Details

166 + v pages.
Size: 23cm by 15cm.
Paperback. 2009
Author: Kenneth Williams
ISBN 978-1-902517-16-2.

Please note that these Manuals do not form a sequence: there is some overlap between the three books.

Reviews

"The author should be commended for his thorough grasp of Krishna Tirthaji's book on Vedic Mathematics, his layout of the divisions of mathematics and various exercises provided for the students which enhance the value of these books [Teacher's Manuals]. They are ideal companions for the teacher and adult students alike."
- Journal of oriental research, Vol. 78-80, 2006-9.

"I am a student of class XI   and planning to appear in IIT entrance exam next year and would like to take this opportunity of telling you what a difference your books have made to my life. I hated maths till class x  and believed that it is the only subject which was a brain freak and I felt that I could never achieve excellent marks in math’s, as I could in other subjects. I am extremely thankful to you for writing these books. They are great source of encouragement for me and many other students like me. I have read your primary level and dying to read next two levels..."
Rajesh Gupta

Preface

PREFACE

 

This Manual is the first of three (elementary, intermediate and advanced) Manuals which are designed for adults with a basic understanding of mathematics to learn or teach the Vedic system. So teachers could use it to learn Vedic Mathematics, though it is not suitable as a text for children (for that the Cosmic Calculator Course is recommended). Or it could be used to teach a course on Vedic Mathematics.

 

The sixteen lessons of this course are based on a series of one week summer courses given at Oxford University by the author to Swedish mathematics teachers between 1990 and 1995. Those courses were quite intensive consisting of eighteen, one and a half hour, lessons.

 

All techniques are fully explained and proofs are given where appropriate, the relevant Sutras are indicated throughout (these are listed at the end of this Manual) and, for convenience, answers are given after each exercise. Cross-references are given showing what alternative topics may be continued with at certain points.

 

It should also be noted that the Vedic system encourages mental work so we always encourage students to work mentally as long as it is comfortable. In the Cosmic Calculator Course pupils are given a short mental test at the start of most or all lessons, which makes a good start to the lesson, revises previous work and introduces some of the ideas needed in the current lesson. In the Cosmic Calculator course there are also many games that help to establish and promote confidence in the ideas used here.

 

Some topics will be found to be missing in this text: for example, there is no section on area, only a brief mention. This is because the actual methods are the same as currently taught so that the only difference would be to give the relevant Sutra(s).

Introduction

 

INTRODUCTION

Vedic Mathematics is an ancient system of mathematics which was rediscovered early last century by Sri Bharati Krsna Tirthaji (henceforth referred to as Bharati Krsna).

The Sanskrit word “veda” means “knowledge”. The Vedas are ancient writings whose date is disputed but which date from at least several centuries BC. According to Indian tradition the content of the Vedas was known long before writing was invented and was freely available to everyone. It was passed on by word of mouth. The writings called the Vedas consist of a huge number of documents (there are said to be millions of such documents in India, many of which have not yet been translated) and these have recently been shown to be highly structured, both within themselves and in relation to each other (see Reference 2). Subjects covered in the Vedas include Grammar, Astronomy, Architecture, Psychology, Philosophy, Archery etc., etc.

A hundred years ago Sanskrit scholars were translating the Vedic documents and were surprised at the depth and breadth of knowledge contained in them. But some documents headed “Ganita Sutras”, which means mathematics, could not be interpreted by them in terms of mathematics. One verse, for example, said “in the reign of King Kamse famine, pestilence and unsanitary conditions prevailed”. This is not mathematics they said, but nonsense.

Bharati Krsna was born in 1884 and died in 1960. He was a brilliant student, obtaining the highest honours in all the subjects he studied, including Sanskrit, Philosophy, English, Mathematics, History and Science. When he heard what the European scholars were saying about the parts of the Vedas which were supposed to contain mathematics he resolved to study the documents and find their meaning. Between 1911 and 1918 he was able to reconstruct the ancient system of mathematics which we now call Vedic Mathematics.

