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This PDF contains :

1. INTRODUCTION TO VEDIC MATHEMATICS.

2. ANALYSIS OF VEDIC MATHEMATICS BY

MATHEMATICIANS AND OTHERS.

3.INTRODUCTION TO BASIC CONCEPTS

AND A NEW FUZZY MODEL.

4.MATHEMATICAL ANALYSIS OF THE

VIEWS ABOUT VEDIC MATHEMATICS USING

FUZZY MODELS.

5.OBSERVATIONS.

1. INTRODUCTION TO VEDIC MATHEMATICS.

2. ANALYSIS OF VEDIC MATHEMATICS BY

MATHEMATICIANS AND OTHERS.

3.INTRODUCTION TO BASIC CONCEPTS

AND A NEW FUZZY MODEL.

4.MATHEMATICAL ANALYSIS OF THE

VIEWS ABOUT VEDIC MATHEMATICS USING

FUZZY MODELS.

5.OBSERVATIONS.

1.
VEDIC MATHEMATICS -

‘VEDIC’ OR ‘MATHEMATICS’:

A FUZZY & NEUTROSOPHIC

ANALYSIS

W. B. VASANTHA KANDASAMY

FLORENTIN SMARANDACHE

2006

‘VEDIC’ OR ‘MATHEMATICS’:

A FUZZY & NEUTROSOPHIC

ANALYSIS

W. B. VASANTHA KANDASAMY

FLORENTIN SMARANDACHE

2006

2.
VEDIC MATHEMATICS -

‘VEDIC’ OR ‘MATHEMATICS’:

A FUZZY & NEUTROSOPHIC

ANALYSIS

W. B. VASANTHA KANDASAMY

e-mail: [email protected]

web: http://mat.iitm.ac.in/~wbv

www.vasantha.net

FLORENTIN SMARANDACHE

e-mail: [email protected]

2006

‘VEDIC’ OR ‘MATHEMATICS’:

A FUZZY & NEUTROSOPHIC

ANALYSIS

W. B. VASANTHA KANDASAMY

e-mail: [email protected]

web: http://mat.iitm.ac.in/~wbv

www.vasantha.net

FLORENTIN SMARANDACHE

e-mail: [email protected]

2006

3.
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

3

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

3

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

4

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

4

5.
Religious extremism has been the root cause of most of the

world problems since time immemorial. It has decided the fates

of men and nations. In a vast nation like India, the imposition of

religious dogma and discrimination upon the people has taken

place after the upsurge of Hindu rightwing forces in the political

arena. As a consequence of their political ascendancy in the

northern states of India, they started to rewrite school textbooks

in an extremely biased manner that was fundamentalist and

revivalist. Not only did they meddle with subjects like history

(which was their main area of operation), but they also imposed

their religious agenda on the science subjects. There was a plan

to introduce Vedic Astrology in the school syllabus across the

nation, which was dropped after a major hue and cry from

secular intellectuals.

This obsession with ‘Vedic’ results from the fundamentalist

Hindu organizations need to claim their identity as Aryan (and

hence of Caucasian origin) and hence superior to the rest of the

native inhabitants of India. The ‘Vedas’ are considered ‘divine’

in origin and are assumed to be direct revelations from God.

The whole corpus of Vedic literature is in Sanskrit. The Vedas

are four in number: Rgveda, Saamaveda, Yajurveda and

Atharvaveda. In traditional Hinduism, the Vedas as a body of

knowledge were to be learnt only by the ‘upper’ caste Hindus

and the ‘lower castes’ (Sudras) and so-called ‘untouchables’

(who were outside the Hindu social order) were forbidden from

learning or even hearing to their recitation. For several

centuries, the Vedas were not written down but passed from

generation to generation through oral transmission. While

religious significance is essential for maintaining Aryan

supremacy and the caste system, the claims made about the

Vedas were of the highest order of hyperbole. Murli Manohar

Joshi, a senior Cabinet minister of the Bharatiya Janata Party

(BJP) that ruled India from 1999-2004 went on to claim that a

cure of the dreaded AIDS was available in the Vedas! In the

5

world problems since time immemorial. It has decided the fates

of men and nations. In a vast nation like India, the imposition of

religious dogma and discrimination upon the people has taken

place after the upsurge of Hindu rightwing forces in the political

arena. As a consequence of their political ascendancy in the

northern states of India, they started to rewrite school textbooks

in an extremely biased manner that was fundamentalist and

revivalist. Not only did they meddle with subjects like history

(which was their main area of operation), but they also imposed

their religious agenda on the science subjects. There was a plan

to introduce Vedic Astrology in the school syllabus across the

nation, which was dropped after a major hue and cry from

secular intellectuals.

