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Page No.Part - 1 11. Mathematics in Vedas 32. Additions 63. Subtractions 94. Multiplications-1 125. Multiplications-2 156. Complements 167. Multiplications-3 178. Multiplications-4 239. Multiplications-5 3010. Multiplications-6 3711. Multiplications-7 4012. Multiplications-8 4213. Multiplications-9 47

Part - 2 5914. Vinculam Numbers 6115. Cube Values-1 7416. Cube Values-2 7817. Divisions-1 8418. Symmetry in the Quotient 8919. Divisions-2 9120. Divisions-3 9321. Divisions-4 9722. Multiplications-10 102

Part - 3 10723. Decimal number System in Vedas 10924. Measures of numbers 11725. Multiplications-11 12126. Multiplications-12 12727. Multiplications-13 13228. Multiplications-14 13429. Multiplications-15 13630. Multiplications-16 13831. Multiplications-17 14032. Multiplications -18 14433. Multiplications-19 147

Index

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34. Multiplications-20 14935. Multiplications-21 15236. Akshahridayam 15337. Positional weightage system 15438. Infinity 15639. ‘AC’ and ‘HAL’ SYMBOLS 157

Part - 4 15940. Methodology for Expressing Numbers in Sanskrit 16141. Katapayadi System - 1 16242. Katapayadi System - 2 16443. Katapayadi System in Vedantasastra 16944. Katapayadi System in Sangita Sastra 17145. Magic Squares with Katapayadi Numbers 17246. Magic Squares with katapayadi Numbers 17547. Katapayadi System - 3 17848. Number of Planetary Revolutions with Katapayadi

System 18149. Logic Behind the Naming of Weekdaya in Sanskrit 18450. Vedic Numerical Codes / Algebraic Notation 18951. Matrhematics in Chandas Sastra (Prosody) 210

Part - 5 21352. Long term calender -1 21553. Long term calender -2 22054. Long term calender -3 23155. Multiplications - Lilavati Ganitam 23556. Method for validation of product of multiplication 24457. About Bhaskaracharya 24658. Divisions - 5 24859. Prime Numbers 25460. Reference to numbers like 19,29,39 and 49 in vedas 25561. Digits & Numbers -their Characteristics -1 25662. Mystic codes in our Philosophy 26163. Digits & Numbers -their Characteristics -2 26264. Divisibility 269

Part - 6 27365. Divisions - 5 27566. Divisions -6 27867. Divisions -7 28568. Divisions -8 286

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69. Ancient Mathematicians of India 28770. Some Contributions of Indian 289

Mathematicians71. Squares - 1 29272. Squares -2 29373. Squares -3 30174. Squares -4 30575. Squares -5 30976. Square Roots - 1 31277. Square Roots - 2 31478. Square Roots - 3 32579. Square Roots - 4 33180. Square Roots - 5 339

Part - 7 34781. Cubes - 3 34882. Cubes - 4 35783. Cubes - 5 36084. Cube Roots - 1 36385. Cube Roots - 2 36486. Cube Roots - 3 38387. Cube Roots - 4 38888. Cube Roots - 5 39589. Fourth Order Roots 40190. Fifth Order Roots - 1 40291. Fifth Order Roots - 2 403

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1. Mathematics in Vedas

Introduction

Veda means “Knowledge”. It is the belief of Hindus that anykind of knowledge, either mundane or divine, can be tracedin the Vedas. It is also opined that the Vedas are eternal anddeal with the Reality. The Vedic Mantras, or the divinestatements, were discovered by the great Rishis during theSamadhi state of their penance.

Several Rishis were involved in the discovery of millions ofMantras, which were compiled by the Sage Veda Vyasa andwere segregated into four Vedas, viz., Rigveda, Yajurveda,Samaveda and Atharvaveda. The same were furthersubdivided into 1,131 branches. This information is found inSrimad Bhagavatham of Veda Vyasa, MahaBhashyam ofPatanjali and the commentary of Sri Vishnu Sahasra NamaStotra of Jagadguru Sankaracharya.

However, as on date, only 13 Vedic branches could belocated, after a search of past 150 years.Among these 13branches, only 7 branches are being studied. The rest of the6 branches are not having the teachers and students. Itmeans that more than 99% of Vedic literature hasdisappeared. Even this remaining part of Vedic literature istoo extensive to understand, as they deal with, in addition tothe main topics of Yajnas and Moksha, the physical sciences,biological sciences, Jyotisha, and Aeoronautics. They alsodeal with the subjects of Medicine, Structural Engineeringetc.

To understand the Vedas, it is a prerequisite to acquire theknowledge of Vedangas. Then only the meanings ofkeywords and other terms of Vedas can be followed. In thiscontext, it may be noted that the Vedas and Vedangas have

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given prominent place for mathematics. Subsequentlyseveral books were written in ancient times about themathematics in Vedas. Some of the authors of that categoryare Boudhayana, Garga, Medhatithi, Parasara, Kasyapa,Maya and Brihaspati. The great mathematicians ofsubsequent times like AryaBhatta, Varahamihira andBhaskaracharya belong to this category only.

From the writings of historians, it is learnt that this subjectwas taken to Arab countries from this country only. It seemsthat a Pandit of Ujjain, by name Kanka, was invited to Bagdadby Kharif Al Mansur in 770 A.D. There he he taught thesubjects of mathematics and Jyotisha to the Arab scholars.Only with his support, the book titled BrahmaSphutaSiddhanta of Brahmagupta could be translated into Arabiclanguage. M.Nev, the French Scientist stated in his book“On new light on our numerals”, that the Hindu numeralswere in use in Syria in 7th century A.D. itself.

Some important topics that are covered in the present seriesof books are as follows:

• Decimal number system

• Names of numbers of high order, as found in Vedas,Ramayana, Lilavati Ganitham etc.

• Even numbers and Odd numbers

• Arithmetical operations

- Additions

- Subtractions

- Multiplications

- Divisions

- Powers of Numbers

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• Positional notation

• Vulgar fractions

• Astronomical calculations

• Statistics

• Geometry

• Set theory

It may please be noted that the main objective of the presentseries is to create awareness among our children about ourrich heritage and the contributions of India to Mathematicsin ancient times. It is also to create interest among the childrenon the subject of mathematics and remove fear complex, tothe extent possible.

References :

The world famous book titled Vedic Mathematics of H.H. SriBharathi Krishna Tirtha Swamiji is the main source for thesebooklets.However,several other old books like AryaBhatiyam,Pavuluri Ganitham, Lilavathi Ganitham and ancient scripturesare referred extensively as and where needed. Simultan-eously the books of contemporary mathematicianslike Sakuntala Devi are also considered for the benefit of thelearners.

Note :

During the process of explanation, certain fundamental topics,which are generally known, are also covered for maintainingthe sequence. If it appears as trivial, a reader may skip thatportion and proceed to the next section unhesitatingly.

Acknowledgements :

We are grateful to Dr. K. Ramesh, Scientific Officer, DAE,Nagarjuna Sagar, for his scrutiny of this book.