Article #9: Greek Vs Indian

Most of us would have heard of Euclid’s great treatise ‘Elements’ but not all of us would have had the opportunity to understand what’s so great about the book and why there is so much of respect for the Greek Mathematicians. In this article, we will see how certain articles stood the test of time and became very much popular while many others died away. We will also see where do look for when we want to see the epitome of literature in different cultures.

To start with, let’s have a quick glimpse on Euclid’s Elements. The treatise is a compilation of whatever knowledge that existed in Greece in Mathematics till then. Probably, no one had done that herculean task of coming up with a definite structure where all the pieces of knowledge in Mathematics till then can fit in perfectly, and also giving a kind of thinking structure where more worlds of Mathematics can be created with the subject structure of Mathematics. It is something similar to what the great Indian Grammarian Panini did with Sanskrit language structure. He gave a very strong structure to the language without creating some new grammar for the subject. Another example somewhat similar to this is Vyasa’s task of compiling the Vedas systematically. One might think why is it that we don’t see many other names of people who must have written books in the respective areas. A possibility could be that the work that these people did became so phenomenal, that most other things before that became redundant and people started following these texts.

Euclid writes the book in Greek (of course) and he write 13 books on Plane and Solid Geometry. In the book #1 (fundamentals of plane geometry involving straight lines), he first gives a set of 23 definitions of terms that he is going to use in his book followed by 5 postulates (some kind of assumptions). This is followed by 5 Common Notions (e.g if a=b and b=c, then a=c). Then he starts giving out propositions with rigourous proofs using this tools – definitions, postulates and common notions. He continues more or less the same thing in the following 12 books. The beauty in the book is the order in which he sequences his propositions. If someone is reading a particular proposition, he won’t be able to refute it based on the earlier propositions that have been proved. E.g. if he has to use the idea of parallelogram in some proposition, he will ensure that the properties of a parallelogram has been discussed and proved in some earlier postulate so that no one can question or counter his proof. It is the typical style of arguing a case in the court of law.

What Euclid did was not just compile a book on Mathematics but bring a structure with which people can argue based on logic and reasoning. The gave a structure on which a theory can be built on to test the ‘truthfulness’ of some things. We see the mode of Scientific pursuing this mode of study and inquiry. But when we look at other civilisations, we do not come across this kind of rigour in Mathematics as much as it is seen in the Greeks and that is where Greeks gain a special place.

Let us come to India. A beautiful comparison that Prof Clemency showed us today was between Baudhayana Sulba Sutra (800 BCE) and Euclid’s Elements (300 BCE). She took the example of the construction of a square. In BSS, it is the first construction that is shown. In the Elements, it is the 46th proposition that is shown probably because certain other things needs to be proved before using those as knowledge and proving the process of construction of a square. Why don’t we see this kind of rigour in Indian Mathematics? Probably because most of the Indian Mathematics was utilitarian and application oriented. For example, BSS was focused in giving geometric construction methods for constructing fire altars. The objective was construction of fire altars and not arguing with reason (we do see that the author knows the reason for certain theorems like Pythagoras Theorem by some implicit remarks). Another difference is the style of writing the texts. In India, at that time, the literature was Sutra literature which involved using of least number of letters and writing in cryptic ways. In Greece it was prose, so there were lesser constraints. Does it mean that Indian treatises missed rigour completely? If we conclude that, perhaps we are look at wrong places.

We see a lot of depth and rigour in philosophical treatises in India. Also, the style of proving wasn’t an axiomatic approach like the Greeks’ which was a cultural difference. We also see that Chandas and Vyakarana (Prosody and Grammar) were very deep and robust in India. Prof. Clemency remarked that perhaps that is because those subjects were considered very important or a kind of epitome of knowledge. She also added that when we want to look for Mathematics, we might have to look for very unassuming places in Indian literature which might be areas generally considered outside the purview of Mathematics (like language, poetry, music, etc.)

Such kind of comparative study helps us understand how the same idea was thought so differently in different cultures. For instance, to draw arcs and straight lines, in India they used a rope. In Greece, they used a compass and a straight edge. Indians didn’t talk about angles in earlier treatises on trigonometry. Can you even imagine doing that today? They were great geometers and they didn’t talk about angles. Can we even digest this fact today? In India, the length of the chord was the function of the arcs and the ratio of the arc lengths to the circumference was the kind of notion for measurement, which today, we know is nothing but the angle measurement. In Greece, the notion of angle and trigonometry was different. Yet, both civilisations pursued the same topics in different ways. The notion of proofs were different. The purpose of doing Mathematics was different, and much much more to think about.

