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Thank you for the questionChemistry Textbooks for Class 11 & Class 12 by NCERTncert-chemistry-textbook-pictureThe NCERT Chemistry text books for Class 11 and Class 12 are really great when it comes to getting started with Chemistry at this level and helps you pick up some of the most important concepts of JEE Mains and Advanced Chemistry in a really great way for all the three branches of chemistry – Physical, Organic and Inorganic.I did a mistake of not referring to these books initially and paid a huge price when I scored bad in my first all India test that was conducted by Brilliant Tutorials. After I saw that my other friends, whom I thought to be smarter than me, were referring to these books I knew where to look next. These books are available in 4 total parts and you can find them here.IIT JEE Chemistry by O.P. AgarwalThis is one book that is recommended by almost any JEE chemistry teacher or even shopkeeper when you speak about Chemistry for JEE Prep. But you gotta take my word on this that you should avoid studying Inorganic and Organic chemistry from this book when you are just starting out.The only reason you should have this book with you is as your chapter-wise and topic-wise guide and also for the vast number of problems at the end of each chapters. But, try to avoid this book as much as possible for concept-building.***Physical Chemistry (Numerical Chemistry) Books for JEEphysical-chemistry-op-tandon-picturePhysical Chemistry by O.P. TandonVarious Physical Chemistry concepts have been explained to great depth in this book and it also accompanies tough problems of various levels with their solutions. If you are preparing with taking JEE Advanced in mind then the problems in this book will be of great help to you however if you are looking to appear only for JEE Mains then you can give this book a miss. More details about this book can be found here.numerical-chemistry-p-bahadur-picturePhysical Chemistry by P.BahadurThis book for Numerical Chemistry by P. Bahadur is as good as the one by O.P Tandon, that I mentioned above and can be used for preparing both JEE Mains and JEE Advanced exams. During my days of JEE preparation, I did refer to the chapters of Chemical Kinetics, Ionic Equilibrium & Thermodynamics from this book and solved most of the numerical problems as daily practice.university-chemistry-bruce-mahan-pictureUniversity Chemistry by Bruce H. MahanThis book is a course text-book higher education level Physical Chemistry in many colleges and provides in-depth explanation and coverage of almost all topics related to the subject. If you are in doubt about any specific topic in Physical chemistry then reading through that part of the book is highly recommended. You can find more details about this book here.modern-approach-to-chemical-calculations-mukherjee-pictureModern Approach to Chemical Calculations by R.C. MukherjeeThis book can easily be termed as the H.C. Verma of Chemistry, well not literally! During my JEE preparation days I used to swear by this book owing to the quality of the numerical problems in it and the explanation of the solutions – simple, to the point and lucid. Without solving questions from this book no preparation for JEE level physical chemistry can be termed as complete. More details about this book can be found here.physical-chemistry-peter-atkins-picturePhysical Chemistry by P.W.AtkinsBefore I say anything about Physical Chemistry by P.W.Atkins, please make sure you do not go into solving the problems in this book. They are of really high level and you do not really need to solve such problems to prepare yourself for the JEE Main exams or even the JEE Advanced exams. Just refer to the book for its solid concepts. Reading through the book was a charm during my JEE Prep days.***JEE Main Organic Chemistry Books (and Advanced)Like I mentioned above, you must go through the Organic Chemistry chapters in the NCERT chemistry text books for Class 11 and Class 12 before you touch any of the books in this list. I am saying this out of experience as it would make it easier for you to have a rough idea of what the author is trying to convey.Organic Chemistry is a highly scoring subject provided you have crystal clear concepts. Wrong information and bad teaching practices have created a false assumption among students that Organic Chemistry is tough, which is not at all true. You should also go through these common mistakes in Organic Chemistry that students make during their JEE Prep.organic-chemistry-op-tandon-pictureOrganic Chemistry by O.P. TandonThis text book for Organic Chemistry, written by O.P. Tandon puts JEE Main and Advanced exams forward, which means you will spend less time finding problems to solve for your daily practice. However, I was not a fan of this book when it comes to studying Organic Chemistry although many of my friends do make this one as their primary reference book.Organic Chemistry by Morrison & Boydorganic-chemistry-morisson-boyd-pictureOrganic Chemistry by Morrison & Boyd is definitely one of the best books for Organic Chemistry out there. Now, if you really want to build some of the best and most solid concepts in Organic Chemistry, this book should be on the top of your lists. Now, mind you the book is really thick which means you will have to put in extra hours. But trust me, if you do all these, you will definitely have an edge over your friends in the Organic Chemistry part of the JEE Main / Advanced exams.You should also spend some time to solve the end-of-chapter questions to check your grasp of various concepts. FIITJEE generally gives tough organic questions in their All India Open Tests and I had scored a 99.99 – percentile in the Organic Chemistry module just be studying Organic Chemistry by Morrison Boyd.advanced-problems-in-organic-chemistry-chauhan-pictureAdvanced Problems In Organic Chemistry For JEE by MS ChauhanThis book has a vast amount of great JEE level problems and tricky questions that really test your Organic Chemistry concepts to the core. A must buy, if you want to gauge your preparation in Organic. However, there is a solution manual available as well for this book but we would suggest not to get the manual before you have put your 110% in solving the problems. More details here.Organic Chemistry by Solomonsorganic-chemistry-solomonsOrganic Chemistry by Solomons and Fryhle is another great book that can be used for JEE Main & Advanced Organic Chemistry preparation. It is more of a choice – you can either choose between this or the Organic Chemistry book by Morrison & Boyd. I used to hear from my friends that the concepts of Stereochemistry has been explained in the simplest yet most effective way possible in this book.The publishers of this book have released a JEE Mains special edition which drops out all the JEE irrelevant matter from the original book. You can find it here.reactions-rearrangements-reagents-sanyalReactions, Rearrangements and Reagents by SanyalThis book is more like a reference textbook and final revision material for Organic Chemistry. It contains all the name reactions, reagents and basic concepts like SN-1 SN-2 +I -I +R -R. It also list the practical use of all the organic chemical reactions so that you do not have to mug up things and can remember reactions by their actual utility.I saw this book with a fellow JEE aspirant on the day of the final exam and regretted for sure that I could have saved so much time had I known about this book before hand. A must have, for sure. You can go through the comments section on Flipkart to see how many students actually swear by this book!***Inorganic Chemistry Books for JEE Mains & AdvancedInorganic Chemistry preparation, in my humble opinion, differs a bit from the two other branches of chemistry. You will have to be more thorough with properties of elements and compounds. Now, for Inorganic Chemistry your NCERT Chemistry text books for Class 11 and Class 12 are really really really indispensable. This was true 9 years back when I took my JEE exam and this is true even till this date!Arihant’s Inorganic ChemistryIn addition to the NCERT Texts, you will need another book that is written with JEE exams in mind just so that you can get a feel of the kind of questions you can expect on your D-Day. The Inorganic Chemistry book from Arihant has a good amount of JEE level problems and solved questions from previous years. You should use this book as the one from O.P. Tandon mentioned below just to solve the problems – try to avoid reading theory from either of them.inorganic-chemistry-op-tandon-picturesInorganic Chemistry by O.P. TandonThis book is an alternative to the Arihant book I mentioned above. Same commentary applies here as well – just use the book for solving the JEE level questions and to go through the questions that have been asked in the previous years’ papers. Avoid reading theory from them. More details about the book can be found here.Concise Inorganic Chemistry by J.D. Leeconcise-inorganic-chemistry-jd-leeLooking for a solid base to your Inorganic Chemistry Preparation? Look no further! Yes, Concise Inorganic Chemistry by JD Lee is the best book for Inorganic Chemistry reference for your prep. The book is extremely relevant to JEE portions and does an awesome job in explaining concepts such as the periodic table properties and chemical bonding.However, some portions in this book might not be your cup of tea so I would strongly recommend to go through the NCERT texts first before you start reading from this book. I used this book on various occasions during my preparation and surely reaped the benefits.