A history of Zero
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One of the commonest questions which the readers of this archive ask is: Who discovered zero? Why then have we not written an article on zero as one of the first in the archive? The reason is basically because of the difficulty of answering the question in a satisfactory form. If someone had come up with the concept of zero which everyone then saw as a brilliant innovation to enter mathematics from that time on, the question would have a satisfactory answer even if we did not know which genius invented it. The historical record, however, shows quite a different path towards the concept. Zero makes shadowy appearances only to vanish again almost as if mathematicians were searching for it yet did not recognise its fundamental significance even when they saw it.
The first thing to say about zero is that there are two uses of zero which are both extremely important but are somewhat different. One use is as an empty place indicator in our place-value number system. Hence in a number like 2106 the zero is used so that the positions of the 2 and 1 are correct. Clearly 216 means something quite different. The second use of zero is as a number itself in the form we use it as 0. There are also different aspects of zero within these two uses, namely the concept, the notation, and the name. (Our name "zero" derives ultimately from the Arabic sifr which also gives us the word "cipher".)
Neither of the above uses has an easily described history. It just did not happen that someone invented the ideas, and then everyone started to use them. Also it is fair to say that the number zero is far from an intuitive concept. Mathematical problems started as 'real' problems rather than abstract problems. Numbers in early historical times were thought of much more concretely than the abstract concepts which are our numbers today. There are giant mental leaps from 5 horses to 5 "things" and then to the abstract idea of "five". If ancient peoples solved a problem about how many horses a farmer needed then the problem was not going to have 0 or -23 as an answer.
One might think that once a place-value number system came into existence then the 0 as an empty place indicator is a necessary idea, yet the Babylonians had a place-value number system without this feature for over 1000 years. Moreover there is absolutely no evidence that the Babylonians felt that there was any problem with the ambiguity which existed. Remarkably, original texts survive from the era of Babylonian mathematics. The Babylonians wrote on tablets of unbaked clay, using cuneiform writing. The symbols were pressed into soft clay tablets with the slanted edge of a stylus and so had a wedge-shaped appearance (and hence the name cuneiform). Many tablets from around 1700 BC survive and we can read the original texts. Of course their notation for numbers was quite different from ours (and not based on 10 but on 60) but to translate into our notation they would not distinguish between 2106 and 216 (the context would have to show which was intended). It was not until around 400 BC that the Babylonians put two wedge symbols into the place where we would put zero to indicate which was meant, 216 or 21 '' 6.
The two wedges were not the only notation used, however, and on a tablet found at Kish, an ancient Mesopotamian city located east of Babylon in what is today south-central Iraq, a different notation is used. This tablet, thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place. There is one common feature to this use of different marks to denote an empty position. This is the fact that it never occured at the end of the digits but always between two digits. So although we find 21 '' 6 we never find 216 ''. One has to assume that the older feeling that the context was sufficient to indicate which was intended still applied in these cases.
If this reference to context appears silly then it is worth noting that we still use context to interpret numbers today. If I take a bus to a nearby town and ask what the fare is then I know that the answer "It's three fifty" means three pounds fifty pence. Yet if the same answer is given to the question about the cost of a flight from Edinburgh to New York then I know that three hundred and fifty pounds is what is intended.
We can see from this that the early use of zero to denote an empty place is not really the use of zero as a number at all, merely the use of some type of punctuation mark so that the numbers had the correct interpretation.
Now the ancient Greeks began their contributions to mathematics around the time that zero as an empty place indicator was coming into use in Babylonian mathematics. The Greeks however did not adopt a positional number system. It is worth thinking just how significant this fact is. How could the brilliant mathematical advances of the Greeks not see them adopt a number system with all the advantages that the Babylonian place-value system possessed? The real answer to this question is more subtle than the simple answer that we are about to give, but basically the Greek mathematical achievements were based on geometry. Although Euclid's Elements contains a book on number theory, it is based on geometry. In other words Greek mathematicians did not need to name their numbers since they worked with numbers as lengths of lines. Numbers which required to be named for records were used by merchants, not mathematicians, and hence no clever notation was needed.
