Ancient Vedic Mathematics

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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-

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For the good of the Hindus who would be benefitted with this document . Also see http://www.esnips.com/web/PUBLIC-DOMAIN for regular updates with new books 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 followingPl visit my website at http://www.esnips.com/web/PUBLIC-DOMAIN , regularly for many documents of ancient India . Comments to [email protected]

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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. Pl visit my website at http://www.esnips.com/web/PUBLIC-DOMAIN , regularly for many documents of ancient India . Comments to [email protected]

For the good of the Hindus who would be benefitted with this document . Also see http://www.esnips.com/web/PUBLIC-DOMAIN for regular updates with new books 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 Pl visit my website at http://www.esnips.com/web/PUBLIC-DOMAIN , regularly for many documents of ancient India . Comments to [email protected]

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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 categories1)Sankheya(countable),2)Asankheya(uncountable) and 3)Anant(infinite). Which clearly indicates that Indian Intellectuals knew "infinite number" very well.

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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

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For the good of the Hindus who would be benefitted with this document . Also see http://www.esnips.com/web/PUBLIC-DOMAIN for regular updates with new books 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 saidThe 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. Pl visit my website at http://www.esnips.com/web/PUBLIC-DOMAIN , regularly for many documents of ancient India . Comments to [email protected]

For the good of the Hindus who would be benefitted with this document . Also see http://www.esnips.com/web/PUBLIC-DOMAIN for regular updates with new books 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 AlKhowarizmi (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. Pl visit my website at http://www.esnips.com/web/PUBLIC-DOMAIN , regularly for many documents of ancient India . Comments to [email protected]

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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) Pl visit my website at http://www.esnips.com/web/PUBLIC-DOMAIN , regularly for many documents of ancient India . Comments to [email protected]

For the good of the Hindus who would be benefitted with this document . Also see http://www.esnips.com/web/PUBLIC-DOMAIN for regular updates with new books 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 -

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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 (15001610 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)

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For the good of the Hindus who would be benefitted with this document . Also see http://www.esnips.com/web/PUBLIC-DOMAIN for regular updates with new books 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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