He wrote sixteen books expounding this system, but unfortunately these have been lost and when the loss was confirmed in 1958 Bharati Krsna wrote a single introductory book entitled “Vedic Mathematics”. This is currently available and is a best-seller (see Reference 1).

There are many special aspects and features of Vedic Mathematics which are better discussed as we go along rather than now because you will need to see the system in action to appreciate it fully. But the main points for now are:

1) The system rediscovered by Bharati Krsna is based on sixteen formulae (or Sutras) and some sub-formulae (sub-Sutras). These Sutras are given in word form: for example By One More than the One Before and Vertically and Crosswise. In this text they are indicated by italics. These Sutras can be related to natural mental functions such as completing a whole, noticing analogies, generalisation and so on.

 
2) Not only does the system give many striking general and special methods, previously unknown to modern mathematics, but it is far more coherent and integrated as a system.

3) Vedic Mathematics is a system of mental mathematics (though it can also be written down).

Many of the Vedic methods are new, simple and striking. They are also beautifully interrelated so that division, for example, can be seen as an easy reversal of the simple multiplication method (similarly with squaring and square roots). This is in complete contrast to the modern system. Because the Vedic methods are so different to the conventional methods, and also to gain familiarity with the Vedic system, it is best to practice the techniques as you go along.

 

Contents

 PREFACE     

 LESSON 1    COMPLETING THE WHOLE

 INTRODUCTION

 THE TEN POINT CIRCLE

 MULTIPLES OF TEN

 DEFICIENCY FROM TEN
       DEFICIENCY AND COMPLETION TOGETHER
 
MENTAL ADDITION
       COMPLETING THE WHOLE
       COLUMNS OF FIGURES

 BY ADDITION AND BY SUBTRACTION
       SUBTRACTING NUMBERS NEAR A BASE

 

LESSON 2    DOUBLING AND HALVING

 DOUBLING
       MULTIPLYING BY 4, 8

 HALVING
       SPLITTING NUMBERS
       DIVIDING BY 4, 8

 EXTENDING YOUR TABLES

 MULTIPLYING BY 5, 50, 25

 DIVIDING BY 5, 50, 25
       DIVIDING BY 5
       DIVIDING BY 50, 25 

 

LESSON 3       DIGIT SUMS

 ADDING DIGITS

 THE NINE POINT CIRCLE

 CASTING OUT NINES

 DIGIT SUM PUZZLES
       MORE DIGIT SUM PUZZLES

 THE DIGIT SUM CHECK
       MULTIPLICATION CHECK

 THE VEDIC SQUARE

 PATTERNS FROM THE VEDIC SQUARE

 NUMBER NINE 

 

LESSON 4      LEFT TO RIGHT

 ADDITION: LEFT TO RIGHT

 MULTIPLICATION: LEFT TO RIGHT

 DOUBLING AND HALVING

 SUBTRACTION: LEFT TO RIGHT

 CHECKING SUBTRACTION SUMS

 MORE SUBTRACTIONS 

 

LESSON 5     ALL FROM 9 AND THE LAST FROM 10

 ALL FROM 9 AND THE LAST FROM 10

 SUBTRACTION
       ADDING ZEROS
       ONE LESS
       ONE MORE
       ONE LESS AGAIN

 MONEY

 

LESSON 6       NUMBER SPLITTING

 ADDITION

 SUBTRACTION

 MULTIPLICATION

 DIVISION

 

LESSON 7       BASE MULTIPLICATION

 TIMES TABLES

 NUMBERS JUST OVER TEN

 MULTIPLICATION TABLE PATTERNS
       RECURRING DECIMALS

 NUMBERS CLOSE TO 100
       MENTALLY
       NUMBERS OVER 100
       MENTAL MATHS
       RUSSIAN PEASANT MULTIPLICATION

 LARGER NUMBERS
       NUMBERS ABOVE THE BASE

 PROPORTIONATELY
       ANOTHER APPLICATION OF PROPORTIONATELY

MULTIPLYING NUMBERS NEAR DIFFERENT BASES

SQUARING NUMBERS NEAR A BASE

A SUMMARY

 