Now there were exceptions to what we have just stated. The exceptions were the mathematicians who were involved in recording astronomical data. Here we find the first use of the symbol which we recognise today as the notation for zero, for Greek astronomers began to use the symbol O. There are many theories why this particular notation was used. Some historians favour the explanation that it is omicron, the first letter of the Greek word for nothing namely "ouden". Neugebauer, however, dismisses this explanation since the Greeks already used omicron as a number - it represented 70 (the Greek number system was based on their alphabet). Other explanations offered include the fact that it stands for "obol", a coin of almost no value, and that it arises when counters were used for counting on a sand board. The suggestion here is that when a counter was removed to leave an empty column it left a depression in the sand which looked like O.
Ptolemy in the Almagest written around 130 AD uses the Babylonian sexagesimal system together with the empty place holder O. By this time Ptolemy is using the symbol both between digits and at the end of a number and one might be tempted to believe that at least zero as an empty place holder had firmly arrived. This, however, is far from what happened. Only a few exceptional astronomers used the notation and it would fall out of use several more times before finally establishing itself. The idea of the zero place (certainly not thought of as a number by Ptolemy who still considered it as a sort of punctuation mark) makes its next appearance in Indian mathematics.
The scene now moves to India where it is fair to say the numerals and number system was born which have evolved into the highly sophisticated ones we use today. Of course that is not to say that the Indian system did not owe something to earlier systems and many historians of mathematics believe that the Indian use of zero evolved from its use by Greek astronomers. As well as some historians who seem to want to play down the contribution of the Indians in a most unreasonable way, there are also those who make claims about the Indian invention of zero which seem to go far too far. For example Mukherjee in [6] claims:-
... the mathematical conception of zero ... was also present in the spiritual form from 17 000 years back in India.
What is certain is that by around 650AD the use of zero as a number came into Indian mathematics. The Indians also used a place-value system and zero was used to denote an empty place. In fact there is evidence of an empty place holder in positional numbers from as early as 200AD in India but some historians dismiss these as later forgeries. Let us examine this latter use first since it continues the development described above.
In around 500AD Aryabhata devised a number system which has no zero yet was a positional system. He used the word "kha" for position and it would be used later as the name for zero. There is evidence that a dot had been used in earlier Indian manuscripts to denote an empty place in positional notation. It is interesting that the same documents sometimes also used a dot to denote an unknown where we might use x. Later Indian mathematicians had names for zero in positional numbers yet had no symbol for it. The first record of the Indian use of zero which is dated and agreed by all to be genuine was written in 876.
We have an inscription on a stone tablet which contains a date which translates to 876. The inscription concerns the town of Gwalior, 400 km south of Delhi, where they planted a garden 187 by 270 hastas which would produce enough flowers to allow 50 garlands per day to be given to the local temple. Both of the numbers 270 and 50 are denoted almost as they appear today although the 0 is smaller and slightly raised.
We now come to considering the first appearance of zero as a number. Let us first note that it is not in any sense a natural candidate for a number. From early times numbers are words which refer to collections of objects. Certainly the idea of number became more and more abstract and this abstraction then makes possible the consideration of zero and negative numbers which do not arise as properties of collections of objects. Of course the problem which arises when one tries to consider zero and negatives as numbers is how they interact in regard to the operations of arithmetic, addition, subtraction, multiplication and division. In three important books the Indian mathematicians Brahmagupta, Mahavira and Bhaskara tried to answer these questions.
Brahmagupta attempted to give the rules for arithmetic involving zero and negative numbers in the seventh century. He explained that given a number then if you subtract it from itself you obtain zero. He gave the following rules for addition which involve zero:-
The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero.
Subtraction is a little harder:-
A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is negative, zero subtracted from a positive number is positive, zero subtracted from zero is zero.
Brahmagupta then says that any number when multiplied by zero is zero but struggles when it comes to division:-
A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.