***Physics Books for JEE Main and JEE AdvancedPhysics used to be one of my most favourite subjects back in the day when I was preparing for JEE exams. Its only because Physics is one subject which you can correlate with your surroundings very easily. However, this is entirely my view point and you may not agree with me which is okay.In addition to the Physics Textbooks for Classes 11 and 12 by NCERT (always keep in mind that JEE Main and Advanced exams are organised by CBSE), here are some of the Physics books for JEE Main and Advanced exams.Concepts of Physics Volume 1 & Volume 2 by H.C. Vermaconcepts-of-physics-hc-vermaThis is a very fine general Physics textbook written by Prof. H.C. Verma, who also happens to be a professor and an experimental physicist at IIT Kanpur. Here is the link to his website. Available in two volumes, this book is more than enough for building your concepts of Physics. All the basic theory and concepts have been explained by the prof in simple and easy to understand language.After studying the chapters, you should attempt the end-of-chapter problems so that you can verify whether or not you thoroughly understood the concepts in there. Some problems are of very easy level and some are tough – try to complete the tough ones as well. In my opinion, if you study both the books well, you should be all set with your Physics preparation for JEE.IIT JEE Main & Advanced – 14 Year’s Objective Solved Papersjee-physics-previous-papersNot a reference book per se, but this one will help you to get an idea of what kind of questions are actually asked in the JEE Mains and Advanced exams.When you finish studying a chapter from H.C. Verma or any of physics books mentioned below, try to attempt problems of those topics from this book so that you know where you stand in your Physics preparation for the JEE exams. I would recommend you to buy this book but any other book that has a listing of previous years’ JEE questions is good enough as well.fundamentals-of-physics-resnick-hallidayFundamentals of Physics by Halliday, Resnick & WalkerThis is an expensive book and a really thick one but it is a charm to read physics from this book. Used as a general physics textbook in American universities, you can refer to this book just for additional study and there is no compulsion for going through this book. The only reason I studied a few chapters (Gravitation, Kinematics & electrostatics) from this book was the author’s ability to explain everything so well from nature’s point of view.Problems in General Physics by I.E. Irodovproblems-in-general-physics-irodovI.E. Irodov was one of the greatest General Physicists of all time (fun fact – he was a soldier in World War 2). His book, Problems in General Physics contains close to 300 problems in various topics of General Physics and in order to solve these problems you will really need to have strong concepts.However, I did not have the time to solve the entire book (and trust me no one actually does that) but I did try to solve the problems from kinematics, gravitation and fluid mechanics. You do not need to buy this book as it is available online along with its solutions at this website.A Collection of Questions & Problems in Physics by L.A. Senacollection-of-questions-and-problems-in-physics-la-senaIf you are doing self-study and have not joined any coaching class or tutorials for your JEE Prep then I would surely recommend you to use this book.Why? This book not only has more than 400 highly conceptual problems but also has the solutions for each of them and the solutions are accompanied by well-illustrated diagrams. You can read more about this book here.I came to know about this book after starting college at IIT Bombay when a couple of my friends from Chhatisgarh were discussing about this book – they had not taken any coaching classes.Physics Volume 1 & 2 by Tiplerphysics-tipler-picturesThe Physics texts (Vol 1 and Vol 2) authored by P.A. Tipler is a textbook for general physics used by students in many countries. One good thing about this book is the way things are laid out such as real-life examples and excerpts from documentaries – helps in the long run to those students who are preparing own their own.In my opinion if you do not want to buy the expensive books by Resnick and Halliday mentioned above you could buy these books instead (provided you have enough time to go through additional material for Physics, after you are done with studying HC Verma).***Mathematics Books for JEE Main and JEE AdvancedFor time immemorial Mathematics has been one of those subject where JEE paper setters have always tried to put the skills of aspirants at test. Mathematics problems can either be very easy to solve or look too deceiving before you can actually figure out how complicated they are. But, with ample practice you can overcome everything.General Books for JEE Mathematics PrepIn my opinion, Cracking JEE Maths is more about practice than concepts. You just have a couple of hours to prove your mettle on the exam day, so when it comes to Mathematics you must make sure you practice a lot and practice a very wide variety of questions. Here are a few general books that you must follow in addition to the NCERT Class 11, 12 Math textbooks.Mathematics for Class 11 & Class 12 by RD Sharma OR Maths for Class 11 & Class 12 by R.S. Agarwalmathematics-class-11-rd-sharmaAs the heading for this section suggests, you should go for either of the books as both give a general outlook to all the required and important topics in Mathematics for JEE Main and Advanced exams. Both the books have an ample amount of solved examples and detailed explanations for various mathematical concepts. The ideal way to prepare is start with one of these books and then move on to the books mentioned below (as required)Link for Mathematics for Class 11 & Class 12 by RD Sharma – Vol 1, Vol 2Link for Maths for Class 11 & Class 12 by R.S. AgarwalTata McGraw Hill (TMH) Mathematicstmh-mathematics-for-jee-mainTMH for JEE Mathematics is a must if you are targeting either JEE Main or JEE Advanced or both. The level of problems in this book go from easy to very difficult and every problem in the book is different from the last one. This makes sure you are exposed to a variety of types of Math problems so that you are well equipped for the JEE Main and Advanced days.One suggestion from my end would be that you should go slow with the book as some of the very tough problems can at times de-motivate you. When you see yourself running into such a situation, take it easy, take a break and come back to it later. I used this book myself and owe my success in JEE math to this one.PRO Tip – Do not use this book for building concepts, coz you won’t find any. Its a pure drill question bank of the top notch level.***Algebra Books for JEE Main & Advancedalgebra-jee-main-advanced-arihantAlgebra By SK Goyal from ArihantThis book is one of the best JEE Level books out there – has the right mix of theory and problems. If you are preparing with JEE Advanced as your target then this one should be your go-to book and you can avoid the books I mentioned in the general section above, not the TMH one of course. I have heard good things about the book but it was not there during my days of JEE Prep so no personal comments. More details.higher-algebra-hall-knight-bookHall & Knight – Higher AlgebraAlthough I did not refer to this book ever but have always heard good things about it from my friends who were preparing with me. If you are looking for a book to build your Algebra concepts and go through more theory then you should have a copy of Higher Algebra by Hall & Knight. However, it being a general algebra book you can stumble on to many topics that do not fall in the purview of JEE syllabus.***Trigonometry Books for JEE Maintrigonometry-sl-loneyPlane Trigonometry by SL LoneyThis book has been used since many years for preparing JEE Trigonometry. You might not be fond of the presentation of topics in the book but has good concepts. What I did was just used this book to go through various necessary Trigonometry concepts and then used the TMH Math book to solve problems. My suggestion – you do not need to buy the book compulsorily.***3D, Vector & Co-Ordinate Geometry for JEE Main & Advancedcoordinate-geometry-sl-loneyCo-ordinate Geometry by S. L. LoneyCo-ordinate geometry are some of the very easy topics in JEE Main and Advanced exams and students usually score pretty high. I used to read a couple of chapters from this book and then used the TMH book to solve the problems. However,I did not refer it to much because of the fact that the co-ordinate geometry problems can be pretty much solved with your class 10 (CBSE) geometry knowledge and some basic common sense.vectors-3d-geometry-arihantVector and 3D Geometry from ArihantThis is another book from Arihant that is favorite among the JEE prep circle and teachers when it comes to studying Vector and 3D geometry. Like Co-Ordinate geometry, these are some really simple sections as well and you aim should be score full. I did not use this book, instead studied Vector and 3D geometry – both from the NCERT Mathematics Text Books. You can find more details about the book here.***Calculus Books for JEE PrepCalculus is a very important topic in both JEE Main and Advanced exams. Make sure you leave no stone unturned in order to master this branch of mathematics. Calculus would be needed in your first year of engineering and the years to come depending on the branch you choose.Differential Calculus from Arihantintegral-calculus-arihantThis is one of the best JEE Main and Advanced focused books in the market for studying Differential Calculations. It has both good amount of theoretical concepts and practice problems which vary between various levels of difficulty. You could start with the NCERT books and then continue with this. More details about the book here.Integral Calculus from ArihantJust like the Differential Calculus book from Arihant, this one is also equally good and will help you in preparing well for the Integral calculus portions for the JEE exams. With calculus you must always remember that practice is the key to success![*] A note about some more books. Problems in Calculus of One Variable by I.A. Maron (link) and J. Edward’s book on Calculus (link) are some great books that you could refer to when you want additional help. However, the calculus books from Arihant cover all these material and also follow it up with JEE level problems and previous years’ questions.***Probability Books for JEE MathThis is the trickiest of all topics in the entire JEE Syllabus. In my experience you can either solve a JEE level probability question in one shot or get stuck on it. Now, in the exams if you are stuck on a Probability question then better leave it for later. You can score way more by concentrating on questions from other areas.Introduction to Probability & Its Applications by W. Feller is one of the best probability books out there. But it is a really vast book and lot of topics in it are not required for the JEE prep. You can refer to it if you want but trust me, you would be better off studying Probability from the NCERT math text books and then solving related questions from the TMH math book (look above)***I hope with this massive list, you can choose 1 (or may be 2) book from each of the sections and begin your JEE preparation or put it back on track if you are already at it. Always remember, success in JEE is a mixture of hard work, conceptual clarity and of course some luck!Huh! Too many booksNow coming to conclusion, I will recommend you to study1.NCERT first full chapter to study basics because Jee Advance is all about basics at higher level .2. Then , coming to higher level,Follow 1 book for concept clearing and one for practice of each subject:Physics : 1.H.C. Verma for concepts2. Irodov for questions.Chemistry:A) Organic: M.S.Chouhan for both concepts and questions.B) Inorganic: J.D.Lee for questions.And. Balaji publications for questions.C) Physical: P.Bahadur for both.Mathematics: Use above mentioned book for concepts . And S.K. GOYAL FOR QUESTIONSThanks again for question.You can follow me for more any advice and text me.Please Upvote if you like my answer.An upvote gives satisfaction for the efforts done in writing answers.Feel free to comment for any advice.Please let me know in the comments section if you like this list. You can also tell me if you feel I missed out any good book or I need to remove a book from the list above.You can follow me for more questions and knowledge as I answer maths questions too.Ask me any question , I will try to resolve your doubts.