This obsession with ‘Vedic’ results from the fundamentalist

Hindu organizations need to claim their identity as Aryan (and

hence of Caucasian origin) and hence superior to the rest of the

native inhabitants of India. The ‘Vedas’ are considered ‘divine’

in origin and are assumed to be direct revelations from God.

The whole corpus of Vedic literature is in Sanskrit. The Vedas

are four in number: Rgveda, Saamaveda, Yajurveda and

Atharvaveda. In traditional Hinduism, the Vedas as a body of

knowledge were to be learnt only by the ‘upper’ caste Hindus

and the ‘lower castes’ (Sudras) and so-called ‘untouchables’

(who were outside the Hindu social order) were forbidden from

learning or even hearing to their recitation. For several

centuries, the Vedas were not written down but passed from

generation to generation through oral transmission. While

religious significance is essential for maintaining Aryan

supremacy and the caste system, the claims made about the

Vedas were of the highest order of hyperbole. Murli Manohar

Joshi, a senior Cabinet minister of the Bharatiya Janata Party

(BJP) that ruled India from 1999-2004 went on to claim that a

cure of the dreaded AIDS was available in the Vedas! In the

5

6.
continuing trend, last week a scientist has announced that

NASA (of the USA) is using a Vedic formula to produce

electricity. One such popular topic of Hindutva imposition was

Vedic Mathematics. Much of the hype about this topic is based

on one single book authored by the Sankaracharya (the highest

Hindu pontiff) Jagadguru Swami Sri Bharati Krsna Tirthaji

Maharaja titled Vedic Mathematics and published in the year

1965, and reprinted several times since the 1990s [51]. This

book was used as the foundation and the subject was

systematically introduced in schools across India. It was

introduced in the official curriculum in the school syllabus in

the states of Uttar Pradesh and Madhya Pradesh. Further,

schools run by Hindutva sympathizers or trusts introduced it

into their curriculum. In this juncture, the first author of this

book started working on this topic five years back, and has since

met over 1000 persons from various walks of life and collected

their opinion on Vedic Mathematics. This book is the result of

those interactions.

In this book the authors probe into Vedic Mathematics (a

concept that gained renown in the period of the religious fanatic

and revivalist Hindutva rule in India): and explore whether it is

really ‘Vedic’ in origin or ‘Mathematics’ in content. The entire

field of Vedic Mathematics is supposedly based on 16 one-to-

three-word sutras (aphorisms) in Sanskrit, which they claim can

solve all modern mathematical problems. However, a careful

perusal of the General Editor’s note in this book gives away the

basic fact that the origin of these sutras are not ‘Vedic’ at all.

The book’s General Editor, V.S. Agrawala, (M.A., PhD.

D.Litt.,) writes in page VI as follows:

“It is the whole essence of his assessment of Vedic

tradition that it is not to be approached from a factual

standpoint but from the ideal standpoint viz., as the

Vedas, as traditionally accepted in India as the repository

of all knowledge, should be and not what they are in

human possession. That approach entirely turns the table

on all critics, for the authorship of Vedic mathematics

need not be labouriously searched for in the texts as

preserved from antiquity. […]

6

NASA (of the USA) is using a Vedic formula to produce

electricity. One such popular topic of Hindutva imposition was

Vedic Mathematics. Much of the hype about this topic is based

on one single book authored by the Sankaracharya (the highest

Hindu pontiff) Jagadguru Swami Sri Bharati Krsna Tirthaji

Maharaja titled Vedic Mathematics and published in the year

1965, and reprinted several times since the 1990s [51]. This

book was used as the foundation and the subject was

systematically introduced in schools across India. It was

introduced in the official curriculum in the school syllabus in

the states of Uttar Pradesh and Madhya Pradesh. Further,

schools run by Hindutva sympathizers or trusts introduced it

into their curriculum. In this juncture, the first author of this

book started working on this topic five years back, and has since

met over 1000 persons from various walks of life and collected

their opinion on Vedic Mathematics. This book is the result of

those interactions.