Really Brahmagupta is saying very little when he suggests that n divided by zero is n/0. Clearly he is struggling here. He is certainly wrong when he then claims that zero divided by zero is zero. However it is a brilliant attempt from the first person that we know who tried to extend arithmetic to negative numbers and zero.
In 830, around 200 years after Brahmagupta wrote his masterpiece, Mahavira wrote Ganita Sara Samgraha which was designed as an updating of Brahmagupta's book. He correctly states that:-
... a number multiplied by zero is zero, and a number remains the same when zero is subtracted from it.
However his attempts to improve on Brahmagupta's statements on dividing by zero seem to lead him into error. He writes:-
A number remains unchanged when divided by zero.
Since this is clearly incorrect my use of the words "seem to lead him into error" might be seen as confusing. The reason for this phrase is that some commentators on Mahavira have tried to find excuses for his incorrect statement.
Bhaskara wrote over 500 years after Brahmagupta. Despite the passage of time he is still struggling to explain division by zero. He writes:-
A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.
So Bhaskara tried to solve the problem by writing n/0 = ∞. At first sight we might be tempted to believe that Bhaskara has it correct, but of course he does not. If this were true then 0 times ∞ must be equal to every number n, so all numbers are equal. The Indian mathematicians could not bring themselves to the point of admitting that one could not divide by zero. Bhaskara did correctly state other properties of zero, however, such as 02 = 0, and √0 = 0.
Perhaps we should note at this point that there was another civilisation which developed a place-value number system with a zero. This was the Maya people who lived in central America, occupying the area which today is southern Mexico, Guatemala, and northern Belize. This was an old civilisation but flourished particularly between 250 and 900. We know that by 665 they used a place-value number system to base 20 with a symbol for zero. However their use of zero goes back further than this and was in use before they introduced the place-valued number system. This is a remarkable achievement but sadly did not influence other peoples.
You can see a separate article about Mayan mathematics.
The brilliant work of the Indian mathematicians was transmitted to the Islamic and Arabic mathematicians further west. It came at an early stage for al-Khwarizmi wrote Al'Khwarizmi on the Hindu Art of Reckoning which describes the Indian place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. This work was the first in what is now Iraq to use zero as a place holder in positional base notation. Ibn Ezra, in the 12th century, wrote three treatises on numbers which helped to bring the Indian symbols and ideas of decimal fractions to the attention of some of the learned people in Europe. The Book of the Number describes the decimal system for integers with place values from left to right. In this work ibn Ezra uses zero which he calls galgal (meaning wheel or circle). Slightly later in the 12th century al-Samawal was writing:-
If we subtract a positive number from zero the same negative number remains. ... if we subtract a negative number from zero the same positive number remains.
The Indian ideas spread east to China as well as west to the Islamic countries. In 1247 the Chinese mathematician Ch'in Chiu-Shao wrote Mathematical treatise in nine sections which uses the symbol O for zero. A little later, in 1303, Zhu Shijie wrote Jade mirror of the four elements which again uses the symbol O for zero.
Fibonacci was one of the main people to bring these new ideas about the number system to Europe. As the authors of [12] write:-
An important link between the Hindu-Arabic number system and the European mathematics is the Italian mathematician Fibonacci.
In Liber Abaci he described the nine Indian symbols together with the sign 0 for Europeans in around 1200 but it was not widely used for a long time after that. It is significant that Fibonacci is not bold enough to treat 0 in the same way as the other numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 since he speaks of the "sign" zero while the other symbols he speaks of as numbers. Although clearly bringing the Indian numerals to Europe was of major importance we can see that in his treatment of zero he did not reach the sophistication of the Indians Brahmagupta, Mahavira and Bhaskara nor of the Arabic and Islamic mathematicians such as al-Samawal.
One might have thought that the progress of the number systems in general, and zero in particular, would have been steady from this time on. However, this was far from the case. Cardan solved cubic and quartic equations without using zero. He would have found his work in the 1500's so much easier if he had had a zero but it was not part of his mathematics. By the 1600's zero began to come into widespread use but still only after encountering a lot of resistance.
