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**Vedic Mathematics Teacher's Manual - Elementary Level**

## 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

## Description

This book is designed for teachers of children in grades 3 to 7. It shows how Vedic Mathematics can be used in a school course but does not cover all school topics (see contents). The book can be used for teachers who wish to learn the Vedic system or to teach courses on Vedic mathematics for this level.

The Manual contains many topics that are not in the other Manuals that are suitable for this age range and many topics that are also in Manual 2 are covered in greater detail here.

**Vertically and Crosswise**

## Contents

Preface

Preface to 'Vedic Mathematics'

1 INTRODUCTION TO THE VERTICALLY AND CROSSWISE SUTRA

2 COMBINED OPERATIONS OF ELEMENTARY ARITHMETIC

3 EVALUATION OF DETERMINANTS

4 THE SOLUTION OF SIMULTANEOUS LINEAR EQUATIONS

5 INVERSION OF MATRICES

6 CURVE-FITTING

7 EVALUATION OF LOGARITHMS AND EXPONENTIALS

8 CHANGE OF ROOTS OF EQUATIONS

9 SINE, COSINE AND INVERSE TANGENT

10 INVERSE SINE AND COSINE AND TANGENT

11 TRANSCENDENTAL EQUATIONS

12 SOLUTION OF CUBIC AND HIGHER ORDER EQUATIONS

13 FUNCTIONS OF POLYNOMIALS

14 FUNCTIONS OF BIPOLYNOMIALS

15 THE SOLUTION OF LINEAR AND NON-LINEAR DIFFERENTIAL, INTEGRAL AND INTEGRO-DIFFERENTIAL EQUATIONS

16 THE SOLUTION OF LINEAR AND NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS

## Description

This is an advanced book of sixteen chapters on one Sutra ranging from elementary multiplication etc. to the solution of non-linear partial differential equations. It deals with (i) calculation of common functions and their series expansions, and (ii) the solution of equations, starting with simultaneous equations and moving on to algebraic, transcendental and differential equations. The text contains exercises and answers.

**GEOMETRY FOR AN ORAL TRADITION**

## Contents

Introduction

Preliminaries

Provisions

Definitions

PROPOSITIONS

Part A: Congruence, magnitudes and Lines

Part B: Angles, parallels, triangles and quadrilaterals

Part C: Concerning area equalities and similar triangles

Part D: Elementary properties of a circle

COMMENTARY

Part I: Some basics

Part II: Language and reason

Part III: Comparisons with Euclid's Elements

Part IV: Movement in geometry

Part V: The valid use of figures

Summary and Conclusions

References

APPENDICES

Appendix 1: Application of the sixteen sutras to the present system of geometry

Appendix 2: Alternative proofs and sequences in Part D

Appendix 3: Further definitions

## Description

This book demonstrates the kind of system that could have existed before literacy was widespread and takes us from first principles to theorems on elementary properties of circles. It presents direct, immediate and easily understood proofs. These are based on only one assumption (that magnitudes are unchanged by motion) and three additional provisions (a means of drawing figures, the language used and the ability to recognise valid reasoning). It includes discussion on the relevant philosophy of mathematics and is written both for mathematicians and for a wider audience.

**INTRODUCTORY LECTURES ON VEDIC MATHEMATICS**

## Details

Authors: A. P. Nicholas, J. Pickles, K. Williams, 1982.

Paperback, 166 pages, A4 size.

Currently out of print.

## Contents

Chapter 8 is on Addition. Chapter 9 is on Subtraction and combined addition and subtraction.

**FREE PRACTICE SHEETS (& Sutras list etc.)**

**Mental Math Workouts**

**Vedic Math Genius**

Free books Hosted on other websites

**Fundamentals & Applications of Vedic Mathematics**

## Details

Published by : State Council of Educational Research & Training, New Delhi

and printed at

Educational Stores, S-5, Bsr. Road Ind. Area, Ghaziabad (U.P.)

