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Discover Vedic Mathematics

Description

Since the reconstruction of this ancient system interest in Vedic mathematics has been growing rapidly. Its simplicity and coherence are found to be astonishing and we begin to wonder why we bother with out modern methods when such easy and enjoyable methods are available

This book gives a comprehensive introduction to the sixteen formulate on which the system is based showing their application in many areas of elementary maths so that a real feel for the formulae is acquired.

Using simple patterns based on natural mental faculties problems normally requiring many steps of working are shown to be easily solved in one often forwards or backwards.

Vedic Mathematics solutions of examination question are also given and in this edition comparisons with the conventional methods are shown. An account of the significance of the Vedic formulate (Sutras) is also included.

Mathematics is universally regarded as the science of all sciences and "the priestess of definiteness and clarity". If, Herbert acknowledges that "everything that the greatest minds of all times have accomplished towards the comprehension of forms by means of concepts is gathered into one great science, Mathematics". In India’s intellectual history and no less in the intellectual history of other civilizations, Mathematics stands forth as that which unites and mediates between Man and Nature, inner and outer world, thought and perception.

Indian Mathematics belongs not only to an hoary antiquity but is a living discipline with a potential for manifold modern applications. It takes its inspiration from the pioneering, though unfinished work of the late Bharati Krishna Tirt aji, a former Shankaracharya of Puri of revered memory who reconstructed a unique system on the basis of ancient Indian tradition of mathematics. British teachers have prepared textbooks of Vedic Mathematics for British Schools. Vedic mathematics is thus a bridge across centuries, civilisations, linguistic barriers and national frontiers.

Vedic mathematics is not only a sophisticated pedagogic and research tool but also an introduction to an ancient civilization. It takes us back to many millennia of India’s mathematical heritage. Rooted in the ancient Vedic sources which heralded the dawn of human history and illumined by their erudite exegesis, India’s intellectual, scientific and aesthetic vitality blossomed and triumphed not only in philosophy, physics, astronomy, ecology and performing arts but also in geometry, algebra and arithmetic. Indian mathematicians gave the world the numerals now in universal use. The crowning glory of Indian mathematics was the invention of zero and the introduction of decimal notation without which mathematics as a scientific discipline could not have made much headway. It is noteworthy that the ancient Greeks and Romans did not have the decimal notation and, therefore, did not make much progress in the numerical sciences. The Arabs first learnt the decimal notation from Indians and introduced it into Europe. The renowned Arabic scholar, Alberuni or Abu Raihan, who was born in 973 A.D. and traveled to India, testified that the Indian attainments in mathematics were unrivalled and unsurpassed. In keeping with that ingrained tradition of mathematics in India, S. Ramanujan, "the man who knew infinity", the genius who was one of the greatest mathematicians of our time and the mystic for whom "a mathematical equation had a meaning because it expressed a thought of God", blazed new mathematical trails in Cambridge University in the second decade of the twentieth century even though he did not himself possess a university degree.

I do not wish to claim for Vedic Mathematics as we know it today the status of a discipline which has perfect answers to every problem. I do however question those who mindlessly decide the very idea and nomenclature of Vedic mathematics and regard it as an anathema. They are obviously affiliated to ideological prejudice and their ignorance is matched only by their arrogance. Their mindsets were bequeathed to them by Macaulay who knew next to nothing of India’s scientific and cultural heritage. They suffer from an incurable lack of self-esteem coupled with an irrational and obscurantist unwillingness to celebrate the glory of Indian achievements in the disciplines of mathematics, astronomy, architecture, town planning, physics, philosophy, metaphysics, metallurgy, botany and medicine. They are as conceited and dogmatic in rejecting Vedic Mathematics as those, who naively attribute every single invention and discovery in human history to our ancestors of antiquity. Let us reinstate reasons as well as intuition and let us give a fair chance to the valuable insights of the past. Let us use that precious knowledge as a building block. To the detractors of Vedic Mathematics I would like to make a plea for sanity, objectivity and balance. They do not have to abuse or disown the past in order to praise the present.

This book consists of a series of examples, with explanations, illustrating the scope and versatility of the Vedic mathematical formulae, as applied in various areas of elementary mathematics. Solutions to ‘O’ and ‘A’ level examination questions by Vedic methods are also given at the end of the book.