Of course there are still signs of the problems caused by zero. Recently many people throughout the world celebrated the new millennium on 1 January 2000. Of course they celebrated the passing of only 1999 years since when the calendar was set up no year zero was specified. Although one might forgive the original error, it is a little surprising that most people seemed unable to understand why the third millennium and the 21st century begin on 1 January 2001. Zero is still causing problems!
History of Ganit (Mathematics)
History of Ganit (Mathematics)
Introduction
Ganit (Mathematics) has been considered a very important subject since ancient times. We find very elaborate proof of this in Vedah (which were compiled around 6000 BC). The concept of division, addition et-cetera was used even that time. Concepts of zero and infinite were there. We also find roots of algebra in Vedah. When Indian Beez Ganit reached Arab, they called it Algebra. Algebra was name of the Arabic book that described Indian concepts. This knowledge reached to Europe from there. And thus ancient Indian Beez Ganit is currently referred to as Algebra.
The book Vedang jyotish (written 1000 BC) has mentioned the importance of Ganit as follows-
Meaning: Just as branches of a peacock and jewel-stone of a snake are placed at the highest place of body (forehead), similarly position of Ganit is highest in all the branches of Vedah and Shastras
Famous Jain Mathematician Mahaviracharya has said the following-
Meaning: What is the use of much speaking. Whatever object exists in this moving and nonmoving world, can not be understood without the base of Ganit(Mathematics).
This fact was well known to intellectuals of India that is why they gave special importance to the development of Mathematics, right from the beginning. When this knowledge was negligible in Arab and Europe, India had acquired great achievements.
People from Arab and other countries used to travel to India for commerce. While doing commerce, side by side, they also learnt easy to use calculation methods of India. Through them this knowledge reached to Europe. From time to time many inquisitive foreigners visited India and they delivered this matchless knowledge to their countries. This will not be exaggeration to say that till 12th century India was the World Guru in the area of Mathematics.
The auspicious beginning on Indian Mathematics is in Aadi Granth (ancient/eternal book) Rigved. The history of Indian Mathematics can be divided into 5 parts, as following.
1) Ancient Time (Before 500 BC)
a)Vedic Time (1000 BC-At least 6000 BC)
a)Later Vedic Time (1000 BC-500BC)
2) Pre Middle Time (500 BC- 400 AD)
3) Middle Time or Golden Age (400 AD - 1200 AD)
4) Later Middle Time (1200 AD - 1800 AD)
5) Current Time (After 1800 AD)
1) Ancient Time (Before 500 BC)
Ancient time is very important in the history of Indian Mathematics. In this time different branches of Mathematics, such as Numerical Mathematics; Algebra; Geometrical Mathematics, were properly and strongly established.
There are two main divisions in Ancient Time. Numerical Mathematics developed in Vedic Time and Geometrical Mathematics developed in Later Vedic Time.
1a) Vedic Time (1000 BC-At least 6000 BC)
Numerals and decimals are cleanly mentioned in Vedah (Compiled at lease 6000 BC). There is a Richa in Veda, which says the following-
In the above Richa , Dwadash (12), Treeni (2), Trishat (300) numerals have been used. This indicates the use of writing numerals based on 10.
In this age the discovery of ZERO and "10th place value method"(writing number based on 10) is great contribution to world by India in the arena of Mathematics.
If "zero" and "10 based numbers" were not discovered, it would not have been possible today to write big numbers.
The great scholar of America Dr. G. B. Halsteed has also praised this. Shlegal has also accepted that this is the second greatest achievement of human race after the discovery of Alphabets.
This is not known for certain that who invented "zero" and when. But it has been in use right from the "vedic" time. The importance of "zero" and "10th place value method" is manifested by their wide spread use in today's world. This discovery is the one that has helped science to reach its current status.
In the second section of earlier portion of Narad Vishnu Puran (written by Ved Vyas) describes "mathematics" in the context of Triskandh Jyotish. In that numbers have been described which are ten times of each other, in a sequence (10 to the power n). Not only that in this book, different methods of "mathematics" like Addition, Subtraction, Multiplication, Addition, Fraction, Square, Square root, Cube root et-cetera have been elaborately discussed. Problems based on these have also been solved.