When we do not introduce history and kindle children to do a comparative study between civilisations, we are not giving them opportunities to think on fundamental questions like – why did they do it? how could they do it with the limited tools that they had? what were the interesting questions they pursued? and so on…

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Article #8 No, Euclid didn’t prove that there are ‘infinitely’ many primes!

If you are a Mathematics high school student (or above), I can bet that you won’t think I am crazy if I say – Euclid didn’t prove that the number of primes are ‘infinite’. And if you are not heard of such a proof, you can just be informed that the number of primes are infinite and in Math classes, a proof given by Euclid is stated as the classical proof that there are infinite number of primes. The proof is not very technical and simple to understand. I would urge Math students to forgive me for the less rigour in the proof as the target audience for this article is the layman.

Assume that there are limited number of prime numbers. Multiply all the primes and we get a number that is now divisible by all the primes. Add 1 to the result. Now, when we divide the resulting number by all the primes, we get a remainder 1. This means that the resulting number is not divisible by any of the earlier primes. This means, it has to be a new prime. But we had assumed that there were only limited number of primes and had multiplied all of them. Our result contradicts our assumption. Hence, our assumption is wrong. Therefore, there are infinite number of primes. This is a clean proof for infinitely many primes and the great Greek Mathematician Euclid (300 BCE) is said to be accredited to this proof. The only problem is, Greeks didn’t talk about ‘infinite’.

Take a pause. Read the statement once again – ‘Euclid proved that there are infinitely many primes.’ Are you feeling confused/uncomfortable? The problem is not with Euclid, but with the English translation. As it often happens with translation of original works, there might not be a suitable vocabulary for certain terms or sometimes we confuse what was meant by the original author to the knowledge that we possess now (but the author didn’t possess at that time). What do I mean by that? Let me elaborate.

Euclid uses the word Greek word ‘apeiron’ which has been loosely translated as ‘infinite’ at many occasions. Some historians opine that ‘aperion’ cannot be translated as ‘infinite’ because Greeks didn’t have the notion of infinity (this is commonly accepted by all historians). Aperion means – ‘indefinitely’ or ‘as much as you please’. In some sense, today we might say that ‘as much as you please’ is the same as ‘infinite’. But Mathematicians wouldn’t agree to that because of the different notions of infinity. Even Euclid might not agree to the translation of infinity because for him, #Geometry is something that could be drawn. The drawing had to be such that it could be drawn (by a human?) on a plane however large that might be. So, when we say, draw a line and extend it as much as you want, it means that you do the task but stop somewhere because there would be a ‘limit’ to it. Otherwise, we cannot draw a line that is infinitely long. This is where we get into a philosophical conflict with Euclid if we use the word ‘infinite’. And in many famous translations of Euclid (like the famous one by Richard Fitzpatrick), there are many instances where the word ‘infinite’ has been used in some translations which is incorrect as the notion of ‘infinite’ didn’t exist among the Greeks. Perhaps, they could have got there as they talk about apeiron as a limitless quantity, some indeterminate, or something that can be done indefinitely, but they couldn’t accept the meaning of ‘infinite’

So, next time someone tells you that Euclid gave a proof that there are infinitely many primes, tell them – No! Euclid said that you can create as many #primes as you want.

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#Elements

Article #7: Did you know some of the biggest numbers ever imagined?

Last week while I was doing an webinar, a 10-11 year old girl asked me very innocently, “What is the biggest number that has been discovered?” I told her, “I don’t think people talk about discovering a ‘biggest’ number.” I didn’t know that I was wrong until today’s lecture. This article is about some instances where people have talked about some really BIG numbers. And how they imagined creating such numbers would only baffle someone.

Story #1: Archimedes
Almost all of us would have heard about the funny story of famous Greek Mathematician Archimedes (300 BCE) where he jumped out of his tub screaming ‘Eureka! Eureka!’ (I saw it! I saw it!) when he discovered the Archimedes’ Principle. Archimedes was one of the best thinkers of antiquity. He wrote a book called ‘Sand Reckoner’ where he asks the question – How many sand grains will fit into the universe? Many of us would even think – why would someone even think such a question?