I don’t know how to have a complete answer for this. But to a certain extent, the research I did on Vedic Mathematics impels me to say that they make sense. And here is what I came across in my research.Vedic MathematicsIn many ancient cultures and civilizations, the development of mathematics was necessitated on account of religious practices and observances. These required an accurate calculation of the times of certain festivals and of the times auspicious for the performances of certain sacrifices and rituals. They also required a correct knowledge of the times of rising and the setting of the sun and the moon, and of the occurrences of the solar and lunar eclipses. All these meant a good knowledge of arithmetic, plane and spherical geometry and trigonometry, and possibly also the know-how of the construction of simple astronomical instruments.[1]The impression that science started only in Europe was deeply embedded in the minds of educated people all over the world until recently. The alchemists of Arab countries were occasionally mentioned, but there was very little reference to India and China. Thanks to the work of the Indian National Science Academy and other learned bodies, the development of science in India during both the ancient and medieval periods has recently been studied. It is becoming clearer from these studies that India has consistently been a scientific country right from Vedic to modern times with the usual fluctuations that can been expected of any country. In fact, a research will not find another civilization except that of ancient Greece, which accorded an exalted place to knowledge and science as in India.[2]It has to be universally acknowledged that much of mathematical knowledge in the world originated in India and moved from East to West. The high degree of sophistication in the use of mathematical symbols and developments in arithmetic, algebra, trigonometry, especially the work attributed to Aryabhatta, is indeed remarkable and should be a source of inspiration to all of us in India. The articles which describe Indian contributions to science from the ancient times to the very modern period bring out quite clearly the continuity of scientific thought as part of our cultural heritage. It is however, unfortunate that the period of decline in India coincided with that of ascendancy of Europe. It is perhaps the contrast during this period that made Europeans believe that all modern science was European.[3]An impending danger which I recognized as part of this study is the fact that Vedic Mathematics is now understood and taught in some institutions merely as an alternate method in facilitating easy and quick mental calculations. So, my study is twofold in nature. The first goal is to have an integral understanding about the domain of Vedic mathematics and second is to have a better understanding about the misconceived notion about Vedic Mathematics, which is becoming prevalent as part of some political or religious agenda. This is a very subjective and personal inquiry towards the disciplines dealt with in Vedic Mathematics which had substantial impact in the Mathematical pursuits of probably all the Indian Mathematicians and Schools of Mathematics.History and Development of Vedic MathematicsThe mathematical tradition in India goes back at least to the Vedas. For compositions with a broad scope covering all aspects of live, spiritual as well as secular, the Vedas show a great fascination for large numbers. As the transmission of the knowledge was oral the numbers were not written, but they were expressed as combinations of powers of 10, and it would be reasonable to believe that when the decimal place value system for written numbers came into being it owed a great deal to the way numbers were discussed in the older compositions.[4] (See the second section of this chapter)It is well known that Geometry was pursued in India in the context of construction of vedis[5] for the yaj̀nas of the Vedic period. The Śulvasūtras[6]contain elaborate descriptions of constructions of vedis and also enunciate various geometric principles. These were composed in the first millennium BC, the earliest Baudhāyana Śulvasūtra dating back to about 800 BC. The Śulvasūtra geometry did not go very far in comparison to the Euclidean geometry developed by the Greeks, who appeared on the scene a little later, in the seventh century BC. It was however an important stage of development in India too. The Śulvasūtra geometers were aware, among other things, of what is now called the Pythagorean theorem, over two hundred years before Pythagoras (all the four major Śulvasūtras contain explicit statement of the theorem), addressed (within the framework of their geometry) issues such as finding a circle with the same area as a square and vice versa and worked out a very good approximation to the square root of 2, in the course of their studies.[7]Though it is generally not recognized, the Śulvasūtras geometry was itself evolving. This is seen in particular from the differences in the contents of the four major extant Śulvasūtras. Certain revisions are especially striking. For instance, in the early Śulvasūtras period the ratio of the circumference to the diameter was, like in other ancient cultures, thought to be 3, as seen in the Sūtra of Baudhāyana, but in the Manava Śulvasūtra a new value was proposed, as 3 and one-fifth; interestingly the Sūtra describing it ends with an exultation “not a hair breadth remains”, and though we see that it is still substantially off the mark, it is a gratifying instance of an advance made. In the Manava Śulvasūtra one also finds an improvement over the method described by Baudhāyana for finding the circle with the same are as that of a given square.[8]Vedic Hindus evinced special interest in two particular branches of Mathematics, namely geometry (Śulva) and astronomy (Jyotiṣa). Sacrifice (Yaj̀na) was their prime religious avocation. Each sacrifice has to be performed on an altar of prescribed size and shape. They were very strict regarding this and thought that even a slight irregularity in the form and size of the altar would nullify the object of the whole ritual and might even lead to adverse effect. So, the greatest care was taken to have the right shape and size of the sacrificial altar. Thus the problems of geometry and consequently the science of geometry originated.[9]As it is evident, available sources of Vedic Mathematics are very poor. Almost all works on the subject have perished. At present we find only a very short treatise on Vedic astronomy in three rescensions, namely, in Ārca Jyotiṣa, Yājuṣa Jyotiṣa and Atharva Jyotiṣa. There are six small treatises on Vedic Geometry belonging to the six schools of the Vedas. Thus for an insight into Vedic Mathematics, we have to now depend more on secondary sources such as the literary works.[10]The study of astronomy began and developed chiefly out of the necessity for fixing the proper time for the sacrifice. This origin of the sciences as an aid to religion is not at all unnatural, for it’s generally found that the interest of a people in a particular branch of knowledge, in all climes and times, has been aroused and guided by specific reasons. In the case of the Vedic Hindu that specific reason was religious. In the course of time, however, those sciences outgrew their original purposes and came to be cultivated for their own sake.[11]The Chāndogya Upaniśad (VII.1.2, 4) mentions among other sciences the science of numbers (rāśi). In the Muņdaka Upaniṣad {I.2. 4-5} knowledge is classified as superior (parā) and inferior (aparā). In the second category is included the study of astronomy (jyotiṣa). In the Mahābhārata (XII.201) we come across a reference to the science of stellar motion (nakṣatragati). The term gaņita, meaning the science of calculation, also occurs copiously in Vedic literature. The Vedāńga jyotiṣa gives it the highest place of honour amongst all the sciences which form the Vedāńga. Thus it was said: ‘As are the crests on the heads of peacocks, as are the gems on the hoods of snakes, so is the gaņita at the top of the sciences known as the Vedāńga’. At that remote period gaņita included astronomy, arithmetic, and algebra, but not geometry. Geometry then belonged to a different group of sciences known as kalpa.[12]Vedic Astronomy – JyotiṣaVedāńga Jyotiṣa is one of the six ancillary branches of knowledge called the Ṣad- Vedāńgas, essentially dealing with astronomy. It must be remarked that although the word Jyotiṣa, in the modern common parlance, is used to mean predictive astrology, in the traditional literature, the word always meant the science of astronomy. Of course, mathematics was considered as a part of this subject. Vedāńga Jyotiṣa is the earliest Indian astronomical work available.[13]The purpose of Vedāńga Jyotiṣa was primarily to fix suitable times for performing the different sacrifices. It is found in two rescensions: the Ŗgveda Jyotiṣa and the Yajurveda Jyotiṣa. Though the contents of both the rescensions are the same, they differ in the number of verses contained in them. While the Ŗgvedic version contains only 36 verses, the Yajurvedic contains 44 verses. This difference in the number of verses is perhaps due to the addition of explanatory verses by the adhvaryu priests with whom it was in use.