In this book the authors probe into Vedic Mathematics (a

concept that gained renown in the period of the religious fanatic

and revivalist Hindutva rule in India): and explore whether it is

really ‘Vedic’ in origin or ‘Mathematics’ in content. The entire

field of Vedic Mathematics is supposedly based on 16 one-to-

three-word sutras (aphorisms) in Sanskrit, which they claim can

solve all modern mathematical problems. However, a careful

perusal of the General Editor’s note in this book gives away the

basic fact that the origin of these sutras are not ‘Vedic’ at all.

The book’s General Editor, V.S. Agrawala, (M.A., PhD.

D.Litt.,) writes in page VI as follows:

“It is the whole essence of his assessment of Vedic

tradition that it is not to be approached from a factual

standpoint but from the ideal standpoint viz., as the

Vedas, as traditionally accepted in India as the repository

of all knowledge, should be and not what they are in

human possession. That approach entirely turns the table

on all critics, for the authorship of Vedic mathematics

need not be labouriously searched for in the texts as

preserved from antiquity. […]

6

7.
In the light of the above definition and approach

must be understood the author’s statement that the

sixteen sutras on which the present volume is based from

part of a Parisista of the Atharvaveda. We are aware that

each Veda has its subsidiary apocryphal text some of

which remain in manuscripts and others have been

printed but that formulation has not closed. For example,

some Parisista of the Atharvaveda were edited by

G.M.Bolling and J. Von Negelein, Leipzig,1909-10. But

this work of Sri Sankaracharyaji deserves to be regarded

as a new Parisista by itself and it is not surprising that

the Sutras mentioned herein do not appear in the hitherto

known Parisistas.

A list of these main 16 Sutras and of their sub-sutras

or corollaries is prefixed in the beginning of the text and

the style of language also points to their discovery by Sri

Swamiji himself. At any rate, it is needless to dwell

longer on this point of origin since the vast merit of

these rules should be a matter of discovery for each

intelligent reader. Whatever is written here by the author

stands on its own merits and is presented as such to the

mathematical world. [emphasis supplied]”

The argument that Vedas means all knowledge and hence

the fallacy of claiming even 20th century inventions to belong to

the Vedas clearly reveals that there is a hidden agenda in

bestowing such an antiquity upon a subject of such a recent

origin. There is an open admission that these sutras are the

product of one man’s imagination. Now it has become clear to

us that the so-called Vedic Mathematics is not even Vedic in

Next, we wanted to analyze the mathematical content and

its ulterior motives using fuzzy analysis. We analyzed this

problem using fuzzy models like Fuzzy Cognitive Maps (FCM),

Fuzzy Relational Maps (FRM) and the newly constructed fuzzy

dynamical system (and its Neutrosophic analogue) that can

analyze multi-experts opinion at a time using a single model.

The issue of Vedic Mathematics involves religious politics,

caste supremacy, apart from elementary arithmetic—so we

7

must be understood the author’s statement that the

sixteen sutras on which the present volume is based from

part of a Parisista of the Atharvaveda. We are aware that

each Veda has its subsidiary apocryphal text some of

which remain in manuscripts and others have been

printed but that formulation has not closed. For example,

some Parisista of the Atharvaveda were edited by

G.M.Bolling and J. Von Negelein, Leipzig,1909-10. But

this work of Sri Sankaracharyaji deserves to be regarded

as a new Parisista by itself and it is not surprising that

the Sutras mentioned herein do not appear in the hitherto

known Parisistas.

A list of these main 16 Sutras and of their sub-sutras

or corollaries is prefixed in the beginning of the text and

the style of language also points to their discovery by Sri

Swamiji himself. At any rate, it is needless to dwell

longer on this point of origin since the vast merit of

these rules should be a matter of discovery for each

intelligent reader. Whatever is written here by the author

stands on its own merits and is presented as such to the

mathematical world. [emphasis supplied]”

The argument that Vedas means all knowledge and hence

the fallacy of claiming even 20th century inventions to belong to

the Vedas clearly reveals that there is a hidden agenda in

bestowing such an antiquity upon a subject of such a recent

origin. There is an open admission that these sutras are the

product of one man’s imagination. Now it has become clear to

us that the so-called Vedic Mathematics is not even Vedic in

Next, we wanted to analyze the mathematical content and

its ulterior motives using fuzzy analysis. We analyzed this

problem using fuzzy models like Fuzzy Cognitive Maps (FCM),

Fuzzy Relational Maps (FRM) and the newly constructed fuzzy

dynamical system (and its Neutrosophic analogue) that can

analyze multi-experts opinion at a time using a single model.