Just as the Roman numerals had a limitation to express really large numbers, the Greek numeration ran out of symbols to express numbers beyond a Myriad (10,000). To answer Archimedes’ question, he knew that he would need a different numbering system. So he created a rhetoric number system without the use of any numerals. And this is what he did. He said, let’s have something called as the First Period. In the First Period, let the numbers from 1 to a myriad Myriad (10^8) be called the ‘First Numbers’. Let the numbers from a myriad Myriad to a myriad myriad myriad Myriad (10^8 to 10^16) be called the ‘Second Numbers’ where the first numbers become the units for the second numbers. The ‘Third Numbers’ would be 10^16 to 10^24. Continue this process upto the myriadth Myriad which will be 1 followed by 800 million zeros! Modern day googol (10^100) is a bit larger than the number of particles in the universe (10^80) and Archimedes’ number was so much larger than that. If you are wonderstruck, then let me give you a shock. He didn’t stop here. This was just the First Period. Now, he starts the Second Period where he takes the First Period as the units for the Second Period and continues with Second Period First Numbers followed by Second Period Second Numbers, and so on until he gets the last number number – a myriad myriad units of the myriad myriadth numbers of the myriad myriadth period. In simple words – 1 followed by 80,000 million million zeros!

Why is Archimedes doing this? Does he need such a big number to answer his original question? Definitely not. He took two pieces of empirical information that the diameter of the universe is < 10,000,000,000 stadia (a unit of measurement) and sphere of diameter 1 finger breadth needed 640,000,000 grains of sand. With that he calculated and got the number of sand grains to fill the universe as 10^63 (modern day number vigintillion). With the number system he created that didn’t involve any symbols it must have been really hard for him to arrive at the answer. But he did it. After all, he was Archimedes!

Story #2: Numbers in Jainism
In Jaina philosophy, they discussed about very large numbers as they formed part of their religious practices. While they did that, a question similar to that of Archimedes arose in their minds – How many mustard seeds can be piled up in a space with Jambu island at the center? This again is a thought experiment. Let’s first understand what is this Jambu dveepa (Jambu island).

Jambu island is circular in shape. Its diameter is fixed at 100,000 yojanas (the measurement of yojanas are given differently by different historians and it is not clear how big is one yojana). Surrounding the Jambu island are concentric rings of bands of sea and land. The width of these rings double each time we move outwards (infinitely). Now they come up with an ingenious idea they come up with to count the number of mustard seeds to pile up in Jambu island at the center.

– Fill the Jambu island with the maximum number of seeds possible. Let the radius of the island be r_0 and number of seeds be s_0.
– Empty the seeds of Jambu island into the successive rings such that you drop one seed in each ring. Thus, you reach s_0th ring in the last drop.
– Consider the resulting disk of radius r_1 that will include the first s_0 rings.
– Fill this disk completely with s_1 seeds.
– Continue the process until you reach the cube of s_0.
They defined that number as the ‘unenumerable of the low enhanced order’.
A famous historian, Prof. R.C.Gupta (to whom we owe a great deal for contributing a lot of articles pertaining to history of Indian Mathematics), calculated this number and found it to be 7.9656 x 10^135. And this was the ‘unenumerable of the low enhanced order’ for the Jains. They were just getting started!

Hope these people from history have turned you numb with their sheer imagination about numbers!

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Article #6: Poetry, Fire Altars and Mathematics!

While the Egyptians, Greeks, Babylonians were busy doing Mathematics and Astronomy, the Indian subcontinent was no less busy. In history, we see the birth and development of various areas in Mathematics and one of the most interesting factors in it is – the motivation to do a certain kind of Mathematics. In India, we find the earliest works on Mathematics in Sulba Sutras written (somewhere between 800 BCE to 200 BCE) by different authors. The earliest available work is said to be of Bodhayana named ‘Baudhayana Sulba Sutra’ followed by many other Sulba Sutrakaras. Sulba Sutras formed part of a bigger treatise called the Kalpam which prescribed instructions to be followed by a citizen for his well-being, his spiritual well-being, societal well-being and so on. In that, Sulba Sutras dealt with construction of fire altars for worshiping. Kalpam also formed a part of a bigger body of knowledge called the Vedanga which had six branches (Shiksha, Kalpam, Vyakaranam, Chandas, Niruktam, Jyothisham)

We restrict this article to some of the significant aspects of Sulba Sutras and Chandas.