[14]Authorship and DateIn one of the verses of the Vedāńga Jyotiṣa, it is said, “I shall write on the lore of Time, as enunciated by sage Lagadha”. Therefore, the Vedāńga Jyotiṣa is attributed to Lagadha.[15]According to the text, at the time of Lagadha, the winter solstice was at the beginning of the constellation Śrāviṣṭhā (Delphini) and the summer solistice was at the midpoint of Āśleṣā. Since Vārāhamihira (505 – 587AD) stated that in his own time the summer solstice was at the end of the first quarter of Punarvasu and the winter solstice at the end of the first quarter of Uttarāṣāḍhā, There had been a precession of one and three quarters of nakṣatra or 230 20`. Since the rate of precession is about a degree in 72 years, the time interval for a precession of 230 20` is 72 * 231/3 = 1680 years prior to Vārāhamihira’s time i.e. around 1150 BC. According to the famous astronomer Prof. T. S. Kuppanna Sastri, if instead of the segment of nakṣatra, the group itself it smeant, which is about 30 within it, Lagadha’s time would be around 1370 BC. Therefore, the composition of Lagadha’s Vedāńga Jyotiṣa can be assigned to the period of 12th to 14th century BC.[16]The Vedāńga Jyotiṣa belongs to the last part of the Vedic age. The text proper can be considered as the records of the essentials of astronomical knowledge needed for the day to day life of the people of those times. The Vedāńga Jyotiṣa is the culmination of the knowledge developed and accumulated over thousands of years of the Vedic period prior to 1400 BC.[17]Furthermore, there is considerable material on astronomy in the Vedic Samhitas. But everything is shrouded in such mystic expressions and allegorical legends that it has now become extremely difficult to discern their proper significance. Hence it is not strange that modern scholars differ widely in evaluating the astronomical achievements of the early Vedic Hindus. Much progress seems, however, to have been made in the Brāhmana period when astronomy came to be regarded as a separate science called nakṣatra-vidyā (the science of stars). An astronomer was called a nakṣatra-darśa (star observer) or ganaka (calculator).[18]Numerous Discoveries AnticipatedNumerous amounts of mental calculations have been done in the Vedic era. The distance of the heaven from the earth have been calculated differently in various works. All of them are figurative expressions indicating that the extent of the universe is infinite. There is speculation in the Ŗg-Veda (V.85.5, VIII.42.1) about the extent of the earth. It appears from passages therein that the earth was considered to be spherical in shape (I.33.8) and suspended freely in the air. (IV.53.3) The Śatapatha Brāhmaņa describes it clearly as parimaņdala (globe or sphere). There is evidence in the Ŗg-Veda of the knowledge of the axial rotation and annual revolution of the earth. It was known that these motions are caused by the sun.[19] In fact, all the postulations and conclusions had been arrived at couple of millenniums prior to the discovery of the same by the westerners.There are also evident mentions about the zodiacal belt, the inclinations of the ecliptic with the equator and the axis of the earth. We see that the apparent annual course of the sun is divided into two halves. We also see that the ecliptic is divided into twelve parts or sings of the zodiac corresponding to the twelve months of the year, the sun moving through the consecutive signs during the successive months. The sun is called by different names at various parts of the zodiac, and thus has originated the idea of twelve ādityas or suns.[20]The Ŗg-Veda (IX.71.9 etc.) says that the moon shines by the borrowed light of the sun. The phases of the moon and their relations to the sun were fully understood. Five planets seem to have been known. The planets Sukra or Vena (Venus) and Manthin are mentioned by name.[21]Knowledge Through ObservationsIt appears from a passage in the Taittrīya Brāhmaņa (I.5.2.1) that Vedic astronomers ascertained the motion of the sun by observing with the naked eye the nearest visible stars rising and setting with the sun from day to day. This passage is considered very important ‘as it describes the method of making celestial observations in old times’. Observations of several solar eclipses are mentioned in the Ŗg-Veda, a passage of which states that the priests of the Atri family observed a total eclipse of the sun caused by its being covered by Svarbhānu, the darkening demon (V.40.5-9).[22] The Atri priests could calculate the occurrence, duration, beginning, and the end of the eclipse. Their descendants were particularly conversant with the calculation of eclipses. At the time of the Ŗg-Veda, the cause of the solar eclipse was understood as the occultation of the sun by the moon. There are also mentions of lunar eclipses.[23]Concept of Time and SeasonsThe Vedic people had considerable knowledge about the seasons of for sowing, reaping etc. Apart from that, they had acquired knowledge required for their religious activities, like the times and periodicity of the full and new moons, the last disappearance of the moon and its first appearance etc. This type of information was necessary for their monthly rites like the darśapūrņamāsa and seasonal rites cāturmāsya.[24]Vedic Hindus counted the beginning of a season on the sun’s entering a particular asterism. After a long interval of time, it was observed that the same season began with the sun entering a different asterism. Thus they discovered the falling back of the seasons with the position of the sun among the asterisms. There are also clear references to the vernal equinox in the asterism Punarvasu. There is also evidence to show that the vernal equinox was once in the asterism Mŗgaśirā from whence, in course of time, it receded to Kŗttikā. Thus there is clear evidence in the Samhitās and Brāhmaņas of the knowledge of the precession of the equinox. Some scholars maintain that Vedic Hindus also knew of the equation of time.[25] The practical way of measuring time is mentioned as the time taken by a specific quantity of water to flow through the orifice of a specified clepsydra (water-clock), as one nāḍikā i.e., 1/60 part of a day. The day is divided into 124 bhāgas (or parts) so that the ending moments of parvas and tithis can be given in whole units. The day is also divided into 603 units called kalās so that the duration of the lunar nakṣatra is given in whole units as 610 kalās. The nakṣatra is divided into 124 amśas so that the nakṣatraspassed at the ends of the parvas may be expressed in whole amśas. [26]During the Yajurveda period, it was known that the solar year had 365 days and a fraction more. In the Kŗṣna Yajurveda (Taittrīya Samhita VII. 2.6) it is mentioned that the extra 11 days over 12 lunar months, Caitra, Vaiśākha etc. (totaling to 354 days), complete the ŗtus by the performance of the ekādaśarātra or eleven day sacrifice. Again, the Taittrīya Samhita says that 5 days more were required over the sāvana year of 360 days to complete the season adding specifically that 4 days are too short and 6 days too long.[27]In the Yajurveda period, they had recognized the six ŗtus (seasons) in a period of 12 tropical months of the year and named them as follows:SeasonsMonthsi.Vasanta ŗtuMadhu and Mādhavaii.Grīṣma ŗtuŚukra and Śuciiii.Varṣa ŗtuNalcha and Nabhasyaiv.Śarad ŗtuIṣa and Ūrjav.Hemanta ŗtuSaha and Sahasyavi.Śiśira ŗtuTapa and TapasyaThe sacrificial year commenced with Vasanta ŗtu. Thy had alos noted that the shortest day was at the winter solstice when the seasonal year Śiśira began with Uttarāyana and rose to a maximum at the summer solstice.[28]Vedic ArithmeticIndia’s recognized contribution to mathematics was chiefly in the fields of arithmetic and algebra. In fact, Indian arithmetic is what is now used world over. The topics discussed in the Hindu Mathematics of early renaissance included the following:-[29]a.Parikarma (The four fundamental operations)b.Vyavahāra (determination)c.Rajju (meaning rope referring to geometry)d.Trairāśi (the rule of three)e.Yāvat tāvat (simple equations)f.Kalasavarņa (operations with fractions)g.Varga and Vargamūla (square and square root)h.Ghana and ghana-mūla (cube and cube root)i.Prastāra and Vikalpa (Permutations and Combinations)Sources of information on Vedic arithmetic being very meagre, it is difficult to define the topics for discussion and their scope of treatment. One problem that appears to have attracted the attention and interest of Vedic Hindus was to divide 1000 into 3 equal parts. It is unknown how the problem could be solved, for 1000 is not divisible by 3. So, an attempt has been made to explain the whole thing as a metaphorical statement. But a passage in the Śatapatha Brāhmaņa (III.3.1.13) seems clearly to belie all such speculations, saying: ‘When Indra and Viṣṇu divided a thousand into three parts, one remained in excess, and that they caused to be reproduced into three parts. Hence even now if any one attempts to divide a thousand by three, one remains over.’ In any case, it was a mathematical exercise.[30]Vedic Hindus developed the terminology of numeration to a high degree of perfection. The highest terminology that ancient Greeks knew was ‘myriad’ which denoted 104 and which came into use only about the fourth century BC. The Romans had to remain content with a ‘mille’ (103). But centuries before them, the Hindus had numerated up to parārdha (1014) which they could easily express without ambiguity or cumbrousness. The whole system is highly scientific and is very remarkable for its precision.[31]Scales of NumerationFrom the time of the Vedas, the Hindus adopted the decimal scale of numeration. They coined separate names for the notational places corresponding to 1, 10, 102, 103, 104, 105, etc. and any number, however big, used to be expressed in terms of them. But in expressing a number greater than 103 (sahasra) it was more usual to follow a centesimal scale. Thus, 50.103 was a more common form than 5.104. In Taittrīya Upaniṣad (II.8) the centesimal scale has been adopted in describing the different orders of bliss. Brahmānanda, or the bliss of Brahman, has been estimated as 10010 times the measure of one unit of human bliss.[32]In cases of actual measurements, the Hindus often followed other scales. For instance, we have in the Śatapatha Brāhmaņa (XII.3.2.5 et seq.) the minute subdivision of time on the scale of 15. The smallest unit prāṇa is given by 1/155 of a day. In the Vedāńga jyotiṣa (verse 31), a certain number is indicated as eka-dvi-saptika. If it really means ‘two-sevenths and one’ as it seems to do, then it will have to be admitted that there was once a septismal scale.