LESSON 8    CHECKING AND DIVISIBILITY

 DIGIT SUM CHECK FOR DIVISION

 THE FIRST BY THE FIRST AND THE LAST BY THE LAST
       THE FIRST BY THE FIRST
       THE LAST BY THE LAST

 DIVISIBILITY BY 4

 DIVISIBILITY BY 11
       REMAINDER AFTER DIVISION BY 11
       ANOTHER DIGIT SUM CHECK

 

LESSON 9     BAR NUMBERS

 REMOVING BAR NUMBERS
       ALL FROM 9 AND THE LAST FROM 10

 SUBTRACTION

 CREATING BAR NUMBERS

 USING BAR NUMBERS

 

LESSON 10  SPECIAL MULTIPLICATION

 MULTIPLICATION BY 11
       CARRIES
       LONGER NUMBERS

 BY ONE MORE THAN THE ONE BEFORE

 MULTIPLICATION BY NINES

 THE FIRST BY THE FIRST AND THE LAST BY THE LAST

 USING THE AVERAGE

 SPECIAL NUMBERS
       REPEATING NUMBERS
       PROPORTIONATELY
       DISGUISES

 

LESSON 11  GENERAL MULTIPLICATION

 REVISION

 TWO-FIGURE NUMBERS
       CARRIES

 MOVING MULTIPLIER

 EXTENSION

 MULTIPLYING BINOMIALS

 MULTIPLYING 3-FIGURE NUMBERS

 WRITTEN CALCULATIONS

 

LESSON 12    SQUARING

 SQUARING NUMBERS THAT END IN 5

 SQUARING NUMBERS NEAR 50

 GENERAL SQUARING
       THE DUPLEX

 NUMBER SPLITTING

 ALGEBRAIC SQUARING

 DIGIT SUMS OF SQUARES

 SQUARE ROOTS OF PERFECT SQUARES

 3 AND 4 FIGURE NUMBERS

 

LESSON 13    EQUATIONS

 ONE-STEP EQUATIONS

 TWO-STEP EQUATIONS

 THREE-STEP EQUATIONS

 

LESSON 14    FRACTIONS

 VERTICALLY AND CROSSWISE

 A SIMPLIFICATION

 COMPARING FRACTIONS

 UNIFICATION OF OPERATIONS

 

LESSON 15    SPECIAL DIVISION

 DIVISION BY 9
       LONGER NUMBERS
       CARRIES
       A SHORT CUT

 DIVISION BY 8 ETC.

 DIVISION BY 99, 98 ETC.

 DIVISOR BELOW A BASE NUMBER
       TWO-FIGURE ANSWERS

 DIVISOR ABOVE A BASE NUMBER

 

LESSON 16    THE CROWNING GEM                 

 SINGLE FIGURE ON THE FLAG

 SHORT DIVISION DIGRESSION

 LONGER NUMBERS

 NEGATIVE FLAG DIGITS

 DECIMALISING THE REMAINDER

 

SUTRAS AND SUB-SUTRAS

 9-POINT CIRCLES

 REFERENCES  

 INDEX OF THE VEDIC FORMULAE 

 INDEX

 

Back Cover

 

VEDIC MATHEMATICS MANUAL

 

ELEMENTARY LEVEL

 

¯ Vedic Mathematics was reconstructed from ancient Vedic texts early last century by Sri Bharati Krsna Tirthaji (1884-1960). It is a complete system of mathematics which has many surprising properties and applies at all levels and areas of mathematics, pure and applied.

 

¯ It has a remarkable coherence and simplicity that make it easy to do and easy to understand. Through its amazingly easy methods complex problems can often be solved in one line.

 

¯ The system is based on sixteen word-formulae (Sutras) that relate to the way in which we use our mind.

 

¯ The benefits of using Vedic Mathematics include more enjoyment of maths, increased flexibility, creativity and confidence, improved memory, greater mental agility and so on.

 

¯ This Elementary Manual is the first of three designed for teachers who wish to teach the Vedic system, either to a class or to other adults/teachers. It is also suitable for anyone who would like to teach themselves the basic Vedic methods.