The fire altars for worship that were prescribed by the texts involved constructions of geometrical shapes like squares, right triangles, circles, etc. The methods to do these constructions involved the knowledge of the Pythagorean Theorem. Bodhayana’s time was atleast a couple of centuries before Pythagoras and in his work, Bodhayana says in verse 1.12 – the rope corresponding to the diagonal of a rectangle makes whatever is made by the lateral and the vertical sides individually. Literally translated, it might even sound absurd. How can the hypotenuse (what is referred to as diagonal of a rectangle) be equal to the sum of the sides? He doesn’t mention about the squaring part that is involved in the theorem of the right angled triangle (a^2 + b^2 = c^2). One might feel that the theorem is not properly stated. But in the very next verse, Bodhayana lists down a few triplets (Pythagorean triplets) which cannot be enunciated if the property is not known correctly.

One needs to note that the notion of proof in Indian Mathematics is not the same as the Greek tradition which is set up by axioms and definitions. In India, Math was more of an applied pursuit. What makes it clear that Bodhayana knew the theorem well is the methods that he prescribes to construct a square that is sum of/difference of two squares. The methods make it very clear that Bodhayana had to know the theorem and not just a few examples to give out such clear methods of constructions.

Bodhayana also gives a very interesting verse that gives the value of square root of 2 correctly upto 5 or 6 decimal places. Remember, this is 800 BCE that is being talked about. He gives out a very interesting formula to calculate the value of root 2. He says, ‘Add 1 to its third and multiply 1/34 subtracted from 1 to one fourth of one third’. By doing this operation we get 577/408 which gives the value of root 2 approximately correct to 5-6 decimal places. He doesn’t leave it there. Bodhayana goes ahead and adds the word ‘savishesha’ which means, ‘close to’ whereby he says that this value is only close to or approximate value of root 2. This is a very strong claim made by him.

Mathematicians after him have tried to trace Bodhayana’s line of thought and tried to figure out how he might have arrived at the formula.

The literature of the Sulba Sutras, Chandas Shastra, etc are Sutra literature (in the form of aphorisms in prose). This literature style died out later on and what took its place was poetry. That is where Chandas comes into play.

The person who systematised the science of study of metres was Pingala Naga (300 BCE). He too, like Panini, was not the originator of Chandas but the one who systematised it so well that there was no need felt to refer to the earlier works on Chandas. Pingala makes syllables the basic component of classification and study of metres. A syllable is one or more consonants combined with a vowel. A syllable can be a short one (laghu) or a long one (guru) depending on how much time it takes to be pronounced. The time taken cannot be at our liberty, i.e. it doesn’t vary from person to person but it pre-defined for different vowels. E.g. Rama has two syllables and both are Gurus. There are two classification of metres – one based on the nature of the syllable and the other based on how much time it takes to be pronounced. The former is called Varna Vritta and the latter Matra Vritta. In the case of a Varna Vritta, the restriction of a metre is on the number of syllables it contains. So if there are n number of syllables, there are 2^n possibilities to have a metre. For a Matra Vritta, the restriction is on the time taken to say the verse. Laghu is considered as 1 and Guru as 2.

With these basic ideas, Pingala proceeds to give algorithms to list down arrays of syllables, gives formulae to count the number of strings in each array, ways to find the structure of a string in a given row number (and vice-versa), heads into binary mathematics, co-efficient of binomial expansion, so-called Fibonacci numbers, and so on. Here we see the first kind of Computer Science in its origin.

It is fascinating to see how construction of fire altars led to the birth of geometry which led into discussions in irrational numbers, number theory and algebraic nature of treating geometry. One of my favourites is the play of Permutations & Combinations in rules of Poetry and a very clever use of binary Mathematics to work out formulae devised in Pingala’s Chandas Shastra. All this only shows how Mathematics was never taught independent of other subjects but as a very clear inter-disciplinary approach. Can we hope to see such approaches in modern day classrooms?

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Article #5: What I do not know I do not think I know

Today’s talk by Prof. Clemency Montelle started with a beautiful well known 16th cent CE painting by Raphael – the School of Athens, where he has put up the pictures of almost all famous astronomers, philosophers and great minds from Greece. In the center, we see the image of two men – Plato (pointing his finger up) and Aristotle to his left. Plato is seen holding a book ‘Timeo/Timaeus’ which is a book that discussees about the origin of the world and Aristotle is holding a book on Ethics. Aristotle was once Plato’s student and later his colleague.