[33]Representation of NumbersThe whole vocabulary of the number-names of the Vedic Hindus consisted mainly of thirty fundamental terms which can be divided into three groups, the ones, tens and the powers of ten. Furthermore, we can understand that Vedic Hindus had a unique and powerful method of their own in representing numbers. Because from the seals and inscriptions of Mohenjo-daro we can see that in the third millennium before the Christian era, numbers were represented in the Indus valley by means of vertical strokes arranged side by side or one group upon another. There were probably other signs for bigger numbers. Those rudimentary and cumbrous devices of rod-numerals were, however, quite useless for the representation of large numbers mentioned in the Vedas. In making calculations with such large numbers, as large as 1012, Vedic Hindus must have found the need for some shorter and more rapid method of representing numbers. This and other considerations give sufficient grounds for concluding that the Vedic Hindus had developed a much better system of numerical symbols.[34]From a reference in the Aṣṭādhyāyī of Pāṇini, we come to know that the letters of the alphabet were used to denote numbers. Another favourite device of Vedic Hindus to indicate a particular number was to employ the names of things permanently connected with that number by tradition or other associations, and sometimes vice versa. Applications of this are found in the earliest Saṁhitās. This practice of recording numbers with the help of letters and words became very popular in later times, especially amongst astronomers and mathematicians.[35]Holiness Attributed to NumbersIt appears that Vedic Hindus used to look upon some numbers as particularly holy. One such number is 3. In the Ṙg-Veda, the gods are grouped in three (I.105.5) and the mystical ‘three dawns’ are mentioned (VIII.41.3, X.67.4). Cases of magic where 3 is employed in a mysterious occult manner occur in the Ṙg-Veda and the Atharva-Veda. Even the number 180 is mentioned in the Ṙg-Veda as three sixties (VIII.96.8) and 210 as three seventies (VIII.19.37). The number regarded as most sacred seems to have been 7. Thus in the Ṙg-Veda, we get ‘seven seas’ (VIII.40.5), ‘seven rays of the sun’ (I.105.9), and ‘seven sages’ (IV.42.8, IX.92.2, etc.); and the number 49 is stated as seven sevens. Instances of combinations of these two numbers also occur. Thus 21 is stated as three sevens in the Ṙg-Veda (I.133.6, 191.12) and the Atharva-Veda (I.1.1), and 1,470 as three seven seventies in the Ṙg-Veda (VIII.46.26).[36]Classification of NumbersNumbers were divided into even (yugma, literally pair) and odd (ayugma, literally non pair), but there is no further subdivisions of numbers. There is an apparent reference to Zero and recognition of the negative number in the Atharvaveda. Zero is called kṣudra (XIX.22.6) meaning trifling’; the negative number is indicated by the epithet anṛca (XIX.23.22), meaning ‘without a hymn’; and the positive number by ṛca (XIX.23.1), meaning ‘a sacred verse’. These designations were replaced in later times by ṛṇa (debt) and dhana (asset).[37] The fractions like half, quarter, one-eighth and one-sixteenth are referred to for the first time, in the history of mathematics, in the Ŗgveda. These fractions are respectively called ardha, pāda, śapha and kalā.[38]Number SeriesVedic Hindus became interested in numbers forming series or progressions. The TaittrīyaSaṁhitā (VII.2.12-17) mentions the following arithmetical series:(i)1, 3, 5,...19,.......99;(ii)2, 4, 6, ...............100;(iii)4, 8, 12, .............100;(iv)5, 10, 15, ............100; and(v)10, 20, 30, ..........100.The arithmetical series are classified into ayugma and yugma. The Vajasaneyi Saṁhitā (XVIII.24.25) has given the following two instances:(i)1, 3, 5, .............31 and(ii)4, 8,12,...............48.The first series occurs also in the Taittrīya Saṁhitā (IV.3.10). The Paňcavimśa Brāhmana (XVIII.3) describes a list of sacrificial gifts forming a geometrical series of some interest and particular nature.12, 24, 48, 96, 192, ..............49152, 98304, 196608, 393216.This series reappears in the Śrauta-sūtras. Some method for the summation of series was also known. Thrice the sum of an arithmetical progression whose first term is 24, the common difference 4 and number of terms 7 is stated correctly in the ŚatapathaBrāhmaņa as 756.[39] . Moreover, from the texts available to us, it is obvious that Vedic Hindus knew how to perform fundamental arithmetical operations even with elementary fractions.[40]Regarding solutions of quadratic equations, we see that problems for solving quadratic equations of the form ax2 = c and ax2 + bx = c are stated and proved. We also find some indeterminate equations of the first degree (wrongly called as Diophantine equations by modern mathematicians). Furthermore, an important feature, peculiar to the Vedic mathematics, was to use geometrical diagrams and methods to solve algebraic problems and identities. Such geometrical techniques are beautifully adopted in the Śulvasūtras to solve algebraic equations.[41]The ŚulvasūtrasThe most ancient mathematical texts of the Vedic lore are the Śulvasūtras which form a part of the Śrauta section of the Kalpa Vedāńga. In the Śulvasūtras are seen very remarkable and rich principles of mathematics, particularly geometry. The word Śulva (or Śulba) is derived from the verb-root, Śulv or Śulb which means to measure. Since for measure length and breadth rope (rajju) was used, the word Śulva, in course of time came to mean a rope. In fact, Geometry (now referred to as Rekhāgaņita) was called Śulva Sāstra or Rajju Sāstra in ancient http://times.It is believed that these Śulvasūtras were composed around eight or nine centuries before Christ. [42]In the Vedic religion, every household man (barring the Sanyasis who would concentrate on meditation for years uninterruptedly) had to do certain acts of worship every day. It would be sinful if he neglected them. For purposes of worship, he would constantly maintain in his house three types of Agnis or fires sheltering them in certain altars of special designs. The required altars had to be constructed with great care so as to conform to certain specific shapes and areas.[43] While the three Agnis were to be used for the daily or routine Pujas or acts of worship, there were more elaborate sacrifices or Pujas for attaining cherished objects or wants. They were called Kamyagnis. The sacrificial altars for these Kamyagnis required more complicated constructions involving combinations of rectangles, triangles and trapeziums. It is clear that these processes require a clear knowledge of the properties of triangles, rectangles and squares, properties of similar figures, and a solution of the problem of ‘squaring the circle’ and its converse, ‘circling the square’ (i.e., to construct a square equal in area to a given circle, and vice versa).[44]Source and OriginOnly seven of the Śulvasūtras are known at present. They are known by the names Baudhāyana, Āpasthamba, Kātyāyana, Manava, Maitrayana, Varaha and Vadhūla after the names of the Rishis or sages who wrote them. The Kātyāyana Sūtra belongs to the section of the Vedas called Sukla Yajurveda while all the rest belong to Krishna Yajurveda. The Bodhayana, Āpasthamba and Kātyāyana Śulvas are of importance from the mathematical point of view.[45]The dates of these Śulvasūtras have been estimated to be between 800 BC and 500 BC. There is no knowledge about the existence of any Śulvas prior to these seven Sūtras. It must be emphasized that the writers of the Śulvasūtras only wrote down and codified the rules for the constructions of the altars, which were in vogue from ancient times. They were not the persons who specified and directed the rules for the constructions of the altars.[46]Simple Theorems in ŚulvasūtrasThe Śulvas explain a large number of simple geometrical constructions- constructions of squares, rectangles, parallelograms and trapeziums. The following geometrical theorems are either explicitly mentioned or clearly implied in the constructions of the altars of the prescribed shapes and sizes.[47]a.The diagonal of a rectangle divides it into two equal parts.b.The diagonal of a rectangle bisect each other and the opposite areas are equal.c.The perpendicular through the vertex of an isosceles triangle on the base divides the triangle into equal halves.d.A rectangle and a parallelogram on the same base and between the same parallels are equal in area.e.The diagonals of a rhombus bisect each other at right angles.f.The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. (The famous theorem known after the name of Pythagoras)g.Properties of similar rectilinear figures.h.If the sum of the squares on two sides of a triangle be equal to the square on the third side, then the triangle is right-angled. (This is the converse of the Pythagoras theorem, nay the Śulva theorem!)How the Mathematical Theorems were derived?These above mentioned theorems of Śulvasūtras cover roughly the first two books and the sixth book of Euclid’s ‘Elements’. How these theorems were actually obtained is a matter for which no definite answer is available. We all know that Euclid’s geometry is based upon certain axioms and postulates as I pointed out in the first chapter[48] and the proofs involve strict logical application of these. The logical methods of Greek Geometry are certainly not discernible in Hindu geometry. No book on Hindu mathematics explains the system of axioms and postulates assumed, and this itself should go some way in refuting the concocted claim that Hindu mathematics is borrowed from the Greeks. At the same time, it may not be correct to conclude that the above theorems were asserted as a matter of experience and measurement. The people who could make out and solve complicated problems of arithmetic, algebra and spherical trigonometry should be credited with some amount of logic in their work. The Śulvas are not formal mathematical treatises. They are only adjuncts to certain religious works. The question has to end with these remarks.