The system of Vedic Mathematics was rediscovered from Vedic texts earlier this century by Sri Bharati Krsna Tirthaji (184l—196O). Bharati Krsna studied the ancient Indian texts between 191 1 and 19 18 and reconstructed a mathematical system based on sixteen Sutras (formulas) and some sub-sutras. He subsequently wrote sixteen volumes, one on each Sutra, but unfortunately these were all lost. Bharati Krsna intended to rewrite the books, but has left us only one introductory volume written in 1957.This is the book "Vedic Mathematics" published in 1965 by Banaras Hindu University and by Motilal Banarsidass.

The Vedic system presents a new approach to mathematics, offering simple, direct, one-line, mental solutions to mathematical problems. The Sutras on which it is based are given in word form, which renders them applicable in a wide variety of situations. They are easy to remember, easy to understand and a delight to use.

The contrast between the Vedic system and conventional mathematics is striking. Modem methods have just one way of doing, say, division and this is so cumbrous and tedious that the students are now encouraged to use a calculating device. This sort of constraint is just one of the factors responsible for the low esteem in which mathematics is held by many people nowadays.

The Vedic system, on the other hand, does not have just one way of solving a particular problem, there are often many methods to choose from. This element of choice in the Vedic system, and even of innovation, together with the mental approach, brings a new dimension to the study and practice of mathematics. The variety and simplicity of the methods brings fun and amusement, the mental practice leads to amore agile, alert and intelligent mind, and innovation naturally follows.

It may seem strange to some people that mathematics could be based on sixteen word •formulae; but mathematics, more patently than other systems of thought, is constructed by internal laws, natural principles inherent in our consciousness and by whose action more complex edifices are constructed. From the very beginning of life there must be some structure in consciousness enabling the young child to organize its perception learn and evolve. It these principles (see appendix) could be formulated and used they would give us the easiest and most efficient system possible for all our mental enquiries. This system of Vedic Mathematics given to us by Sri Bharati Krsna Tirthaji points towards a new basis for mathematics and a unifying principle by which we can simultaneously extend our understanding of the world and of our self.

This book was first published in 1984 one hundred years since the birth of bharati Krsna. In this edition many new variations have been added as well as many comparisons with the conventional methods so that readers can clearly see the contrast between the two systems. An appendix has been added that describes each of the sixteen sutras as a principle or natural law. In this edition also is a proof of a class of equations coming under the Samuccaya Sutra by Thomas Dahl of Kristianstad University Sweden (see Chapter 10)

Foreword | vii | |

Preface | ix | |

Illustrative Examples | xv | |

1 | All From Nine and the Last From Ten | 1 |

Subtraction | 1 | |

Multiplication | 2 | |

One Number above and one number below the base | 4 | |

Multiplying Numbers Near Different Bases | 4 | |

Using other bases | 5 | |

Multiplications of three or more numbers | 7 | |

First corollary squaring and cubing of numbers near a base . | 9 | |

Second Corollary Squaring of numbers beginnings or ending in 5 etc | 10 | |

Third Corollary Multiplication by nines | 12 | |

Division | 12 | |

The Vinculum | 17 | |

Simple applications of the Vinculum | 18 | |

Exercise on Chapter 1 | 20 | |

2 | Vertically and crosswise | 25 |

Multiplication | 25 | |

Number of Zeros after the Decimal Point | 28 | |

Using the Vinculum | 28 | |

Multiplying from left to right | 29 | |

Using the Vinculum | 30 | |

Algebraic Products | 31 | |

Using Pairs of Digits | 31 | |

The Position of the Multiplier | 31 | |

Multiplying a Long Number by a short Number the moving Multiplier Method | 32 | |