In the painting we see many other great figures like Xeno (famous for his paradoxes which got solved only after the invention of calculus), Ptolemy (the famous Greek astronomer whose works were studied by the Arabs, famous for Ptolemy’s theorem in cyclic quadrilaterals), Euclid/Archimedes (on the bottom right there’s a person bending down to draw something. Some say, it is Euclid, some Archimedes), Pythagoras (on the left bottom sitting in a white cloth writing something), Hypatia to the left of Pythagoras standing in a white cloth (the famous woman Mathematician who had to face a tragic ending. There’s a wonderful movie ‘Agora’ on her life starred by Monica Bellucci), Socrates (dressed in green in the middle left), Diogenes (lying down on the stairs. Raphael probably placed him there because nobody liked Diogenes much), Heraclitus and Raphael (placed himself in the bottom right facing us). Of course, all of these minds weren’t contemporaries. But Raphael puts a lot of thought behind the painting and gets all of the on board with a beautiful title. Not to miss the beauty of the painting where it is made keeping in mind things like the ‘vanishing point’ (an effect that seems to take the painting to a depth in the background where the depth of the painting seems to vanish at a single point).

After putting a perspective about the Greeks, the talk progressed into a variety of aspects from the Greek tradition. One of the things that I loved the most was the philosophical question that some of the philosophers asked – What is the ultimate substance underlying the creation? To Thales, it was ‘water’. To Anaximander it was ‘apeiron’ which meant ‘limitless quantity’. To Pythagoras it was ‘Numbers’. And the rationale was that he could see a way in which everything could be quantified…even aesthetics could be quantified. In some sense, it is something that we see getting applied even today. We love to see statistical analysis, show even things like assessment in numbers. In some sense, numbers do seem to rule the world. This was the pre-Socrates thinkers.

Then comes Plato and his teacher, Socrates. Plato believed that the Ultimate Creator had to be a Mathematician. Socrates believed that ‘the one thing he knows is that he knows nothing’ which is an often paraphrased sentence of the title of this article. The talks about both Socrates and Plato are very interesting. Socrates is Plato’s teacher and is a very interesting character. Socrates’ writings aren’t available. One knows about the teachings of Socrates through Plato’s anecdotes of his teacher. Plato, in some sense, seems to show Socrates as a character that people might get irritated with mainly because he was good in convincing people that they don’t know what they think they know. He believed, that if we aren’t sure of something, we shouldn’t think that we know it and take the stand of being ignorant very comfortably. Only then can one learn something in the process. So, if someone would come to Socrates and show off that they know something, then Socrates would ask them questions in such a way that they would be nudged to answer and finally give up and attain a ‘aporia’ (waylessness) state where they will through up their hands and accept that they don’t know anything about it. Socrates believed that one had to reach this state to start learning something. He believed that knowledge existed in everyone’s ‘soul’ and one had to be just reminded of it and it would be possible to bring out the knowledge that one possessed. For this to happen, teaching shouldn’t be done…one just needs to be nudged to bring out his/her learning. Some of his pedagogical aspects can be seen in the works of the great recent Mathematician – George Polya.

It was very interesting to see how these philosophical aspects percolated into Mathematics and the fundamental axioms they had in Mathematics and Science in ancient Greece.

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Article #3: Sherlock Holmes in Babylon

As I was mentioning in the first article of this series, the Babylons were using clay tablets to write/record. While the archaeologists excavated and brought out artifacts, every artifact was numbered. E.g. A problem from BM 13901 would mean, a (Mathematical) problem from artifact number 13901 from the British Museum. One such piece was ‘Plimpton 322’ (where Plimpton is possibly the name of the trader and the artifact number could be 322) which caught a lot of attention. Many people gave different interpretations and translations of it. One of the articles about Pimpton 322 was titled ‘Sherlock Holmes in Babylon’ by Creighton Buck in 1980 where he tries to decode the tablet like a detective. Buck dates back the tablet to 1700 BCE and finds an interesting appearance of trigonometric tables. The arguments are interesting, but not always can everything interesting be true. There’s another research article countering the paper by Buck which titles – Neither Sherlock nor Babylon by Eleanor Robson who criticizes everyone like Buck who tries to study history of Math or Science like a detective novel. Robson says, ‘Ancient mathematical texts and artefacts, if we are to understand them fully, must be viewed in the light of their mathematico-historical context, and not treated as artificial, self-contained creations in the style of detective stories. I take as a dramatic case study the famous cuneiform tablet Plimpton 322. I show that the popular view of it as some sort of trigonometric table cannot be correct, given what is now known of the concept of angle in the Old Babylonian period.’

 

Robson gives six points which are very important while a historian (or anyone) would try to analyse Mathematical history.