This proves at that time various mathematical methods were not in concept stage, rather those were getting used in a methodical and expanded manner.
"10th place value method" dispersed from India to Arab. From there it got transferred to Western countries. This is the reason that digits from 1-9 are called "hindsa" by the people of Arab. In western countries 0,1,2,3,4,5,6,7,8,9 are called Hindu-Arabic Numerals.
1b) Later Vedic Time (1000 BC - 500 BC)
1b.1) Shulv and Vedang Jyotish Time
Vedi was very important while performing rituals. On the top of "Vedi" different type of geomit(geometry: as you notice this word is derived from a Sanskrit word)) were made. To measure those geometry properly, "geometrical mathematics" was developed. That knowledge was available in form of Shulv Sutras (Shulv Formulae). Shulv means rope. This rope was used in measuring geometry while making vedis.
In that time we had three great formulators-Baudhayan, Aapstamb and Pratyayan. Apart from them Manav, Matrayan, Varah and Bandhul are also famous mathematician of that time.
The following excerpt from "Baudhayan Sulv Sutra (1000 BC)" is today known as Paithogorus Theorem (amazing, isn't it ?)
In the above formula , the following has been said. In a Deerghchatursh (Rectangle) the Chetra (Square) of Rajju (hypotenuse) is equal to sum of squares of Parshvamani (base) and Triyangmani (perpendicular).
In the same book Baudhayan has discussed the method of making a square equal to difference of two squares. He has also described method of making a square shape equal to addition of two squares. He has also mentioned the formula to find the value (upto five decimal places) of a root (square root, cube root ...) a number, according to that the square root of 2 can be found as below-
While Geometric Mathematics was developed for making Vedi in Yagya , in parallel there was a need to find appropriate timing for Yagya. This need led to development of Geotish Shastra (Astrology) In Geotish Shastra (Astrology) they calculated time, position and motion of stars. By reading the book Vedanga Jyotish (At least 1000 BC) we find that astrologers knew about addition, multiplication, subtraction et-cetera. For example please read below-
Meaning: Multiply the date by 11, then add to it the "Bhansh" of "Parv" and then divide it by "Nakshatra" number. In this way the "Nakshtra" of date should be told.
1b.2) Surya Pragyapti Time
We find elaborated description of Mathematics in the Jain literature. In fact the clarity and elaboration by which Mathematics is described in Jain literature, indicates the tendency of Jain philosophy to convey the knowledge to the language and level of common people (This is in deviation to the style of Veda which told the facts indirectly).
Surya Pragyapti and Chandra Pragyapti (At least 500 BC) are two famous scriptures of Jain branch of Ancient India. These describe the use of Mathematics.
Deergha Vritt (ellipse) is clearly described in the book titled Surya Pragyapti. "Deergha Vritt" means the outer circle (Vritta) on a rectangle(Deergha), that was also known as Parimandal.
This is clear that Indians had discovered this at least 150 years before Minmax (150 BC). As this history was not known to the West so they consider Minmax as the first time founder of ellipse.
This is worth mentioning that in the book Bhagvati Sutra (Before 300 BC) the word Parimandal has been used for Deergha Vritt (ellipse). It has been described to have two types 1) Pratarparimandal and 2)Ghanpratarparimandal.
Jain Aacharyas contributed a lot in the development of Mathematics. These gurus have described different branches of mathematics in a very through and interesting manner. They are examples too.
They have described fractions, algebraic equations, series, set theory, logarithm, and exponents .... Under the set theory they have described with examples- finite, infinite, single sets. For logarithm they have used terms like Ardh Aached , Trik Aached, Chatur Aached. These terms mean log base 2, log base 3 and log base 4 respectively. Well before Joan Napier (1550-1617 AD), logarithm had been invented and used in India which is a universal truth.