Varun Marg, Defence Colony, New Delhi-110024

**Chief Advisor**

Anita Satia Director, SCERT**Guidance**

Dr. Pratibha Sharma,

Joint Director, SCERT**Contributors**

Dr. Anil Kumar Teotia (Sr. Lecturer, DIET Dilshad Garden)

Neelam Kapoor (Retired PGT, Directorate of Education)

Chander Kanta Chabria (PGT, RPVV Tyagraj Nagar, Lodhi Road)

Rekha Jolly (TGT, RPVV Vasant Kunj)

Dr. Satyavir Singh (Principal SNI College Pilana)

**Editor**

Dr. Anil Kumar Teotia

Sr. Lecturer, DIET Dilshad Garden

**Publication Officer**

Ms. Sapna Yadav

**Publication Team**

Navin Kumar, Ms. Radha, Jai Baghwan

Pages 65

## Contents

Preface 03-04

Introduction 07-11

Chapter-1 Addition and Subtraction 12-24

1. Addition - Completing the whole

2. Addition from left to right

3. Addition of list of numbers - Shudh method

4. Subtraction - Base method

5. Subtraction - Completing the whole

6. Subtraction from left to right

Chapter-2 Digit Sums, Casting out 9s, 9-Check Method 25-28

Chapter-3 11-Check method 29-31

Chapter-4 Special Multiplication methods 32-52

1. Base Method

2. Sub Base Method

3. Vinculum

4. Multiplication of complimentary numbers

5. Multiplication by numbers consisting of all 9s

6. Multiplication by 11

7. Multiplication by two-digit numbers from right to left

8. Multiplication by three and four-digit numbers from right to left.

Chapter-5 Squaring and square Roots 53-57

Squaring

1. Squaring numbers ending in 5

2. Squaring Decimals and Fraction

3. Squaring Numbers Near 50

4. Squaring numbers near a Base and Sub Base

5. General method of Squaring - from left to right

6. Number splitting to simplify Squaring Calculation

7. Algebraic Squaring

Square Roots

1. Reverse squaring to find Square Root of Numbers ending in 25

2. Square root of perfect squares

3. General method of Square Roots

Chapter-6 Division 58-64

1. Special methods of Division

2. Straight Division

**Vedic Math Presentation**

**Vedic Mathematics**

**Vedic Mathematics Methods**

## Contents

Vedic Mathematics - Methods

Preface 1

I. Why Vedic Mathematics? 3

II. Vedic Mathematical Formulae 5

1. Ekadhikena Purvena 7

2. Nikhilam navatascaramam Dasatah 18

3. Urdhva - tiryagbhyam 31

4. Paravartya Yojayet 41

5. Sunyam Samya Samuccaye 53

6. Anurupye - Sunyamanyat 64

7. Sankalana - Vyavakalanabhyam 65

8. Puranapuranabhyam 67

9. Calana - Kalanabhyam 68

10. Ekanyunena Purvena 69

11. Anurupyena 75

12. Adyamadyenantya - mantyena 82

13. Yavadunam Tavadunikrtya Varganca Yojayet 86

14. Antyayor Dasakepi 93

15. Antyayoreva 96

16. Lopana Sthapanabhyam 101

17. Vilokanam 106

18. Gunita Samuccayah : Samuccaya Gunitah 113

III Vedic Mathematics - A briefing 115

1. Terms and Operations 116

2. Addition and Subtraction 130

3. Multiplication 139

4. Division 144

5. Miscellaneous Items 151

IV Conclusion 158

**Vedic Mathematical Concepts and Their Application ****to Unsolved Mathematical Problems: ****Three Proofs of Fermat's Last Theorem**

**MAHARISHI'S VEDIC MATHEMATICS AT THE ELEMENTARY LEVEL**

## Details

Author John M Muehlman, 1993.

Publisher: Dissertation Information Service

Pages 72 (although pages are numbered from 36 for some reason)

ASIN: B0006PAVSA

## Contents

Introduction 39

Maharishi's Description of the Veda 41

Self-Interacting Dynamics of Consciousness 41

Laws of Nature 42

Pure Knowledge and Infinite Organizing Power 43

Development of Consciousness through MaharishiÕs Vedic Technologies 43

Experience of Pure Consciousness 45

Higher States of Consciousness 47

Nature of All Knowingness in MaharishiÕs Teaching 49

Infinite Correlation 49

Principle of Least Action 50

Mathematics Without Steps 50

Introduction to Vedic Sªtra Based Computation 55

Literature Review on Characteristics of Vedic Sªtra Based Computation

Relevant to the Experimental Study 61

Efficient Algorithms and Ease of Learning 61

Flexible Format 62

Conducive to Mental Computation 63

Upsurges of Joy 63

Pilot Research on Vedic Sªtra Based Computation 63

Pilot Study Number One 63

Pilot Study Number Two 64

Pilot Study Number Three 65

An Experimental Study on Vedic Sªtra Based Multiplication and Checking 69

Subjects and Setting 69

Overview of the Curricula 70

Comparison of the Algorithms 71

Instruments Used to Evaluate Hypotheses 75

Design 76

Discussion of Results from the Study 76

Hypothesis One: Multiplication Skill 77

Hypothesis Two: Checking Skill 80

Hypothesis Three: Multiplication and Checking Affect 81

Hypothesis Four: Mental Mathematics 86

Extension of Results to Explain Possible Growth in the Direction of

All Knowingness 87

Appendix: Review of Additional Characteristics of a Full Vedic Sªtra Based

Computation Program 90

**Vedic Mathematics: Vedic or Mathematics: A Fuzzy and Neutrosophic Analysis**

## Details

W. B. Vasantha Kandasamy (Author)

Florentin Smarandache (Author)

