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Discover Vedic Mathematics

Discover Vedic Mathematics
Item Code: NAB932
Author: Kenneth R Williams
Publisher: Motilal Banarsidass Publishers Pvt. Ltd.
Edition: 2006
ISBN: 8120830962
Pages: 216
Cover: Hardcover
Other Details: 9.5 inch X 6.5 inch
weight of the book: 550 gms
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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