Buddha literature has also given due importance to Mathematics. They have divided Mathematics under two categories- 1) Garna (Simple Mathematics) and 2)Sankhyan (Higher Mathematics). They have described numbers under three categories-1)Sankheya(countable),2)Asankheya(uncountable) and 3)Anant(infinite). Which clearly indicates that Indian Intellectuals knew "infinite number" very well.
2) Pre Middle Time (500 BC- 400 AD)
This is unfortunate that except for the few pages of the books Vaychali Ganit, Surya Siddhanta and Ganita Anoyog of this time, rest of the writings of this time are lost. From the remainder pages of this time and the literature of Aryabhatt, Brahamgupt et-cetera of Middle Time, we can conclude that in this time too Mathematics underwent sufficient development.
Sathanang Sutra, Bhagvati Sutra and Anoyogdwar Sutra are famous books of this time. Apart from these the book titled Tatvarthaadigyam Sutra Bhashya of Jain philosopher Omaswati (135 BC) and the book titled Tiloyapannati of Aacharya (Guru) Yativrisham (176 BC) are famous writings of this time.
The book titled Vaychali Ganit discusses in detail the following -the basic calculations of mathematics, the numbers based on 10, fraction, square, cube, rule of false position, interest methods, questions on purchase and sale... The book has given the answers of the problems and also described testing methods. Vachali Ganit is a proof of the fact that even at that time (300 BC) India was using various methods of the current Numerical Mathematics. This is noticeable that this book is the only written Hindu Ganit book of this time that was found as a few survived pages in village Vaychat Gram (Peshawar) in 1000 AD.
Sathanang Sutra has mentioned five types of infinite and Anoyogdwar Sutra has mentioned four types of Pramaan (Measure). This Granth(book) has also described permutations and combinations which are termed as Bhang and Vikalp .
This is worth mentioning that in the book Bhagvati Sutra describes the following. From n types taking 1-1,2-2 types together the combinations such made are termed as Akak, Dwik Sanyog and the value of such combinations is mentioned as n(n-1)/2 which is used even today.
Roots of the Modern Trignometry lie in the book titled Surya Siddhanta . It mentions Zya(Sine), Otkram Zya(Versesine), and Kotizya(Cosine). Please remember that the same word (Zia) changed to "Jaib" in Arab. The translation of Jaib in Latin was done as "Sinus". And this "Sinus" became "Sine" later on.
This is worth mentioning that Trikonmiti word is pure Indian and with the time it changed to Trignometry. Indians used Trignometry in deciding the position , motion et-cetera of the spatial planets.
In this time the expansion of Beezganit (When this knowledge reached Arab from India it became Algebra)was revolutionary. The roots of Modern Algebra lie in the book Vaychali Ganit. In this book while describing Isht KarmaIsht Karm "Rule of False" as the origin of expansion of Algebra. Thus Algebra is also gifted to world by Indians
Although almost all ancient countries used quantities of unknown values and using them found the result of Numerical Mathematics. However the the expansion of Beez Ganit (Now known as Alzebra) became possible when right denotion method was developed. The glory for this goes to Indians who for the first time used Sanskrit Alphabet to denote unknown quantities. Infact expansion of Beez Ganit (Now known as Alzebra) became possible when Indians realized that all the calculations of Numerical Mathematics could be done by notations. And that +, - these signs can be used with those notations.
Indians developed rules of addition, subtraction, multiplication with these signs (+,-,x). In this context we can not forget the contribution of great mathematician Brahmgupt (628 AD). He said-
The multiplication of a positive number with a negative number comes out to be a negative number and multiplication of a positive number with a positive number comes out to be a positive number.
He further told:
When a positive number is divided by a positive number the result is a positive number and when a positive number is divided by a negative number or a negative number is divided by a positive number the result is a negative number.
Indians used notations for squares, cube and other exponents of numbers. Those notations are used even today in the mathematics. They gave shape to Beezganit Samikaran(Algebraic Equations). They made rules for transferring the quantities from left to right or right to left in an equation. Right from the 5th century AD, Indians majorly used aforementioned rules.
In the book titled Anoyogdwar Sutra has described some rules of exponents in Beez Ganit (Later the name Algebra became more popular). Please find below a few examples.
