And they imbibed them and in fact developed them further. Then it spread to the west from there. The Arabs sort through Arab through Spain and then to Italy. And in the 13th, 14th century, this arithmetic from Indo-Arabic arithmetic was in fact a big thing in Italy. Because the trade was expanding and in fact, I mean, this was still not a part of the curricular education either in school or college. But people used to pay fair amounts to get a tutor who teach their words, their arithmetic in the style that we do today. In the first instance, when you repeat the number again from 11 to 19, the 10 was not represented by two digits. In fact, it starts in Babylon again with the place value notation originates from. It started by leaving a gap over there. So there was a place for a number, but it was left blank indicating there was no number. So the problem comes up is what happens if what we understand is 0 or the absence of a number is at the end of it. Now that space becomes very difficult to understand that there is a blank after this. And if it is in the middle of two numbers, well, you can approximate absence of a number. There is no concept of 0 in such a system. And that's why in our, if you see what used to be called the AD, BC way of counting years, there was no 0 year. Okay, there is one AD, there is one BC, but there is no 0 year because there's no concept of an absence of a year as such. So this problem arises with the place value notation when you try to write it. And then they decided that first let's indicate a sum marker to indicate there was, there is something here, but it's actually has no value. A symbol for the concept of nothing. I mean, why do you need to count nothing? But this really kick-started some extraordinary mathematics, things like the calculus, even our digital age is dependent on things like 1s and 0s. Well, amazingly, mathematics was really vibrant in the Indian subcontinent and we find 0 being talked about for the first time there. From say about 5th century BC to about 5th century AD, the most active mathematically seem to be the Jain monks or the Jains who did a lot of these enquiries and also talked about it and wrote it down. They start talking about the place value which is left empty as you need. And that's sort of the origin of even the word sifra, then it becomes 0. So all this cipher 0, a lot of these things originate from really in different ways. It comes logically from actually the void as the concept. Although the earliest use of 0 has been recorded in several historical locations in South Asia, the oldest so far is a series of birch bark manuscripts discovered in 1881 at the Bakhshali village in Khaybar area of the present day Pakistan. It is the oldest representation of 0. The modern symbol of 0 shall we say anywhere else. But here it must be understood when you look at the Bakhshali manuscripts and what is the mathematics that is there. The third century what have been dated in the third century text or we still don't have dated the all the pages of the Bakhshali manuscript to only a sample of them. They seem to indicate that in this phase, the third century phase because Bakhshali manuscripts is a combination of texts from different centuries. In the third century text it seems it was used basically as a symbol. Still the rules of arithmetic were not applied to the symbol 0. The Bakhshali manuscript is actually quite a complicated document because it isn't just one piece of mathematics. It's many pieces written over, it appears now, quite a few centuries. And it seems to be a document helping merchants to do mathematical calculations. So there are lots of examples and lots of numbers. And this is where we see the 0 because in some of these numbers we see a dot. This is a dot saying there is nothing here. And so this dot is the one that will eventually open up to be the 0 that we use today. The idea of 0 as a placeholder is actually very ancient. The ancient Babylonians, the Mayans had an idea of a 0 as a placeholder. So something like 101 means there are no tens. So we see 0 in the Bakhshali manuscript being used as a placeholder. But why is this so important as a moment in the history of mathematics? Because this symbol eventually gives rise to 0 in its own right. A number for counting absolutely nothing. And this is a revolutionary moment in mathematics. These could be applied to operations with 0. It's clearly enunciated first really in India. It seems that Arya Bhatta may have been aware of it. But clearly enunciation of the rules of arithmetic of the sense is really with Brahma Gupta. Brahma Gupta, Bhaskara Wan, Varaha Nihir. All these now are getting into enunciating the rules of arithmetic. With 0 as a number. So this is something which happens between 5th and 7th century. 0 gave us access to the negative numbers. How? Because just like we can say that 5 minus 3 is 2. But you could not say what is 3 minus 5. Now from 3 minus 5 you can go to and minus 2. You can go to the left of the line. So if you think of the line with 1 starting here, 2 here, 3 here, 4 here etc. What you have done is just added 1 to the left of it and called it 0. Minus 1 is the number minus 1. Just go beyond. Negative numbers were considered valid in a lot of the Indian works. But somehow in Europe there was a lot of resistance or lack of willingness to admit negative numbers. If you are doing borrowing and lending, if you are in trade, how much have you borrowed? How much have you given back? Chinese for instance had red and black sticks for negative and positive numbers. But it was mathematically very difficult for them to think about taking out something bigger than something that existed. As late as 20th century people wrote papers or articles saying that negative numbers did not make sense. Now it sounds like people believe in flat earth etc. But as late as 20th century there are European writers reluctant to admit, think of negative numbers. But on the other hand in India negative numbers are found from very old times. We talked about the evolution of numbers, the place value notation and the use of 0 besides the negative numbers. The story of Ganita or mathematics cannot be completed without exploring the developments in geometry and proof system. That is something we will discuss in the next part of our film. One of the oldest and widespread mathematical developments after the basic arithmetic and geometry was a theorem named after Pythagoras. Pythagoras theorem is