Base five Product | 33 | |

Straight Division | 33 | |

Two or More Figures on the Flag | 36 | |

Argumental division | 38 | |

Numerical Application | 39 | |

Squaring | 40 | |

Square Roots | 42 | |

Working two digits at a time | 44 | |

Algebraic Square Roots | 44 | |

Fractions | 45 | |

Algebraci Fractions | 47 | |

Left to Right Calculations | 48 | |

Pythagoras Theorem | 48 | |

Equation of a line | 49 | |

Exercise on Chapter 2 | 50 | |

3 | Proportionately | 57 |

Multiplication and division | 57 | |

Cubing | 58 | |

Factorising quadratics | 58 | |

Ratios in Triangles | 60 | |

Transformation of Equations | 61 | |

Number Bases | 62 | |

Miscellaneous | 63 | |

Exercises on Chapter 3 | 64 | |

4 | By Addition and by Subtraction | 57 |

Simultaneous Equations | 67 | |

Divisibility | 68 | |

Miscellaneous | 69 | |

Exercises on Chapter 4 | 70 | |

5 | By Alternate elimination and retention | 71 |

Highest Common Factor | 71 | |

Algebraic H.C.F | 72 | |

Factorizing | 73 | |

Exercises on Chapter 5 | 74 | |

6 | By Mere Observation | 75 |

Multiplication | 75 | |

Additional and subtraction from left to right | 76 | |

Miscellaneous | 77 | |

Exercises on Chapter 6 | 78 | |

7 | Using the average | 79 |

Exercises on Chapter 7 | 82 | |

8 | Transpose and Apply | 83 |

Division | 83 | |

Algebraic division | 83 | |

Numerical division | 86 | |

The Remainder Theorem | 89 | |

Solution of Equations | 90 | |

Linear Equations in which ‘x’ Appears more than once | 91 | |

Literal Equations | 93 | |

Mergers | 93 | |

Transformation of Equations | 94 | |

Differentiation and integrations | 95 | |

Simultaneous Equations | 95 | |

Partial fractions | 96 | |

Odd and Even Functions | 99 | |

Exercises on Chapter 8 | 99 | |

9 | One Ratio: The Other One Zero | 102 |

Exercise on Chapter 9 | 103 | |

10 | When the Samuccaya is the Same it is Zero | 104 |

Samuccaya as a Common Factor | 104 | |

Samuccaya as the Product of the Independent terms | 104 | |

Samuccaya as the sum of the denominators of two fractions having the same | 105 | |

Numerical Numerator | 105 | |

Samuccaya as a Combination or Total | 105 | |

Cubic Equations | 108 | |

Quartic Equations | 108 | |

The Ultimate and twice the Penultimate | 109 | |

Exercise on Chapter 10 | 109 | |

11 | The First by the first and the last by the last | 111 |

Factorizing | 112 | |

12 | By the Completion or Non Completion | 114 |

Exercise on Chapter 12 | 116 | |

13 | By One more than the one Before | 118 |

Recurring Decimals | 118 | |

Auxiliary Fractions A.F. | 121 | |

Denominators not ending in 1,3,7,9 | 124 | |

Groups of Digits | 126 | |

Remainder Patterns | 127 | |

Remainders by the Last Digit . | 128 | |

Divisibility . | 129 | |

Osculating From left to right . | 131 | |

Finding the Remainder | 132 | |

Writing a Number divisible by a given number | 132 | |

Divisor not ending in 9 | 132 | |

The Negative Osculator Q | 133 | |

P+Q = D | 134 | |

Divisor not ending in 1,3,7,9. | 134 | |

Groups of Digits | 135 | |

Exercises on Chapter 13. | 136 | |

14 | The Product of the Sum is the sum of the products | 138 |

15 | Only the Last terms | 142 |

Summation of Series | 143 | |

Limits. | 144 | |

Coordinate Geometry | 148 | |

16 | Calculus | 149 |

Integration | 153 | |

Differential Equations | 154 | |

Binomial and Maclaurin Theoremss | 155 | |

‘O’ and ‘A’ Level Examination Papers | 157 | |

‘O’ Level Multiple Choice Paper 1 | 158 | |

‘O’ Level Multiple Choice Paper 2 | 164 | |

‘A’ Level Multiple Choice Paper 1 | 168 | |

‘A’ Level Multiple Choice Paper 2 | 172 | |

Answers to Exercise | 177 | |

List of Vedic Sutra | 187 | |

List of Vedic Sub Sutras | 188 | |

Index of the Vedic Sutras | 189 | |

References | 191 | |

Appendix | 193 | |

Index | 197 |

Item Code:

NAB932

Cover:

Hardcover

Edition:

2006

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ISBN:

8120830962

Size:

9.5 inch X 6.5 inch

Pages:

216

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Weight of the Book: 550 gms

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Since the reconstruction of this ancient system interest in Vedic mathematics has been growing rapidly. Its simplicity and coherence are found to be astonishing and we begin to wonder why we bother with out modern methods when such easy and enjoyable methods are available

This book gives a comprehensive introduction to the sixteen formulate on which the system is based showing their application in many areas of elementary maths so that a real feel for the formulae is acquired.

