Price $42 (including postage)

## Description

This book shows how the Vedic system applies in a large number of areas of elementary mathematics, covering arithmetic, algebra, geometry, calculus etc. Each chapter concentrates on one Vedic Sutra or Sub-sutra and shows many applications. This gives a real feel for the Vedic Sutras each of which has its own unique character. It covers much of the content of Bharati Krsna's book above but in more detail and with more applications and explanations. It also contains Vedic solutions to GCSE and 'A' level examination questions, and an explanation of the Vedic Sutras themselves.

## Details

197 + xi pages.

Size: 24cm by 17cm.

Paperback. 2009

Author: Kenneth Williams

ISBN 978-1-902517-20-9.

## Reviews

*Discover Vedic Mathematics was tremendous - it is a system, and makes so many things perfectly comprehensible* - Matthew Kirk, teacher*Just a quick note to say that your book, Discover Vedic Mathematics, is absolutely wonderful! Your examples and explanations are comprehensive in their scope; upon reading the text, working out the sample problems, and completing the corresponding exercises, I feel that I am well on my VM journey! -* Dawn Dee Ahem, Maths teacher

## Preface

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 (1884-1960). Bharati Krsna studied the ancient Indian texts between 1911 and 1918 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. Modern 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 a more 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 organise its perceptions, learn and evolve. If 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.

In the chapters that follow each example shows a different application of the formula which is the subject of that chapter. A letter with a page number at the end of a section of a chapter indicates that an exercise on that section will be found at the end of the chapter.

This book was first published in 1984, one hundred years since the birth of Bharati Krsna. In this edition some 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).

## Contents

**PREFACE v****ILLUSTRATIVE EXAMPLES viii****1 All from Nine and the Last from Ten 1**

SUBTRACTION 1

MULTIPLICATION 2

One number above and one below the Base 4

Multiplying numbers near different bases 4

Using other bases 5

Multiplication of three or more numbers 7

First corollary: squaring and cubing of numbers near a base 9

Second corollary: squaring of number beginning or ending in 5 etc. 10

Third corollary: multiplication of nines 12

DIVISION 12

THE VINCULUM 16

Simple applications of the vinculum 18

Exercises on Chapter 1: 20-**2 Vertically and Crosswise 25**

MULTIPLICATION 25

Number of zeros after the decimal point 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

Algebraic Fractions 47

LEFT TO RIGHT CALCULATIONS 48

Pythagoras theorem 48

Equation of a line 49

Exercises 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 67**

SIMULTANEOUS EQUATIONS 67

DIVISIBILITY 68

MISCELLANEOUS 69

Exercise on Chapter 4: 70**5 By Alternate Elimination and Retention 71**

HIGHEST COMMON FACTOR 71

Algebraic H.C.F. 72

FACTORISING 73

Exercises on Chapter 5: 74**6 By Mere Observation 75**

MULTIPLICATION 75

ADDITION AND SUBTRACTION FROM LEFT

TO RIGHT 76

MISCELLANEOUS 77

Exercise on Chapter 6: 78**7 Using the Average 79**

Exercise 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 INTEGRATION 95

SIMULTANEOUS EQUATIONS 95

PARTIAL FRACTIONS 96

ODD AND EVEN FUNCTIONS 99

Exercises on Chapter 8: 99-**9 One in 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 NUMERICAL NUMERATOR 105

SAMUCCAYA AS A COMBINATION OR TOTAL 105

Cubic equations 108

Quartic equations 108

THE ULTIMATE AND TWICE THE PENULTIMATE 109

Exercises on Chapter 10: 109-**11 The First by the First and the Last by the Last 111**

FACTORISING 112**12 By the Completion or Non-Completion 114**

Exercises 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**'O' AND 'A' LEVEL EXAMINATION PAPERS 157**

'O' Paper 1: 158

'O' Paper 2: 164

'A' Paper 1: 168

'A' Paper 2: 172**ANSWERS TO EXERCISES 177****LIST OF VEDIC SUTRAS 188-9****INDEX OF THE VEDIC FORMULAE 190-1****REFERENCES 191****APPENDIX 192****INDEX 196-7**

## Back Cover

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 our modern methods when such easy and enjoyable methods are available.

This book gives a comprehensive introduction to the sixteen formulae 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 line, often forwards or backwards

Vedic Mathematics solutions of examination questions are also given, and in this edition comparisons with the conventional methods are shown, and also an account of the significance of the Vedic formulae (Sutras).