6. DISCOVER VEDIC MATHEMATICS - details

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. 216 pages, paperback.. Author: K. Williams, Originally published, 1984, latest edition 2006. ISBN 81-208-3097-0. Price 13.75 pounds. 

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

"Discover Vedic Mathematics" - 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

"Discover Vedic Mathematics" - 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).