Using simple patterns based on natural mental faculties problems normally requiring many steps of working are shown to be easily solved in one often forwards or backwards.

Vedic Mathematics solutions of examination question are also given and in this edition comparisons with the conventional methods are shown. An account of the significance of the Vedic formulate (Sutras) is also included.

Mathematics is universally regarded as the science of all sciences and "the priestess of definiteness and clarity". If, Herbert acknowledges that "everything that the greatest minds of all times have accomplished towards the comprehension of forms by means of concepts is gathered into one great science, Mathematics". In India’s intellectual history and no less in the intellectual history of other civilizations, Mathematics stands forth as that which unites and mediates between Man and Nature, inner and outer world, thought and perception.

Indian Mathematics belongs not only to an hoary antiquity but is a living discipline with a potential for manifold modern applications. It takes its inspiration from the pioneering, though unfinished work of the late Bharati Krishna Tirt aji, a former Shankaracharya of Puri of revered memory who reconstructed a unique system on the basis of ancient Indian tradition of mathematics. British teachers have prepared textbooks of Vedic Mathematics for British Schools. Vedic mathematics is thus a bridge across centuries, civilisations, linguistic barriers and national frontiers.

Vedic mathematics is not only a sophisticated pedagogic and research tool but also an introduction to an ancient civilization. It takes us back to many millennia of India’s mathematical heritage. Rooted in the ancient Vedic sources which heralded the dawn of human history and illumined by their erudite exegesis, India’s intellectual, scientific and aesthetic vitality blossomed and triumphed not only in philosophy, physics, astronomy, ecology and performing arts but also in geometry, algebra and arithmetic. Indian mathematicians gave the world the numerals now in universal use. The crowning glory of Indian mathematics was the invention of zero and the introduction of decimal notation without which mathematics as a scientific discipline could not have made much headway. It is noteworthy that the ancient Greeks and Romans did not have the decimal notation and, therefore, did not make much progress in the numerical sciences. The Arabs first learnt the decimal notation from Indians and introduced it into Europe. The renowned Arabic scholar, Alberuni or Abu Raihan, who was born in 973 A.D. and traveled to India, testified that the Indian attainments in mathematics were unrivalled and unsurpassed. In keeping with that ingrained tradition of mathematics in India, S. Ramanujan, "the man who knew infinity", the genius who was one of the greatest mathematicians of our time and the mystic for whom "a mathematical equation had a meaning because it expressed a thought of God", blazed new mathematical trails in Cambridge University in the second decade of the twentieth century even though he did not himself possess a university degree.

I do not wish to claim for Vedic Mathematics as we know it today the status of a discipline which has perfect answers to every problem. I do however question those who mindlessly decide the very idea and nomenclature of Vedic mathematics and regard it as an anathema. They are obviously affiliated to ideological prejudice and their ignorance is matched only by their arrogance. Their mindsets were bequeathed to them by Macaulay who knew next to nothing of India’s scientific and cultural heritage. They suffer from an incurable lack of self-esteem coupled with an irrational and obscurantist unwillingness to celebrate the glory of Indian achievements in the disciplines of mathematics, astronomy, architecture, town planning, physics, philosophy, metaphysics, metallurgy, botany and medicine. They are as conceited and dogmatic in rejecting Vedic Mathematics as those, who naively attribute every single invention and discovery in human history to our ancestors of antiquity. Let us reinstate reasons as well as intuition and let us give a fair chance to the valuable insights of the past. Let us use that precious knowledge as a building block. To the detractors of Vedic Mathematics I would like to make a plea for sanity, objectivity and balance. They do not have to abuse or disown the past in order to praise the present.