The issue of Vedic Mathematics involves religious politics,

caste supremacy, apart from elementary arithmetic—so we

7

8.
cannot use simple statistics for our analysis. Further any study,

when scientifically carried out using fuzzy models has more

value than a statistical approach to the same. We used linguistic

questionnaires for our data collection; experts filled in these

questionnaires. In many cases, we also recorded our interviews

with the experts in case they did not possess the technical

knowledge of working with our questionnaire. Apart from this,

several group discussions and meetings with various groups of

people were held to construct the fuzzy models used to analyze

this problem.

This book has five chapters. In Chapter I, we give a brief

description of the sixteen sutras invented by the Swamiji.

Chapter II gives the text of select articles about Vedic

Mathematics that appeared in the media. Chapter III recalls

some basic notions of some Fuzzy and Neutrosophic models

used in this book. This chapter also introduces a fuzzy model to

study the problem when we have to handle the opinion of multi-

experts. Chapter IV analyses the problem using these models.

The final chapter gives the observations made from our study.

The authors thank everybody who gave their opinion about

Vedic Mathematics. Without their cooperation, the book could

not have materialized. We next thank Dr.K.Kandasamy for

proof-reading the book. I thank Meena and Kama for the layout

and formatting of this book. Our thanks are also due to Prof.

Praveen Prakash, Prof. Subrahmaniyam, Prof. E. L.

Piriyakumar, Mr. Gajendran, Mr. S. Karuppasamy, Mr.

Jayabhaskaran, Mr. Senguttuvan, Mr. Tamilselvan, Mr. D.

Maariappan, Mr. P. Ganesan, Mr. N. Rajkumar and Ms.

Rosalyn for the help rendered in various ways that could

convert this book into a solid reality. We also thank the students

of All India Students Federation (AISF) and the Students

Federation of India (SFI) for their help in my work.

The authors dedicate this book to the great philosopher and

intellectual Rahul Sangridyayan who revealed and exposed to

the world many of the truths about the Vedas.

We have given a long list of references to help the

interested reader.

W.B.VASANTHA KANDASAMY

FLORENTIN SMARANDACHE

8

when scientifically carried out using fuzzy models has more

value than a statistical approach to the same. We used linguistic

questionnaires for our data collection; experts filled in these

questionnaires. In many cases, we also recorded our interviews

with the experts in case they did not possess the technical

knowledge of working with our questionnaire. Apart from this,

several group discussions and meetings with various groups of

people were held to construct the fuzzy models used to analyze

this problem.

This book has five chapters. In Chapter I, we give a brief

description of the sixteen sutras invented by the Swamiji.

Chapter II gives the text of select articles about Vedic

Mathematics that appeared in the media. Chapter III recalls

some basic notions of some Fuzzy and Neutrosophic models

used in this book. This chapter also introduces a fuzzy model to

study the problem when we have to handle the opinion of multi-

experts. Chapter IV analyses the problem using these models.

The final chapter gives the observations made from our study.

The authors thank everybody who gave their opinion about

Vedic Mathematics. Without their cooperation, the book could

not have materialized. We next thank Dr.K.Kandasamy for

proof-reading the book. I thank Meena and Kama for the layout

and formatting of this book. Our thanks are also due to Prof.

Praveen Prakash, Prof. Subrahmaniyam, Prof. E. L.

Piriyakumar, Mr. Gajendran, Mr. S. Karuppasamy, Mr.

Jayabhaskaran, Mr. Senguttuvan, Mr. Tamilselvan, Mr. D.

Maariappan, Mr. P. Ganesan, Mr. N. Rajkumar and Ms.

Rosalyn for the help rendered in various ways that could

convert this book into a solid reality. We also thank the students

of All India Students Federation (AISF) and the Students

Federation of India (SFI) for their help in my work.

The authors dedicate this book to the great philosopher and

intellectual Rahul Sangridyayan who revealed and exposed to

the world many of the truths about the Vedas.

We have given a long list of references to help the

interested reader.

W.B.VASANTHA KANDASAMY

FLORENTIN SMARANDACHE

8

9.
Chapter One

INTRODUCTION TO

VEDIC MATHEMATICS

In this chapter we just recall some notions given in the book on

Vedic Mathematics written by Jagadguru Swami Sri Bharati

Krsna Tirthaji Maharaja (Sankaracharya of Govardhana Matha,

Puri, Orissa, India), General Editor, Dr. V.S. Agrawala. Before

we proceed to discuss the Vedic Mathematics that he professed

we give a brief sketch of his heritage [51].