Thus it proves that Beez Ganit (Later the name Algebra became more popular) was well expanded by the mathematicians of Pre-middle Time. This was more expanded in the Middle Time.
It is without doubt that like Aank Ganit (Numerical Mathematics) Beez Ganit (Later the name Algebra became more popular) reached Arab from India. Arab mathematician Al-Khowarizmi (780-850 AD) has described topics based on Indian Beez Ganit in his book titled "Algebr". And when it reached Europe it was called Algebra.
As for as other countries are concerned we find that in the golden time of Greece Mathematics there was no sign of Algebra with respect to modern concept of Algebra. In classical period Greece people had ability to solve tough questions of Beez Ganit (Later the name Algebra became more popular) but there all solutions were based on Geometrical Mathematics. For the first time in Greece world, the concept of Beez Ganit (Later the name Algebra became more popular) is described in a books of Diofantus (275 AD). By that time Indians were far ahead. This is worth noting that the shape and form of current Beez Ganit (Later the name Algebra became more popular) is originally Indian.
3) Middle Time or Golden Age 400 AD- 1200 AD)
This period is called golden age of Indian Mathematics. In this time great mathematicians like Aryabhatt, Brahmgupt, Mahaveeracharya, Bhaskaracharya who gave a broad and clear shape to almost all the branches of mathematics which we are using today. The principles and methods which are in form of Sutra(formulae) in Vedas were brought forward with their full potential, in front of the common masses. To respect this time India gave the name "Aryabhatt" to its first space satellite.
The following is the description about great mathematicians and their creations.
Aryabhatt (First) (490 AD)
He was a resident of Patna in India. He has described, in a very crisp and concise manner, the important fundamental principles of Mathematics only in 332 Shlokas. His book is titled Aryabhattiya. In the first two sections of Aryabhattiya, Mathematics is described. In the last two sections of Aryabhattiya, Jyotish (Astrology) is described. In the first section of the book, he has described the method of denoting big decimal numbers by the alphabets.
In the second section of the book Aryabhattiya we find difficult questions from topics such as Numerical Mathematics, Geometrical Mathematics, Trignometry and Beezganit (Algebra). He also worked on indeterminate equations of Beezganit (Later in West it was called Algebra). He was the first to use Vyutkram Zia (Which was later known as Versesine in the West) in Trignometry. He calculated the value of pi correct upto four decimal places.
He was first to find that the sun is stationary and the earth revolves around it. 1100 years later, this fact was accepted by Coppernix of West in 16th century. Galileo was hanged for accepting this.
Bhaskar (First) (600 AD)
He did matchless work on Indeterminate equations. He expanded the work of Aryabhatt in his books titled Mahabhaskariya, Aryabhattiya Bhashya and Laghu Bhaskariya .
Brahmgupt (628 AD)
His famous work is his book titled Brahm-sfut. This book has 25 chapters. In two chapters of the book, he has elaborately described the mathematical principles and methods. He threw light on around 20 processes and behavior of Mathematics. He described the rules of the solving equations of Beezganit (Algebra). He also told the solution of indeterminate equations with two exponent. Later Ailer in 1764 AD and Langrez in 1768 described the same.
Brahmgupt told the method of calculating the volume of Prism and Cone. He also described how to sum a GP Series. He was the first to tell that when we divide any positive or negative number by zero it becomes infinite.
Mahaveeracharya (850 AD)
He wrote the book titled "Ganit Saar Sangraha". This book is on Numerical Mathematics. He has described the currently used method of calculating Least Common Multiple (LCM) of given numbers. The same method was used in Europe later in 1500 AD. He derived formulae to calculate the area of ellipse and quadrilateral inside a circle.
Shridharacharya (850 AD)
He wrote books titled "Nav Shatika", "Tri Shatika", "Pati Ganit". These books are on Numerical Mathematics. His books on Beez Ganit (Algebra) are lost now, but his method of solving quadratic equations is still used. This is method is also called "Shridharacharya Niyam". The great thing is that currently we use the same formula as told by him. His book titled "Pati Ganit" has been translated into Arabic by the name "Hisabul Tarapt".