This book consists of a series of examples, with explanations, illustrating the scope and versatility of the Vedic mathematical formulae, as applied in various areas of elementary mathematics. Solutions to ‘O’ and ‘A’ level examination questions by Vedic methods are also given at the end of the book.

The system of Vedic Mathematics was rediscovered from Vedic texts earlier this century by Sri Bharati Krsna Tirthaji (184l—196O). Bharati Krsna studied the ancient Indian texts between 191 1 and 19 18 and reconstructed a mathematical system based on sixteen Sutras (formulas) and some sub-sutras. He subsequently wrote sixteen volumes, one on each Sutra, but unfortunately these were all lost. Bharati Krsna intended to rewrite the books, but has left us only one introductory volume written in 1957.This is the book "Vedic Mathematics" published in 1965 by Banaras Hindu University and by Motilal Banarsidass.

The Vedic system presents a new approach to mathematics, offering simple, direct, one-line, mental solutions to mathematical problems. The Sutras on which it is based are given in word form, which renders them applicable in a wide variety of situations. They are easy to remember, easy to understand and a delight to use.

The contrast between the Vedic system and conventional mathematics is striking. Modem methods have just one way of doing, say, division and this is so cumbrous and tedious that the students are now encouraged to use a calculating device. This sort of constraint is just one of the factors responsible for the low esteem in which mathematics is held by many people nowadays.

The Vedic system, on the other hand, does not have just one way of solving a particular problem, there are often many methods to choose from. This element of choice in the Vedic system, and even of innovation, together with the mental approach, brings a new dimension to the study and practice of mathematics. The variety and simplicity of the methods brings fun and amusement, the mental practice leads to amore agile, alert and intelligent mind, and innovation naturally follows.

It may seem strange to some people that mathematics could be based on sixteen word •formulae; but mathematics, more patently than other systems of thought, is constructed by internal laws, natural principles inherent in our consciousness and by whose action more complex edifices are constructed. From the very beginning of life there must be some structure in consciousness enabling the young child to organize its perception learn and evolve. It these principles (see appendix) could be formulated and used they would give us the easiest and most efficient system possible for all our mental enquiries. This system of Vedic Mathematics given to us by Sri Bharati Krsna Tirthaji points towards a new basis for mathematics and a unifying principle by which we can simultaneously extend our understanding of the world and of our self.

This book was first published in 1984 one hundred years since the birth of bharati Krsna. In this edition many new variations have been added as well as many comparisons with the conventional methods so that readers can clearly see the contrast between the two systems. An appendix has been added that describes each of the sixteen sutras as a principle or natural law. In this edition also is a proof of a class of equations coming under the Samuccaya Sutra by Thomas Dahl of Kristianstad University Sweden (see Chapter 10)

Foreword | vii | |

Preface | ix | |

Illustrative Examples | xv | |

1 | All From Nine and the Last From Ten | 1 |

Subtraction | 1 | |

Multiplication | 2 | |

One Number above and one number below the base | 4 | |

Multiplying Numbers Near Different Bases | 4 | |

Using other bases | 5 | |

Multiplications of three or more numbers | 7 | |

First corollary squaring and cubing of numbers near a base . | 9 | |

Second Corollary Squaring of numbers beginnings or ending in 5 etc | 10 | |

Third Corollary Multiplication by nines | 12 | |

Division | 12 | |

The Vinculum | 17 | |

Simple applications of the Vinculum | 18 | |

Exercise on Chapter 1 | 20 | |

2 | Vertically and crosswise | 25 |

Multiplication | 25 | |

Number of Zeros after the Decimal Point | 28 | |

Using the Vinculum | 28 | |

Multiplying from left to right | 29 | |

Using the Vinculum | 30 | |

Algebraic Products | 31 | |

Using Pairs of Digits | 31 | |

The Position of the Multiplier | 31 | |

Multiplying a Long Number by a short Number the moving Multiplier Method | 32 | |