Meena Kandasamy (Illustrator)

Perfect Paperback: 220 pages

Publisher: Automaton; 1 edition (December 5, 2006)

Language: English

ISBN-10: 1599730049

ISBN-13: 978-1599730042

Product Dimensions: 8.7 x 6 x 0.5 inches

## Contents

Preface 5

Chapter One

INTRODUCTION TO VEDIC MATHEMATICS 9

Chapter Two

ANALYSIS OF VEDIC MATHEMATICS BY MATHEMATICIANS AND OTHERS 31

2.1 Views of Prof. S.G.Dani about Vedic

Mathematics from Frontline 33

2.2 Neither Vedic Nor Mathematics 50

2.3 Views about the Book in Favour and Against 55

2.4 Vedas: Repositories of Ancient Indian Lore 58

2.5 A Rational Approach to Study Ancient Literature 59

2.6 Shanghai Rankings and Indian Universities 60

2.7 Conclusions derived on Vedic Mathematics and the

Calculations of Guru Tirthaji - Secrets of

Ancient Maths 61

Chapter Three

INTRODUCTION TO BASIC CONCEPTS AND A NEW FUZZY MODEL 65

3.1 Introduction to FCM and the Working of this Model 65

3.2 Definition and Illustration of

Fuzzy Relational Maps (FRMS) 72

3.3 Definition of the New Fuzzy Dynamical System 77 4

3.4 Neutrosophic Cognitive Maps with Examples 78

3.5 Description of Neutrosophic Relational Maps 87

3.6 Description of the new Fuzzy Neutrosophic model 92

Chapter Four

MATHEMATICAL ANALYSIS OF THE VIEWS ABOUT VEDIC MATHEMATICS USING FUZZY MODELS 95

4.1 Views of students about the use of Vedic

Mathematics in their curriculum 97

4.2 Teachers views on Vedic Mathematics and

its overall influence on the Students Community 101

4.3 Views of Parents about Vedic Mathematics 109

4.4 Views of Educationalists about Vedic Mathematics 114

4.5 Views of the Public about Vedic Mathematics 122

Chapter Five

OBSERVATIONS 165

5.1 Students’ Views 165

5.2 Views of Teachers 169

5.3 Views of Parents 180

5.4 Views of the Educated 182

5.5 Observations from the Views of the Public 193

REFERENCE 197

INDEX 215

ABOUT THE AUTHORS 220

## Description

The Vedas are considered divine in origin and are assumed to be revelations from God. In traditional Hinduism, the Vedas were to be learnt only by the upper caste Hindus. The lower castes (Sudras) and so-called untouchables (who were outside the Hindu social order) were forbidden from even hearing to its recitation. In recent years, there have been claims that the Vedas contain the cure to AIDS and the production of electricity. Here the authors probe into Vedic Mathematics (that gained renown during the revivalist Hindutva rule in India and was introduced into school syllabus in several states); and explore if it is really Vedic in origin or Mathematics in content. To gain a better understanding of its imposition, we interviewed students, teachers, parents, educationists and activists. We analyze this problem using models like Fuzzy Cognitive Maps (FCM), Fuzzy Relational Maps (FRM) and newly constructed Fuzzy Dynamical System (and their Neutrosophic Analogues). The issue of imposition of Vedic Mathematics into the school curriculum involves religious politics, caste supremacy, apart from elementary arithmetic so we use fuzzy and neutrosophic techniques to gain acute insight into how students have been affected because of this politically motivated syllabus revision.

## Our Comments

We believe this paper was a reaction by some in the academic community to certain elements of the Hindu community trying to use Vedic Mathematics to promote Hinduism by teaching Vedic Mathematics in schools. It is our understanding that much of this initial teaching of Vedic Mathematics may have been implemented poorly by people who did not have a good grounding in their subject matter as they may have been more interested in promoting Hinduism than Vedic Mathematics, hence the academic reaction.

This paper was produced in 2006 and often appears in searches for Vedic Mathematics. The understanding and teaching of Vedic Mathematics has moved on a lot since this point in time.