Aryabhatta Second (950 AD)
He wrote a book titled Maha Siddhanta. This book discusses Numerical Mathematics (Ank Ganit) and Algebra. It describes the method of solving algebraic indeterminate equations of first order. He was the first to calculate the surface area of a sphere. He used the value of pi as 22/7.
Shripati Mishra (1039 AD)
He wrote the books titled Siddhanta Shekhar and Ganit Tilak. He worked mainly on permutations and combinations. Only first section of his book Ganit Tilak is available.
Nemichandra Siddhanta Chakravati (1100 AD)
His famous book is titled Gome-mat Saar. It has two sections. The first section is Karma Kaand and the second section is titled Jeev Kaand. He worked on Set Theory. He described universal sets, all types of mapping, Well Ordering Theorems et-cetera.One to One Mapping was used by Gailileo and George Kanter(1845-1918) after many centuries.
Bhaskaracharya Second (1114 AD)
He has written excellent books namely Siddhanta Shiromani,Leelavati Beezganitam,Gola Addhaya,Griha Ganitam and Karan Kautoohal. He gave final touch to Numerical Mathematics, Beez Ganit (Algebra), and Trikonmiti (Trignometry).
The concepts which were in the form of formulae in Vedah. He has also described 20 methods and 8 behaviors of Brahamgupt.
Great Hankal has praised a lot Bhaskaracharya's Chakrawaat Method of solving indeterminate equations of Beezganit (Algebra). This Bhaskaracharya's Chakrawaat Method was used by Ferment in 1667 to solve indeterminate equations.
In his book Siddhanta Shiromani, he has described in length the concepts of Trignometry. He has described Sine, Cosine, Versesine,... Infinitesimal Calculus and Integration. He wrote that earth has gravitational force.
3) Later Middle Period (1200 AD- 1800 AD)
Not much original work was done after Bhaskaracharya Second. Comments on ancient texts are the main contribution of this period.
In his book (1500 AD), the mathematician Neel Kantha of Kerla has given the following formula to calculate Sine r -
The same formula is given in the Malyalam book Mookti Bhaas. These days this series is called Greygeries Series. The following is a descriptions of the famous mathematicians of this period.
Narayan Pundit (1356 AD)
He wrote the book titled Ganit Kaumidi. This book deals with Permutations and Combinations, Partition of Numbers, Magic Squares.
Neel Kanta (1587 AD)
He wrote the book titled Tagikani Kanti. This book deals with Zeotish Ganit(Astrological Mathematics).
Kamalakar (1608 AD)
He wrote a book titled Siddhanta Tatwa Viveka.
Samraat Jagannath (1731 AD)
He wrote two books titled Samraat Siddhanta and Rekha Ganit (Line Mathematics)
Apart from the above-mentioned mathematicians we have a few more worth mentioning mathematicians. From Kerla we have Madhav (1350-1410 AD). Jyeshta Deva (1500-1610 AD) wrote a book titled Ukti Bhasha. Shankar Paarshav (1500-1560 AD) wrote a book titled Kriya Kramkari.
3) Current Period (1800 AD- Current)
Please find below a list of famous mathematicians and their writings.
Nrisingh Bapudev Shastri (1831 AD)
He wrote books on Geometrical Mathematics, Numerical Mathematics and Trignometry.
Sudhakar Dwivedi (1831 AD)
He wrote books titled Deergha Vritta Lakshan(which means characteristics of ellipse), Goleeya Rekha Ganit(which means sphere line mathematics),Samikaran Meemansa(which means analysis of equations) and Chalan Kalan.
Ramanujam (1889 AD)
Ramanujam is a modern mathematics scholar. He followed the vedic style of writing mathematical concepts in terms of formulae and then proving it. His intellectuality is proved by the fact it took all mettle of current mathematicians to prove a few out of his total 50 theorems.
Swami Bharti Krishnateerthaji Maharaj (1884-1960 AD)
He wrote the book titled Vedic Ganit.
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