UDAYAN S. PATANKAR & SUNIL M. PATANKAR Elements of Vedic Mathematics Tallinn 2018 1 Cover design, Layout and Setup by Parimal Kshama Sunil Patankar Copyright Udayan S. Patankar, 2018 ISBN 978-9949-83-215-6 (publication) ISBN 978-9949-83-216-3 (pdf) 2 3 4 SWAMI BHARATI KRISHANA TIRTH (1884 – 1960) (PIONEER OF VEDIC MATHEMATICS) Swami Bharati Krishana Tirth was the author of the first available book “Vedic Mathematics”. He was Jagadguru Shankaracharya of Govardhan Peeth, Puri (Odhisha). He was highly a learned person both in philosophy and in the field of science. He delivered many lectures and demonstrations on Vedic Maths in America and United Kingdom. He appeared at the M.A. examination of the American College of sciences, Rochester, New York, from Bombay Centre in 1903. He passed M.A. examination in seven subjects simultaneously securing highest honours in all, which is perhaps the all time world record of academic excellence and brilliance. His subjects included Sanskrit, Philosophy, English, Mathematics and Science. During 1919 – 1927 he developed deep into Vedas and other scriptures. During this period, by meditating and by intuition, Swamiji formulated a concept of Vedic Maths. He postulated sixteen sutra of Vedic Mathematics. These sutras are easy to understand, apply and remember as well as the entire work can be summarized in one word as “MENTAL”. Swamiji had written in all sixteen volumes on Vedic Mathematics but at present only one introductory volume is available. Although he intended to rewrite all the previous volumes but his health did not permit it. A spiritual seer of highest order, Swami Bharati Krishana Tirth was also a philosopher, great mathematician and a scientist. His work on Vedic Mathematics is a gem of gift to the world of mathematics and to the entire mankind. 5 PREFACE Om Gurudevay Namha, I Am Thankful To God for Giving Me This Opportunity. I felt Mathematics like a GOD, No one has seen it, but it is there in everything. Being an Electronic Engineer I felt that Mathematics is the base of all sciences but thought to be a hard nut to crack by many students because of the monotonous, lengthy methods of conventional mathematics. We have witnessed that Vedic mathematics is already started gaining its spin in digital electronics. Now a days many researchers are working on how to use Vedic mathematic terms in different calculations, Computer Programs, Digital Designs etc. I had come across many interesting projects Like ALU based on Vedic Mathematics, which has given a very good possibilities of utilizing Vedic mathematics into various digital operations, securities etc. Our great ancestors had a great knowledge about it and developed many easiest n simplest ways of doing it. So I felt necessity to have a single book for Vedic Mathematics which covers all the topics of three subjects- Vedic Arithmetic, Vedic Algebra and Vedic Geometry. While performing various critical operations I always thought to make it interesting. Vedic Maths encourages student’s creativity. The element of choice and flexibility at each stage keeps his mind alert and lively. In this way holistic development of human brain automatically occurs. Vedic Mathematics will help slow learners to know the basic concepts while inducing creativity amongst intelligent. Vedic Mathematics is a uniquely modern and extremely user friendly system. These Vedic Mathematics methods will be helpful in developing not only an insight into mathematics but also helpful to work in different subject. These methods are based on the scientific footings and practically ease the work without sacrificing anything. The credit to create utmost interest in mathematics goes to Swamiji Bharati Krishan Tirth – Shankaracharya of Govardhan Peeth (Puri). His blessings are always with us for learning Vedic Mathematics. During my study I came across various personalities starting from my home my Mother, Father and Big Brother. They always supported me to explore new things. My Mentors from sports to study, Dr. Vilas Nitnaware, Prof. Toomas Rang, Dr. Ants Koel and many more who has taught me many things about how to keep myself motivated and focused. I am thankful for their valuable guidance and support. Let us enjoy! Udayan Sunil Patankar 6 PREFACE I am glad to present this text book “Elements of Vedic Mathematics” for certificate course in Vedic Mathematics affiliated to KAVIKULAGURU KALIDAS SANSKRIT VISHWAVIDYALAYA, RAMTEK (NAGPUR). I felt necessity to have a single book for certificate course in Vedic Mathematics which covers all the topics of three subjects- Vedic Arithmetic, Vedic Algebra and Vedic Geometry. Regarding Vedic Mathematics Bharati Krishana Tirtha Shankaracharya Swamiji has said “It is magic until you understand it and mathematics