something which all of you learnt in class 5. It says that if you take a right angle triangle with C being the hypotenuse and A and B the two other sides, then A squared plus B squared equals C squared. So suppose think of yourself as 10 year olds, you want to play football, you want to mark a field. So this is a typical football field, the world boxes, the line in the middle, the central line and the field. Suppose you have just a long rope and you want to do this. So what you do is you take the rope and you tie knots to divide it into 12 equal pieces. Once you've done that, then you take three of your friends, tell one of them to be on the third knot, another one on the fourth knot after that, and the other one on the end, who holds the two ends together. What you have now is a right angle triangle because one of the sides is 3, the other is 4, hypotenuse is 5. So 3 squared plus 4 squared equals 5 squared. So you've been able to draw a right angle. The first known evidence of Pythagorean theorem actually comes from Babylonia. The Babylonian civilization succeeded the Sumerian civilization and the Sumerians actually, their civilization was quite advanced. They had an administrative and legal system, they built irrigation condensers. They introduced writing using a cuneiform script, cuneiform means corners. They made clay and on clay with a reed marker, they marked things and then they baked it. And that's why Babylonian and Mesopotamian tablets still survive and we know so much about them now. So this is a typical clay tablet which exhibit lots of geometric pictures and there are various such things found. So in one such tablet which is again in the British Museum, there is a nice problem that 4 is the length, 5 is the diagonal, what is the breadth? Its size is not known and they are also given an answer to it in the same tablet. 4 times 4 is 16, 5 times 5 is 25, you take 16 from 25, there remains 9. What times what is 9? 3 times 3 is 9, so 3 is the breadth. There is also another Babylonian tablet, the cuneiform tablet, which in which there is a show the computation of square root of 2 represented in this hexadecimal system which they were following and that certainly involves the Pythagoras theorem to understand the magnitude, one had to know the Pythagoras theorem. There are marks on the side on the left and two on the horizontal diagonal of the square. So there are two marks on the horizontal diagonal and one marked there. What are they? They are the numbers 30, then 1, 24, 51, 10, 42, 25, 35. This is the hexadecimal system, so 30 is exactly like 5 in a decimal system and what we can think of 5 as being half of 10. So if you translate this and if you remember that 30 is half of 60, what is the red line is just half and below there are two things, one is the sixth decimal approximation of square root 2 with this approximation divided by 2. If you do an elementary Pythagoras, so the red line is half, the other line to its left is also of length half, half square plus half square, then square root of that becomes root 2 by 2. It is well known that the Babylonians had the table of Pythagorean triples. Now some people have debated it may not exactly correspond to Pythagorean theorem but on the other hand could be. It is called the Plympton 322. This is just a list of Pythagorean triplets. A 3, 4, 5 is called the Pythagorean triplet because 3 square plus 4 square equals 5 square and these are probably used in practical applications in land measurements and various other things in irrigation canals. So the Chinese civilization is the only civilization which has been uninterrupted. So we know a lot about it because of the continuity. Around 2700 before the Christian era, there was a Chinese emperor, Wang Ti, who took an active interest in mathematics and he is supposed to have composed the text but of course that's lost now but these are on astronomy and mathematics. Chinese call this theorem the Gou-Gou theorem. Gou is the hypotenuse and Gou is the side. So one of the pictures from the book relates to two proofs of the Pythagoras theorem. But till now the Mesopotamian everything, there was no mention of a proof. Here they do a mathematical proof in the modern day sense. One very interesting problem which they have in this book is called the Bent Bamboo Problem. You can see how global mathematics is from here. The Bent Bamboo Problem which says that there's a bamboo which is 10 chi. Chi is the unit of length in China. The upper end of which being broken touches the ground. So it was a rod like this, it broke here and so it has become a triangle now. And it says it touches the ground 3 chi from the foot of the step. They ask what is the height of the break. So you know it was 10 chi, it broke. So what is this height now? So of course you have a right angle triangle. You know that the hypotenuse plus this height is 10. You know this length is 3. You can set up the equation A plus C if C is the hypotenuse. A plus C is 10, the base is 3. So these are the two equations you get. And you also have the equation from the Pythagoras theorem A squared plus B squared equals C squared. So you can solve this quadratic equation and you get the answer A equals 91 by 20. It appears exactly same in Mahavira's Ganita Shastra which was around 850 Christian era. And later in Europe Arabs in this Philippi Kalandri's book called Arithmetic from 1491. The earliest explicitly stated statement of the theorem comes from the Shlva Sutras. Especially Bodhana being the earliest among the Shlva Sutras. Shlva Sutras are part of the Vedanga Kalpasutras of the Vedas. And Vedangas are the so-called appendices of the Vedas. There are nine Shlva Sutras of which three ascribed to Bodhiana, Agastamba and Kadyayana, the Vedangas which are rich in mathematics. The Shlva Sutras specify the construction of Yagye Bhumika or fire altars. Very elaborate structures were described in verse but without diagrams. The most famous of these is the falcon altar which comprises of certain geometrical shapes like parallelograms, rectangles, triangles and various other types. The Shlva Sutras are very specific about these sizes so that the brick makers could bake the exact shapes. There is this Sanskrit sloka there which basically translates to the chord placed along the diagonal of a square, produces an area double the size of the square. If I see this diagram AC square is double the size of the area of the square inside. The square inside is of length AB times BC