1. Historical Sensitivity

2. Cultural Consistency (if someone is talking of discovery/invention of calculus before the time when numeration has evolved in the particular civilisation, then it is not culturally consistent)

3. Calculational Plausibility

4. Physical Reality

5. Textual Completeness

6. Tabular Order

 

Many a times there are biases that come as a filter for research, especially when it comes to history where everyone wants to get a higher rank or wants to declare how better their civlisation was in comparison to the other.

 

The objective of this post was mainly to highlight the point that:

1. When we analyse things (especially with history), we might not be able to solve it like a detective novel by taking it independently. We should rather look at what is the work done in the past, surrounding the topic, and then look at it.

2. Let’s not be biased.

 

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Article #2: Modern day clay tablet of an ancient day writing

When I held this clay tablet today, I felt as though I was holding an ancient artefact. It was a replica of a tablet dating to 560 BCE from somewhere in Mesopotamia. We can see how minutely the matter is carved out by the scribe. The content reads about freedom granted to a slave woman and her children.

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Article #1: Discoveries about Mathematics buried under the sands

I am going to write some short articles from the understanding of the course that I am presently attending in IIT Bombay on History of Mathematics (Course name: Mathematics in Perspective) which discusses about Mathematics developed in India and in other civilisations.

Article #1: Discoveries about Mathematics buried under the sands

The region called Mesopotamia (near modern day Syria, Iraq) was once a place where there was a great civilisation. But this was completely unknown to the world until 150 years ago when some British and French explorers went to do some excavations in those regions based on some hunch and hints. The region was under conquests and invasions for many years and when the explorers went there, they found just sands….and huge cities buried under those sands. These cities had temples, schools, houses and whatever a city would need. But they laid there for two thousand years unnoticed by anyone.

Prof. Clemency Montelle from University of Canterbury, New Zealand, explained to us about how the people who lived there used clay tablets, how language developed in that region over thousands of years, how Mathematics developed in the region, the nature of Mathematics, and so on. The Babylonians (residents of Mesopotamia), as they are called, used Base 60 to represent numbers. Today we use Base 10 (a system with 10 symbols 0 to 9) and Base 60 had 59 symbols. This might make some of us faint, but they too felt the same and hence didn’t introduce 59 symbols but just one symbol that could be used to represent more numbers (something like Tally marks). There was also a detailed discussion on why ‘base 60’ was used by them. The arguments/questions from the participants was also very interesting. Someone asked if it could be that earlier people had fixed the number of days for earth’s revolution as 360 that resulted in the number 60 for time in some way? To that, the speaker replied that the reckoning of time is recorded much later in the history than use of base 60 in representing numbers.

Another important point that was discussed is that when we say …’the Babylonians or any ancient civilization used something or did certain kind of Math’, it is very important to note that the civilisation was spread across millenniums. So, it is important to look at which time period the discussion is about and what was the development of Mathematics at that time. We cannot make a general statement that a civilization knew such and such without considering the time period.

Coming to the notebooks that have survived the test of time…all the records are found in the form of clay tablets. However, most of it don’t have a name or date. These were mostly tablets used as notebooks by teachers and students in the classroom and not for the purpose of recording knowledge. Hence, it becomes all the more difficult for historians to decode from such tablets especially when the language used is not even in use. In comparison to that, it’s really great to see that Indian treatises that has survived thousands of years are written in Sanskrit which is very well understood by atleast a considerable number of population and there are atleast a few scholars to decipher it.

For a few moments, let’s try to port ourselves to a classroom 3000 years ago in Babylon. There was no writing. There was only creating an impression on clay with a stylus. The instrument used (like a stylus) was mostly reed from the river which would be cut on its edge in a shape what they call as a ‘wedge’. There might have been a lump of clay in every classroom where the student goes and gets a lump that can be shaped into some sort of an oval shape within the palms and starts making impressions using this stylus. Since there was only one ‘digit’ used as tally marks to make all the numbers, they didn’t need different styluses to make different shapes. There would be multiplication tables made on clay tablets that people would use while performing calculations. Can you make a guess on the multiplication tables that they would have to learn by-heart if they had to? It would be something like tables upto 59 x 59! That would have been such a miserable life! So, they made tables handy in smart ways. If they would have to write the times table of 9, they wouldn’t do it upto 59. They would do it upto 9 x 20 and follow it by 9 x 30, 9 x 40 and so on. So, if someone had to find out what is 9 x 47, they would do 9 x 40 and add it to 9 x 7. Smarties! However, it wouldn’t be that easy for us today because the system was Base 60 and not 10. There were some more challenges, which I will write in another article.

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