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Discover Vedic Mathematics


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Discover Vedic Mathematics

Discover Vedic Mathematics

Código del Artículo: NAB932

por Kenneth R Williams

Hardcover (Edición: 2006)

Motilal Banarsidass Publishers
ISBN 8120830962

Tamaño: 9.5 inch X 6.5 inch
Páginas: 216
Weight of the Book: 550 gms

Precio: Euro 22.87

Back of the Book

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)



Preface ix

Illustrative Examples xv
1 All From Nine and the Last From Ten 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 Roots42

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


Factorising quadratics 58

Ratios in Triangles 60

Transformation of Equations 61

Number Bases62

Miscellaneous 63

Exercises on Chapter 364
4 By Addition and by Subtraction 57

Simultaneous Equations67

Divisibility 68

Miscellaneous 69

Exercises on Chapter 470
5 By Alternate elimination and retention 71

Highest Common Factor71

Algebraic H.C.F72

Factorizing 73

Exercises on Chapter 574
6 By Mere Observation 75

Multiplication 75

Additional and subtraction from left to right 76

Miscellaneous 77

Exercises on Chapter 678
7 Using the average 79

Exercises on Chapter 782
8 Transpose and Apply 83


Algebraic division83

Numerical division86

The Remainder Theorem89

Solution of Equations90

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

Literal Equations93

Mergers 93

Transformation of Equations 94

Differentiation and integrations 95

Simultaneous Equations 95

Partial fractions 96

Odd and Even Functions 99

Exercises on Chapter 899
9 One Ratio: The Other One Zero 102

Exercise on Chapter 9103
10 When the Samuccaya is the Same it is Zero 104

Samuccaya as a Common Factor104

Samuccaya as the Product of the Independent terms104

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

Numerical Numerator105

Samuccaya as a Combination or Total105

Cubic Equations108

Quartic Equations 108

The Ultimate and twice the Penultimate 109

Exercise on Chapter 10109
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 12116
13 By One more than the one Before 118

Recurring Decimals118

Auxiliary Fractions A.F.121

Denominators not ending in 1,3,7,9124

Groups of Digits126

Remainder Patterns127

Remainders by the Last Digit .128

Divisibility .129

Osculating From left to right .131

Finding the Remainder132

Writing a Number divisible by a given number 132

Divisor not ending in 9132

The Negative Osculator Q133

P+Q = D134

Divisor not ending in 1,3,7,9.134

Groups of Digits135

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


Coordinate Geometry148
16 Calculus 149


Differential Equations 154

Binomial and Maclaurin Theoremss155

‘O’ and ‘A’ Level Examination Papers157

‘O’ Level Multiple Choice Paper 1158

‘O’ Level Multiple Choice Paper 2164

‘A’ Level Multiple Choice Paper 1168

‘A’ Level Multiple Choice Paper 2172

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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