Base five Product | 33 | |

Straight Division | 33 | |

Two or More Figures on the Flag | 36 | |

Argumental division | 38 | |

Numerical Application | 39 | |

Squaring | 40 | |

Square Roots | 42 | |

Working two digits at a time | 44 | |

Algebraic Square Roots | 44 | |

Fractions | 45 | |

Algebraci Fractions | 47 | |

Left to Right Calculations | 48 | |

Pythagoras Theorem | 48 | |

Equation of a line | 49 | |

Exercise on Chapter 2 | 50 | |

3 | Proportionately | 57 |

Multiplication and division | 57 | |

Cubing | 58 | |

Factorising quadratics | 58 | |

Ratios in Triangles | 60 | |

Transformation of Equations | 61 | |

Number Bases | 62 | |

Miscellaneous | 63 | |

Exercises on Chapter 3 | 64 | |

4 | By Addition and by Subtraction | 57 |

Simultaneous Equations | 67 | |

Divisibility | 68 | |

Miscellaneous | 69 | |

Exercises on Chapter 4 | 70 | |

5 | By Alternate elimination and retention | 71 |

Highest Common Factor | 71 | |

Algebraic H.C.F | 72 | |

Factorizing | 73 | |

Exercises on Chapter 5 | 74 | |

6 | By Mere Observation | 75 |

Multiplication | 75 | |

Additional and subtraction from left to right | 76 | |

Miscellaneous | 77 | |

Exercises on Chapter 6 | 78 | |

7 | Using the average | 79 |

Exercises on Chapter 7 | 82 | |

8 | Transpose and Apply | 83 |

Division | 83 | |

Algebraic division | 83 | |

Numerical division | 86 | |

The Remainder Theorem | 89 | |

Solution of Equations | 90 | |

Linear Equations in which ‘x’ Appears more than once | 91 | |

Literal Equations | 93 | |

Mergers | 93 | |

Transformation of Equations | 94 | |

Differentiation and integrations | 95 | |

Simultaneous Equations | 95 | |

Partial fractions | 96 | |

Odd and Even Functions | 99 | |

Exercises on Chapter 8 | 99 | |

9 | One Ratio: The Other One Zero | 102 |

Exercise on Chapter 9 | 103 | |

10 | When the Samuccaya is the Same it is Zero | 104 |

Samuccaya as a Common Factor | 104 | |

Samuccaya as the Product of the Independent terms | 104 | |

Samuccaya as the sum of the denominators of two fractions having the same | 105 | |

Numerical Numerator | 105 | |

Samuccaya as a Combination or Total | 105 | |

Cubic Equations | 108 | |

Quartic Equations | 108 | |

The Ultimate and twice the Penultimate | 109 | |

Exercise on Chapter 10 | 109 | |

11 | The First by the first and the last by the last | 111 |

Factorizing | 112 | |

12 | By the Completion or Non Completion | 114 |

Exercise on Chapter 12 | 116 | |

13 | By One more than the one Before | 118 |

Recurring Decimals | 118 | |

Auxiliary Fractions A.F. | 121 | |

Denominators not ending in 1,3,7,9 | 124 | |

Groups of Digits | 126 | |

Remainder Patterns | 127 | |

Remainders by the Last Digit . | 128 | |

Divisibility . | 129 | |

Osculating From left to right . | 131 | |

Finding the Remainder | 132 | |

Writing a Number divisible by a given number | 132 | |

Divisor not ending in 9 | 132 | |

The Negative Osculator Q | 133 | |

P+Q = D | 134 | |

Divisor not ending in 1,3,7,9. | 134 | |

Groups of Digits | 135 | |

Exercises on Chapter 13. | 136 | |

14 | The Product of the Sum is the sum of the products | 138 |

15 | Only the Last terms | 142 |

Summation of Series | 143 | |

Limits. | 144 | |

Coordinate Geometry | 148 | |

16 | Calculus | 149 |

Integration | 153 | |

Differential Equations | 154 | |

Binomial and Maclaurin Theoremss | 155 | |

‘O’ and ‘A’ Level Examination Papers | 157 | |

‘O’ Level Multiple Choice Paper 1 | 158 | |

‘O’ Level Multiple Choice Paper 2 | 164 | |

‘A’ Level Multiple Choice Paper 1 | 168 | |

‘A’ Level Multiple Choice Paper 2 | 172 | |

Answers to Exercise | 177 | |

List of Vedic Sutra | 187 | |

List of Vedic Sub Sutras | 188 | |

Index of the Vedic Sutras | 189 | |

References | 191 | |

Appendix | 193 | |

Index | 197 |

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