thereafter”. Mathematics is the base of all sciences but thought to be a hard nut to crack by many students because of unwillingness attitude to change the monotonous, lengthy methods of conventional mathematics. Vedic Maths encourages student’s creativity. The element of choice and flexibility at each stage keeps his mind alert and lively. In this way holistic development of human brain automatically occurs. Mathematics will be popular as an interesting subject to majority students by including easiest and quick methods of Vedic Mathematics in the syllabus from primary level itself. Vedic Mathematics will help slow learners to know the basic concepts while inducing creativity amongst intelligent. Vedic Mathematics is a uniquely modern and extremely user friendly system. These Vedic Mathematics methods will be helpful in developing an insight into mathematics first and other subjects thereafter. These methods will not only provide quicker calculation but also increase the creativity of mind, develop the brain and will provide the eagerness to learn more and more about mathematics. These methods are based on the scientific footings and practically ease the work without sacrificing anything. The credit to create utmost interest in mathematics goes to Swamiji Bharati Krishan Tirth – Shankaracharya of Govardhan Peeth (Puri). His blessings are always with us for learning Vedic Mathematics. During my study towards Vedic Mathematics I came across various personalities in this field- Dr.A.W.Vyawahare (Nagpur), Dr.S.D. Mohogaonkar (Nagpur), Dr.Kailash Vishwakarma (Rath), V.G.Unkalkar (Bengaluru), M.Sitaramarao (Hydrabad), Shriram Chouthaiwale (Akola), Prof. Bachchubhai Rawal (Ahamadabad). I am thankful for their valuable guidance. I am thankful to Mrs.Maya Madhukar Kondalwadikar for performing laborious proof reading task of the notes. I am thankful to Mrs.Maya Madhukar Kondalwadikar and Shri Umesh – K - Bhonsule for their valuable guidance and assistance in preparation of this book. Definitely, Vedic Mathematics methods will be a boon for various competitive examinations to save time, space and energy. Let us enjoy! Sunil Manohar Patankar 7 Special Thanks for Family, Guru’s and Friends For their Strong belief, Guidance and Blessings. This milestone is the beginning of New Journey. 8 DON’T FEAR JUST DO THE DARE GOD IS HERE TO TAKE CARE… - UDAYAN S. PATANKAR 9 10 Contents CHAPTER ONE: ADDITION AND SUBTRACTION .............................................................15 CHAPTER TWO: VINCULUM NUMBER ...........................................................................20 CHAPTER THREE: MULTIPLICATION ...............................................................................27 CHAPETER FOUR: DIVISION ...........................................................................................64 CHAPTER FIVE: HCF OR GCD ..........................................................................................79 CHAPTER SIX: DIVISIBILITY .............................................................................................85 CHAPTER SEVEN: AWARENESS OF 1 – 5 VEDIC SUTRA ................................................107 CHAPTER EIGHT: CONTRIBUTION OF ANCIENT INDIAN MATHEMATICIANS IN ARITHMETIC .................................................................................................................140 CHAPTER NINE: ALGEBRIC MULTIPLICATION ..............................................................155 CHAPTER TEN: ALGEBRAIC DIVISION ...........................................................................158 CHAPTER ELEVEN: ALGEBRIC GCD OR HCF ..................................................................161 CHAPTER TWELVE: FACTORISATION ...........................................................................165 CHAPTER THIRTEEN: PARTIAL FRACTION ....................................................................174 CHAPTER FOURTEEN: DETERMINANT .........................................................................181 CHAPTER FIFTEEN: MATRICES .....................................................................................190 CHAPTER SIXTEEN: SIMULTANEOUS EQUATION WITH 3 VARIABLES ..........................197 CHAPTER SEVENTEEN: MERU – PRASTAR ...................................................................203 CHAPTER EIGHTEEN: AWARENESS OF 6-11 VEDIC SUTRA ..........................................217 CHAPTER NINTEEN: CONTRIBUTION OF ANCIENT INDIAN MATHEMATICIANS IN ALGEBRA ......................................................................................................................231 CHAPTER TWENTY: BODHAYAN GEOMETRY ...............................................................237 CHAPTER TWENTY ONE: COMPLEX NUMBER..............................................................263