[49] And probably from those very remarks does spring up, valid support for the idea, hypothesis and ultimate destiny of my attempt in this research paper.To make the view clearer, it could be stated that what Pythagoras’ theorem states was known, proved and applied in cases by the Vedic Hindus even before Pythagoras was born in the 6th century BC. Of course, the Vedic scholars did not prove the theorem though they stated and used it. But then there is no evidence that Pythagoras proved it either! The well known proof given by Euclid in this regard may be his own. But the theorem itself was known and widely used from very early times as we mentioned some 5 or 6 centuries prior to the birth of Euclid. It is interesting to note that A. Burk even argues that the much travelled Pythagoras borrowed the result from India![50]Geometrical Constructions contained in ŚulvasThe Vedis discussed in the Śulvasūtras are of various forms. Their constructions require a good knowledge of the properties of the square, the rectangle, the rhombus, the trapezium, the triangle, and of course, the circle. The following are among the important geometrical constructions used in the Śulvasūtras.[51](i)To divide a line-segment into any number of equal parts(ii)To divide a circle into any number of equal areas by drawing diameters (Baudhāyana, II. 73-74, Āpasthamba VII. 13-14)(iii)To divide a triangle into a number of equal and similar areas (Baudhāyana, III. 256)(iv)To draw a straight line at right-angles to a given line (Kātyāyana I. 3)(v)To draw a straight line at right-angles to a given line from a given point on it. (Kātyāyana I. 3)(vi)To construct a square on a given side(vii)To construct a rectangular of given length and breadth. (Baudhāyana I. 36-40)(viii)To construct an isosceles trapezium of given altitude, face and base. (Baudhāyana I. 41, Āpasthamba V. 2-5)(ix)To construct a parallelogram having the given sides at a given inclination. (Āpasthamba XIX. 5)(x)To construct a square equal to the sum of two different squares. (Baudhāyana I. 51-52, Āpasthamba II. 4-6, Kātyāyana II. 22)(xi)To construct a square equivalent to two given triangles.(xii)To construct a square equivalent to two given pentagons. (Baudhāyana III. 68, 288, Kātyāyana IV. 8)(xiii)To construct a square equal to a given rectangle in area. (Baudhāyana I. 58, Āpasthamba II. 7, Kātyāyana III. 2)(xiv)To construct a rectangle having a given side and equivalent to a given square.(xv)To construct an isosceles trapezium having a given face and equivalent to a given square or rectangle. (Baudhāyana I. 55)(xvi)To construct a triangle equivalent to a given square. (Baudhāyana I. 56)(xvii)To construct a square equivalent to a given isosceles triangle. (Kātyāyana IV. 5)(xviii)To construct a rhombus equivalent to a given square or rectangle. (Baudhāyana I. 57, Āpasthamba XII. 9)(xix)To construct a square equivalent to a given rhombus. (Kātyāyana IV. 6)Squaring a CircleOne of the greatest problems that had remained unsolved for centuries in the history of mathematics, till recently, was what is popularly known as “squaring the circle”, i.e., to construct – using only ruler and compasses – a square whose area is equal to that of a given circle. IT is really remarkable that this very problem and vice versa, was tackled by the authors of the Śulvasūtras. They gave practical methods for constructing a square whose area is equal to that of a given circle and vice versa. Of course, their constructions involved approximating the value of the well-known constant number Pi to 3.088 (Pi is the ratio of the circumference of any circle to its diameter). The approximation made by the Vedic Ŗṣis is quite justifiable and admirable in the light of crude mathematical tools and methods available to them thousands of years ago. IT is now, however, in modern mathematics established that an exact construction to “Square a circle” or “to circle a square” (in the sense of the areas) is impossible. Incidentally, it must be pointed out that the approximate value of Pi used in the Śulvasūtras is certainly far better than the Biblical value, 3 (see Kings VII. 23 and Chronicles IV. 2) given many centuries later! In fact, it is only as late as in 1761 AD that Lambert proved that Pi is “irrational” and only in 1882 AD that Lindemann established further that Pi is transcendental.[52]Vedic Knowledge of Irrational NumbersThe essentially arithmetical background of the Śulva mathematics must be contrasted with the essentially geometrical background characteristic of Greek mathematics. Simple fractions and operations on them are available in the Śulvas. Moreover, we find the use of Irrational numbers in them. Surds (Irrational numbers) of the form √2, √3 etc. are called Karanis, thus √2 is dwi-karani, √3 is tri-karani, √1/3 is triteeya karani, √1/7 is saptama karani, so on.[53]The shape of Ashwamedhiki Vedika is an isosceles trapezium whose head, foot and altitude are respectively 24√2, 30√2 , 36√2 prakramas. Its area is stated to be 1944 prakramas (square is to be understood).Area = 24√2 * 1/2(30√2 * 36√2) = 1944.This indicates knowledge of the method of finding the area of a trapezium and simple operations on surds.[54]A remarkable approximation of √2 occurs in each of the three Śulvas Bodhayana, Āpasthamba and Kātyāyana.√2 = 1 + 1/3 + 1/ (3.4) – 1/(3.4 34)....This gives √2 = 1.4142156........., whereas the true value is 1.414213............ The approximation is thus correct to five decimal places, and is expressed by means of simple unit fractions. The problem evidently arises in the construction of a square double a given square in area. The Śulvas contain no clue at all as to the manner in which this remarkable approximation was arrived at. Many theories or plausible explanations have been proposed.[55]This point to the fact that the India, thanks to Vedic mathematics, was the first nation to use irrational numbers. It has to be believed that the Vedic Hindus knew its irrationality. The Greeks also used irrational numbers. If AB is a given segment, Pythagoras and others described the methods of constructing the segments of length√2AB, √3AB, √5AB, etc. But no rational approximation to √2, √3 etc. are found in Greek mathematics, nor are there any problems of the arithmetical operations on irrational numbers. This is easily explained, because the requisite knowledge of arithmetic was not available to the Greeks. It will also be borne in mind that according to unprejudiced estimates, the Śulvasūtras are about two or three centuries prior to Pythagoras.[56][1] Iyengar, The History of Ancient Indian Mathematics, 6.[2]Raja, “The Cultural Heritage of India”,1.[3] Raja, “The Cultural Heritage of India”, 2.[4] Dani, “Ancient Indian Mathematics - A conspectus”, 236.[5] The Vedic Culture had the unique feature of performances of five rituals, known as yajnas. These involved well laid out altars, the vedis, and fire platforms, known as citi or agni, elaborately constructed in the form of birds, tortoise, wheel, etc.[6] The SulvaSūtras are compositions, in the form of manuals for construction of vedis and citis, but they also discuss the geometric principles involved.[7] Dani, “Ancient Indian Mathematics - A conspectus”, 238.[8] Dani, “Ancient Indian Mathematics - A conspectus”, 239.[9] Datta, “Vedic Mathematics”, 18.[10] Datta, “Vedic Mathematics”, 19.[11] Datta, “Vedic Mathematics”, 18.[12] Datta, “The Scope and Development of Hindu Ganita”, 480.[13] Rao, Indian Mathematics and Astronomy, 23.[14] Rao, Indian Mathematics and Astronomy, 23.[15] Rao, Indian Mathematics and Astronomy, 23.[16] Rao, Indian Mathematics and Astronomy, 24.[17] Rao, Indian Mathematics and Astronomy, 24.[18] Datta, “Vedic Mathematics”, 19.[19] Ghosh, “Studies on Rig-Vedic Deities”, 11.[20] Tilak, “The Orion or Researches into the Antiquity of the Vedas, 25.[21] Datta, “Vedic Mathematics”, 20.[22] Datta, “Vedic Mathematics”, 21.[23] Tilak, “The Orion or Researches into the Antiquity of the Vedas, 26.[24] Rao, Indian Mathematics and Astronomy, 25.[25] Mookerjee, “Notes on Indian Astronomy”, 137.[26] Rao, Indian Mathematics and Astronomy, 25.[27] Rao, Indian Mathematics and Astronomy, 25.[28] Rao, Indian Mathematics and Astronomy, 26.[29] Rao, Indian Mathematics and Astronomy, 3.[30] Datta, “Vedic Mathematics”, 30.[31] Rao, Indian Mathematics and Astronomy, 3..[32] Datta, “Vedic Mathematics”, 31.[33] Datta, “Vedic Mathematics”, 31.[34] Datta, “Vedic Mathematics”, 33.[35] Datta, “Vedic Mathematics”, 34.[36] Hopkins, “Numerical Formulae in the Veda and their Bearing on Vedic Criticism”, 279.[37] Shukla, “The Deceptive Title of Swamiji’s Book”, 36.[38] Rao, Indian Mathematics and Astronomy, 14.[39] Shukla, “The Deceptive Title of Swamiji’s Book”, 37.[40] Mehta, Positive Sciences in the Vedas, 114.[41] Rao, Indian Mathematics and Astronomy, 12.[42] Rao, Indian Mathematics and Astronomy, 12.[43] Mehta, Positive Sciences in the Vedas, 117.[44] Iyengar, The History of Ancient Indian Mathematics, 6.[45] Iyengar, The History of Ancient Indian Mathematics, 7.[46] Iyengar, The History of Ancient Indian Mathematics, 8.[47] Rao, Indian Mathematics and Astronomy, 14.[48] Chapter I (1.2.1) Geometry, 16-21.[49] Iyengar, The History of Ancient Indian Mathematics, 9.[50] Rao, Indian Mathematics and Astronomy, 15.[51] Rao, Indian Mathematics and Astronomy, 15-17.[52] Rao, Indian Mathematics and Astronomy, 18.[53] Iyengar, The History of Ancient Indian Mathematics, 13.[54] Iyengar, The History of Ancient Indian Mathematics, 13.[55] Iyengar, The History of Ancient Indian Mathematics, 14.[56] Iyengar, The History of Ancient Indian Mathematics, 15.