but AB and BC are the same. Two AB times BC is two AB square which is the same as AB square plus BC square which is exactly the Pythagoras theorem for a square. They also talk about a rectangle. Here they say that the chord placed along the diagonal of a rectangle produces both the areas which the chords along the longer and the shorter sides produce. It's exactly the same thing. The main objectives of the Shlva Sutras was to convey how this structure, elaborate structure is to be constructed and the stipulations of various kinds first of all it had to be in five layers of bricks. Each layer was made up of 200 tiles. They could be of different shapes. There could be square, triangle and then sometimes even pentagonal and so on. For brick building purposes the sutras had to be more specific. So as in the Plympton 322 tablet where there was this list of numbers in the Vedas also in Baudhiana's universal sutra there are lists also given. So they write 3 and 4, the 5 will come 12 and 5, 15 and 8. These are all part of Pythagorean triplets. In the process of doing this kind of work they realized various geometric principles. One of the fundamental things in basic geometry is to be able to draw a perpendicular. Given one line and a point on that you need to draw a perpendicular. These are some of the first things that one learns in school. The method that we use in the school that is taught in the schools was in fact known at the time of Shiva Sutras. You know the Pythagorean triplets, because they are so basic probably originated in each of these three places independently. There is no reason to believe that there was a monogenetic origin for Pythagoras theorem. Unfortunately we do not know much about the Egyptian civilization in the sense that we do not have evidence of Pythagorean triplets but I am pretty sure that they would have had it because if they could build such elaborate pyramids and everything they would be sure they had it. If we could also get evidence from the civilizations in America because they were completely disjoint. Unfortunately again we do not have much evidence but there also they built pyramids, the Mayans, the Incas, the Steris farming and everything. So these are local variations but I would say that each of them are true in their own right that yes these are the areas which reached this conclusion and it is neither that the Greeks we gave the world the Pythagoras theorem and it is also an appropriation of the West to claim it is a Pythagoras theorem. It actually belongs to nature not to Pythagoras. The Greeks as I said because they have geometricized their mathematics more than others therefore they also were the first to establish a formal way of writing down geometry. When you go for statements especially in geometry which are a little more complicated if somebody makes them there is two things firstly there is some way in which they would have arrived at it and secondly having arrived it they would have some way of confirming or validation I would call validation. So proof has the main component in proof is validation. And if you look at what proof we have talked about the Pythagoras theorem it is essentially a visual proof. You move certain squares around you fit it in from one set of squares to another set of squares and you see okay you know they really fit in and therefore they are visually a proof therefore that the squares on the two sides are equal to the square of the hypotenuse. A mathematician will not accept a statement if it is not accompanied by a proof. The complete idea of proof as evolved from the Greek tradition is that you should be able to give a proof which is sort of independent of whom we are trying to convince including an adversarial participant of this debate. You cannot go back and back and back to prove so there are certain things you have to assume they are called axioms or postulates and then we have a certain system of accepted principles of what implies what and then you enter from there to deduce statements so those would be valid statements if you are hired by intuition at some statement being true then it was kind of a duty for you to be able to produce an argument which would lead to that. One such example is the infiniteness of prime numbers. What is a prime number? A prime number is a positive integer which is not divisible by any other positive integer except itself and one. Three is a prime, seven is a prime but fifteen is not because fifteen is divisible by three and five. So a question which is attributed to Euclid which he asked is how many prime numbers are there? There are infinitely many prime numbers. There is no limit. So one way of proving is what is called reductured absurdum or proved by contradiction. He says okay suppose there is an end to it. Suppose there is an end to the primes and there has to be a largest prime. Suppose for argument sake seventeen is the largest prime. You multiply all the numbers from one to seventeen. Then the product is divisible by one to three all the way up to seventeen but you add one to it. Now when you divide by anything there would always be a remainder of one. So this number plus one is a number which is not divisible by any of the primes either it itself is a prime or there is some other prime number larger than seventeen which will divide. So this is a proof. So in the Indian tradition they did go beyond the obvious statements and then they had means of validating that but the validation was not based on this systematic procedure or no systematic procedure was in fact laid down. So in that sense it is true that Indian mathematics did not have proofs. That does not mean that they simply contemplated and found every statement. They had means of, they had ways of arriving at statement. They had ways of convincing their counterparts. The world upapatti which means more like explanation or something like that which played the role of proof in Indian mathematics. If we give credit to the South Asian cultural area for the number system place value notation and zero as it is known to the modern world though each of them might have been discovered in different ways we have to similarly give credit to the Greeks for having reached this more model of proof which is really a powerful method of proof. The mathematics is something which is just completely universal. Nobody is shy of taking ideas from elsewhere or contributing to that. Having a geographical ownership of mathematics doesn't make any sense.