Thank you for the questionChemistry Textbooks for Class 11 & Class 12 by NCERTncert-chemistry-textbook-pictureThe NCERT Chemistry text books for Class 11 and Class 12 are really great when it comes to getting started with Chemistry at this level and helps you pick up some of the most important concepts of JEE Mains and Advanced Chemistry in a really great way for all the three branches of chemistry – Physical, Organic and Inorganic.I did a mistake of not referring to these books initially and paid a huge price when I scored bad in my first all India test that was conducted by Brilliant Tutorials. After I saw that my other friends, whom I thought to be smarter than me, were referring to these books I knew where to look next. These books are available in 4 total parts and you can find them here.IIT JEE Chemistry by O.P. AgarwalThis is one book that is recommended by almost any JEE chemistry teacher or even shopkeeper when you speak about Chemistry for JEE Prep. But you gotta take my word on this that you should avoid studying Inorganic and Organic chemistry from this book when you are just starting out.The only reason you should have this book with you is as your chapter-wise and topic-wise guide and also for the vast number of problems at the end of each chapters. But, try to avoid this book as much as possible for concept-building.***Physical Chemistry (Numerical Chemistry) Books for JEEphysical-chemistry-op-tandon-picturePhysical Chemistry by O.P. TandonVarious Physical Chemistry concepts have been explained to great depth in this book and it also accompanies tough problems of various levels with their solutions. If you are preparing with taking JEE Advanced in mind then the problems in this book will be of great help to you however if you are looking to appear only for JEE Mains then you can give this book a miss. More details about this book can be found here.numerical-chemistry-p-bahadur-picturePhysical Chemistry by P.BahadurThis book for Numerical Chemistry by P. Bahadur is as good as the one by O.P Tandon, that I mentioned above and can be used for preparing both JEE Mains and JEE Advanced exams. During my days of JEE preparation, I did refer to the chapters of Chemical Kinetics, Ionic Equilibrium & Thermodynamics from this book and solved most of the numerical problems as daily practice.university-chemistry-bruce-mahan-pictureUniversity Chemistry by Bruce H. MahanThis book is a course text-book higher education level Physical Chemistry in many colleges and provides in-depth explanation and coverage of almost all topics related to the subject. If you are in doubt about any specific topic in Physical chemistry then reading through that part of the book is highly recommended. You can find more details about this book here.modern-approach-to-chemical-calculations-mukherjee-pictureModern Approach to Chemical Calculations by R.C. MukherjeeThis book can easily be termed as the H.C. Verma of Chemistry, well not literally! During my JEE preparation days I used to swear by this book owing to the quality of the numerical problems in it and the explanation of the solutions – simple, to the point and lucid. Without solving questions from this book no preparation for JEE level physical chemistry can be termed as complete. More details about this book can be found here.physical-chemistry-peter-atkins-picturePhysical Chemistry by P.W.AtkinsBefore I say anything about Physical Chemistry by P.W.Atkins, please make sure you do not go into solving the problems in this book. They are of really high level and you do not really need to solve such problems to prepare yourself for the JEE Main exams or even the JEE Advanced exams. Just refer to the book for its solid concepts. Reading through the book was a charm during my JEE Prep days.***JEE Main Organic Chemistry Books (and Advanced)Like I mentioned above, you must go through the Organic Chemistry chapters in the NCERT chemistry text books for Class 11 and Class 12 before you touch any of the books in this list. I am saying this out of experience as it would make it easier for you to have a rough idea of what the author is trying to convey.Organic Chemistry is a highly scoring subject provided you have crystal clear concepts. Wrong information and bad teaching practices have created a false assumption among students that Organic Chemistry is tough, which is not at all true. You should also go through these common mistakes in Organic Chemistry that students make during their JEE Prep.organic-chemistry-op-tandon-pictureOrganic Chemistry by O.P. TandonThis text book for Organic Chemistry, written by O.P. Tandon puts JEE Main and Advanced exams forward, which means you will spend less time finding problems to solve for your daily practice. However, I was not a fan of this book when it comes to studying Organic Chemistry although many of my friends do make this one as their primary reference book.Organic Chemistry by Morrison & Boydorganic-chemistry-morisson-boyd-pictureOrganic Chemistry by Morrison & Boyd is definitely one of the best books for Organic Chemistry out there. Now, if you really want to build some of the best and most solid concepts in Organic Chemistry, this book should be on the top of your lists. Now, mind you the book is really thick which means you will have to put in extra hours. But trust me, if you do all these, you will definitely have an edge over your friends in the Organic Chemistry part of the JEE Main / Advanced exams.You should also spend some time to solve the end-of-chapter questions to check your grasp of various concepts. FIITJEE generally gives tough organic questions in their All India Open Tests and I had scored a 99.99 – percentile in the Organic Chemistry module just be studying Organic Chemistry by Morrison Boyd.advanced-problems-in-organic-chemistry-chauhan-pictureAdvanced Problems In Organic Chemistry For JEE by MS ChauhanThis book has a vast amount of great JEE level problems and tricky questions that really test your Organic Chemistry concepts to the core. A must buy, if you want to gauge your preparation in Organic. However, there is a solution manual available as well for this book but we would suggest not to get the manual before you have put your 110% in solving the problems. More details here.Organic Chemistry by Solomonsorganic-chemistry-solomonsOrganic Chemistry by Solomons and Fryhle is another great book that can be used for JEE Main & Advanced Organic Chemistry preparation. It is more of a choice – you can either choose between this or the Organic Chemistry book by Morrison & Boyd. I used to hear from my friends that the concepts of Stereochemistry has been explained in the simplest yet most effective way possible in this book.The publishers of this book have released a JEE Mains special edition which drops out all the JEE irrelevant matter from the original book. You can find it here.reactions-rearrangements-reagents-sanyalReactions, Rearrangements and Reagents by SanyalThis book is more like a reference textbook and final revision material for Organic Chemistry. It contains all the name reactions, reagents and basic concepts like SN-1 SN-2 +I -I +R -R. It also list the practical use of all the organic chemical reactions so that you do not have to mug up things and can remember reactions by their actual utility.I saw this book with a fellow JEE aspirant on the day of the final exam and regretted for sure that I could have saved so much time had I known about this book before hand. A must have, for sure. You can go through the comments section on Flipkart to see how many students actually swear by this book!***Inorganic Chemistry Books for JEE Mains & AdvancedInorganic Chemistry preparation, in my humble opinion, differs a bit from the two other branches of chemistry. You will have to be more thorough with properties of elements and compounds. Now, for Inorganic Chemistry your NCERT Chemistry text books for Class 11 and Class 12 are really really really indispensable. This was true 9 years back when I took my JEE exam and this is true even till this date!Arihant’s Inorganic ChemistryIn addition to the NCERT Texts, you will need another book that is written with JEE exams in mind just so that you can get a feel of the kind of questions you can expect on your D-Day. The Inorganic Chemistry book from Arihant has a good amount of JEE level problems and solved questions from previous years. You should use this book as the one from O.P. Tandon mentioned below just to solve the problems – try to avoid reading theory from either of them.inorganic-chemistry-op-tandon-picturesInorganic Chemistry by O.P. TandonThis book is an alternative to the Arihant book I mentioned above. Same commentary applies here as well – just use the book for solving the JEE level questions and to go through the questions that have been asked in the previous years’ papers. Avoid reading theory from them. More details about the book can be found here.Concise Inorganic Chemistry by J.D. Leeconcise-inorganic-chemistry-jd-leeLooking for a solid base to your Inorganic Chemistry Preparation? Look no further! Yes, Concise Inorganic Chemistry by JD Lee is the best book for Inorganic Chemistry reference for your prep. The book is extremely relevant to JEE portions and does an awesome job in explaining concepts such as the periodic table properties and chemical bonding.However, some portions in this book might not be your cup of tea so I would strongly recommend to go through the NCERT texts first before you start reading from this book. I used this book on various occasions during my preparation and surely reaped the benefits.***Physics Books for JEE Main and JEE AdvancedPhysics used to be one of my most favourite subjects back in the day when I was preparing for JEE exams. Its only because Physics is one subject which you can correlate with your surroundings very easily. However, this is entirely my view point and you may not agree with me which is okay.In addition to the Physics Textbooks for Classes 11 and 12 by NCERT (always keep in mind that JEE Main and Advanced exams are organised by CBSE), here are some of the Physics books for JEE Main and Advanced exams.Concepts of Physics Volume 1 & Volume 2 by H.C. Vermaconcepts-of-physics-hc-vermaThis is a very fine general Physics textbook written by Prof. H.C. Verma, who also happens to be a professor and an experimental physicist at IIT Kanpur. Here is the link to his website. Available in two volumes, this book is more than enough for building your concepts of Physics. All the basic theory and concepts have been explained by the prof in simple and easy to understand language.After studying the chapters, you should attempt the end-of-chapter problems so that you can verify whether or not you thoroughly understood the concepts in there. Some problems are of very easy level and some are tough – try to complete the tough ones as well. In my opinion, if you study both the books well, you should be all set with your Physics preparation for JEE.IIT JEE Main & Advanced – 14 Year’s Objective Solved Papersjee-physics-previous-papersNot a reference book per se, but this one will help you to get an idea of what kind of questions are actually asked in the JEE Mains and Advanced exams.When you finish studying a chapter from H.C. Verma or any of physics books mentioned below, try to attempt problems of those topics from this book so that you know where you stand in your Physics preparation for the JEE exams. I would recommend you to buy this book but any other book that has a listing of previous years’ JEE questions is good enough as well.fundamentals-of-physics-resnick-hallidayFundamentals of Physics by Halliday, Resnick & WalkerThis is an expensive book and a really thick one but it is a charm to read physics from this book. Used as a general physics textbook in American universities, you can refer to this book just for additional study and there is no compulsion for going through this book. The only reason I studied a few chapters (Gravitation, Kinematics & electrostatics) from this book was the author’s ability to explain everything so well from nature’s point of view.Problems in General Physics by I.E. Irodovproblems-in-general-physics-irodovI.E. Irodov was one of the greatest General Physicists of all time (fun fact – he was a soldier in World War 2). His book, Problems in General Physics contains close to 300 problems in various topics of General Physics and in order to solve these problems you will really need to have strong concepts.However, I did not have the time to solve the entire book (and trust me no one actually does that) but I did try to solve the problems from kinematics, gravitation and fluid mechanics. You do not need to buy this book as it is available online along with its solutions at this website.A Collection of Questions & Problems in Physics by L.A. Senacollection-of-questions-and-problems-in-physics-la-senaIf you are doing self-study and have not joined any coaching class or tutorials for your JEE Prep then I would surely recommend you to use this book.Why? This book not only has more than 400 highly conceptual problems but also has the solutions for each of them and the solutions are accompanied by well-illustrated diagrams. You can read more about this book here.I came to know about this book after starting college at IIT Bombay when a couple of my friends from Chhatisgarh were discussing about this book – they had not taken any coaching classes.Physics Volume 1 & 2 by Tiplerphysics-tipler-picturesThe Physics texts (Vol 1 and Vol 2) authored by P.A. Tipler is a textbook for general physics used by students in many countries. One good thing about this book is the way things are laid out such as real-life examples and excerpts from documentaries – helps in the long run to those students who are preparing own their own.In my opinion if you do not want to buy the expensive books by Resnick and Halliday mentioned above you could buy these books instead (provided you have enough time to go through additional material for Physics, after you are done with studying HC Verma).***Mathematics Books for JEE Main and JEE AdvancedFor time immemorial Mathematics has been one of those subject where JEE paper setters have always tried to put the skills of aspirants at test. Mathematics problems can either be very easy to solve or look too deceiving before you can actually figure out how complicated they are. But, with ample practice you can overcome everything.General Books for JEE Mathematics PrepIn my opinion, Cracking JEE Maths is more about practice than concepts. You just have a couple of hours to prove your mettle on the exam day, so when it comes to Mathematics you must make sure you practice a lot and practice a very wide variety of questions. Here are a few general books that you must follow in addition to the NCERT Class 11, 12 Math textbooks.Mathematics for Class 11 & Class 12 by RD Sharma OR Maths for Class 11 & Class 12 by R.S. Agarwalmathematics-class-11-rd-sharmaAs the heading for this section suggests, you should go for either of the books as both give a general outlook to all the required and important topics in Mathematics for JEE Main and Advanced exams. Both the books have an ample amount of solved examples and detailed explanations for various mathematical concepts. The ideal way to prepare is start with one of these books and then move on to the books mentioned below (as required)Link for Mathematics for Class 11 & Class 12 by RD Sharma – Vol 1, Vol 2Link for Maths for Class 11 & Class 12 by R.S. AgarwalTata McGraw Hill (TMH) Mathematicstmh-mathematics-for-jee-mainTMH for JEE Mathematics is a must if you are targeting either JEE Main or JEE Advanced or both. The level of problems in this book go from easy to very difficult and every problem in the book is different from the last one. This makes sure you are exposed to a variety of types of Math problems so that you are well equipped for the JEE Main and Advanced days.One suggestion from my end would be that you should go slow with the book as some of the very tough problems can at times de-motivate you. When you see yourself running into such a situation, take it easy, take a break and come back to it later. I used this book myself and owe my success in JEE math to this one.PRO Tip – Do not use this book for building concepts, coz you won’t find any. Its a pure drill question bank of the top notch level.***Algebra Books for JEE Main & Advancedalgebra-jee-main-advanced-arihantAlgebra By SK Goyal from ArihantThis book is one of the best JEE Level books out there – has the right mix of theory and problems. If you are preparing with JEE Advanced as your target then this one should be your go-to book and you can avoid the books I mentioned in the general section above, not the TMH one of course. I have heard good things about the book but it was not there during my days of JEE Prep so no personal comments. More details.higher-algebra-hall-knight-bookHall & Knight – Higher AlgebraAlthough I did not refer to this book ever but have always heard good things about it from my friends who were preparing with me. If you are looking for a book to build your Algebra concepts and go through more theory then you should have a copy of Higher Algebra by Hall & Knight. However, it being a general algebra book you can stumble on to many topics that do not fall in the purview of JEE syllabus.***Trigonometry Books for JEE Maintrigonometry-sl-loneyPlane Trigonometry by SL LoneyThis book has been used since many years for preparing JEE Trigonometry. You might not be fond of the presentation of topics in the book but has good concepts. What I did was just used this book to go through various necessary Trigonometry concepts and then used the TMH Math book to solve problems. My suggestion – you do not need to buy the book compulsorily.***3D, Vector & Co-Ordinate Geometry for JEE Main & Advancedcoordinate-geometry-sl-loneyCo-ordinate Geometry by S. L. LoneyCo-ordinate geometry are some of the very easy topics in JEE Main and Advanced exams and students usually score pretty high. I used to read a couple of chapters from this book and then used the TMH book to solve the problems. However,I did not refer it to much because of the fact that the co-ordinate geometry problems can be pretty much solved with your class 10 (CBSE) geometry knowledge and some basic common sense.vectors-3d-geometry-arihantVector and 3D Geometry from ArihantThis is another book from Arihant that is favorite among the JEE prep circle and teachers when it comes to studying Vector and 3D geometry. Like Co-Ordinate geometry, these are some really simple sections as well and you aim should be score full. I did not use this book, instead studied Vector and 3D geometry – both from the NCERT Mathematics Text Books. You can find more details about the book here.***Calculus Books for JEE PrepCalculus is a very important topic in both JEE Main and Advanced exams. Make sure you leave no stone unturned in order to master this branch of mathematics. Calculus would be needed in your first year of engineering and the years to come depending on the branch you choose.Differential Calculus from Arihantintegral-calculus-arihantThis is one of the best JEE Main and Advanced focused books in the market for studying Differential Calculations. It has both good amount of theoretical concepts and practice problems which vary between various levels of difficulty. You could start with the NCERT books and then continue with this. More details about the book here.Integral Calculus from ArihantJust like the Differential Calculus book from Arihant, this one is also equally good and will help you in preparing well for the Integral calculus portions for the JEE exams. With calculus you must always remember that practice is the key to success![*] A note about some more books. Problems in Calculus of One Variable by I.A. Maron (link) and J. Edward’s book on Calculus (link) are some great books that you could refer to when you want additional help. However, the calculus books from Arihant cover all these material and also follow it up with JEE level problems and previous years’ questions.***Probability Books for JEE MathThis is the trickiest of all topics in the entire JEE Syllabus. In my experience you can either solve a JEE level probability question in one shot or get stuck on it. Now, in the exams if you are stuck on a Probability question then better leave it for later. You can score way more by concentrating on questions from other areas.Introduction to Probability & Its Applications by W. Feller is one of the best probability books out there. But it is a really vast book and lot of topics in it are not required for the JEE prep. You can refer to it if you want but trust me, you would be better off studying Probability from the NCERT math text books and then solving related questions from the TMH math book (look above)***I hope with this massive list, you can choose 1 (or may be 2) book from each of the sections and begin your JEE preparation or put it back on track if you are already at it. Always remember, success in JEE is a mixture of hard work, conceptual clarity and of course some luck!Huh! Too many booksNow coming to conclusion, I will recommend you to study1.NCERT first full chapter to study basics because Jee Advance is all about basics at higher level .2. Then , coming to higher level,Follow 1 book for concept clearing and one for practice of each subject:Physics : 1.H.C. Verma for concepts2. Irodov for questions.Chemistry:A) Organic: M.S.Chouhan for both concepts and questions.B) Inorganic: J.D.Lee for questions.And. Balaji publications for questions.C) Physical: P.Bahadur for both.Mathematics: Use above mentioned book for concepts . And S.K. GOYAL FOR QUESTIONSThanks again for question.You can follow me for more any advice and text me.Please Upvote if you like my answer.An upvote gives satisfaction for the efforts done in writing answers.Feel free to comment for any advice.Please let me know in the comments section if you like this list. You can also tell me if you feel I missed out any good book or I need to remove a book from the list above.You can follow me for more questions and knowledge as I answer maths questions too.Ask me any question , I will try to resolve your doubts.