He was born in March 1884 to highly learned and pious

parents. His father Sri P Narasimha Shastri was in service as a

Tahsildar at Tinnivelly (Madras Presidency) and later retired as

a Deputy Collector. His uncle, Sri Chandrasekhar Shastri was

the principal of the Maharajas College, Vizianagaram and his

great grandfather was Justice C. Ranganath Shastri of the

Madras High Court. Born Venkatraman he grew up to be a

brilliant student and invariably won the first place in all the

subjects in all classes throughout his educational career. During

his school days, he was a student of National College

Trichanapalli; Church Missionary Society College, Tinnivelli

and Hindu College Tinnivelly in Tamil Nadu. He passed his

matriculation examination from the Madras University in 1899

topping the list as usual. His extraordinary proficiency in

Sanskrit earned him the title “Saraswati” from the Madras

Sanskrit Association in July 1899. After winning the highest

place in the B.A examination Sri Venkataraman appeared for

9

INTRODUCTION TO

VEDIC MATHEMATICS

In this chapter we just recall some notions given in the book on

Vedic Mathematics written by Jagadguru Swami Sri Bharati

Krsna Tirthaji Maharaja (Sankaracharya of Govardhana Matha,

Puri, Orissa, India), General Editor, Dr. V.S. Agrawala. Before

we proceed to discuss the Vedic Mathematics that he professed

we give a brief sketch of his heritage [51].

He was born in March 1884 to highly learned and pious

parents. His father Sri P Narasimha Shastri was in service as a

Tahsildar at Tinnivelly (Madras Presidency) and later retired as

a Deputy Collector. His uncle, Sri Chandrasekhar Shastri was

the principal of the Maharajas College, Vizianagaram and his

great grandfather was Justice C. Ranganath Shastri of the

Madras High Court. Born Venkatraman he grew up to be a

brilliant student and invariably won the first place in all the

subjects in all classes throughout his educational career. During

his school days, he was a student of National College

Trichanapalli; Church Missionary Society College, Tinnivelli

and Hindu College Tinnivelly in Tamil Nadu. He passed his

matriculation examination from the Madras University in 1899

topping the list as usual. His extraordinary proficiency in

Sanskrit earned him the title “Saraswati” from the Madras

Sanskrit Association in July 1899. After winning the highest

place in the B.A examination Sri Venkataraman appeared for

9

10.
the M.A. examination of the American College of Sciences,

Rochester, New York from the Bombay center in 1903. His

subject of examination was Sanskrit, Philosophy, English,

Mathematics, History and Science. He had a superb retentive

In 1911 he could not anymore resist his burning desire for

spiritual knowledge, practice and attainment and therefore,

tearing himself off suddenly from the work of teaching, he went

back to Sri Satcidananda Sivabhinava Nrisimha Bharati Swami

at Sringeri. He spent the next eight years in the profoundest

study of the most advanced Vedanta Philosophy and practice of

the Brahmasadhana.

After several years in 1921 he was installed on the

pontifical throne of Sharada Peetha Sankaracharya and later in

1925 he became the pontifical head of Sri Govardhan Math Puri

where he served the remainder of his life spreading the holy

spiritual teachings of Sanatana Dharma.

In 1957, when he decided finally to undertake a tour of the

USA he rewrote from his memory the present volume of Vedic

Mathematics [51] giving an introductory account of the sixteen

formulae reconstructed by him. This is the only work on

mathematics that has been left behind by him.

Now we proceed on to give the 16 sutras (aphorisms or

formulae) and their corollaries [51]. As claimed by the editor,

the list of these main 16 sutras and of their sub-sutras or

corollaries is prefixed in the beginning of the text and the style

of language also points to their discovery by Sri Swamiji

himself. This is an open acknowledgement that they are not

from the Vedas. Further the editor feels that at any rate it is

needless to dwell longer on this point of origin since the vast

merit of these rules should be a matter of discovery for each

intelligent reader.

Now having known that even the 16 sutras are the

Jagadguru Sankaracharya’s invention we mention the name of

the sutras and the sub sutras or corollaries as given in the book

[51] pp. XVII to XVIII.

10

Rochester, New York from the Bombay center in 1903. His

subject of examination was Sanskrit, Philosophy, English,

Mathematics, History and Science. He had a superb retentive

In 1911 he could not anymore resist his burning desire for

spiritual knowledge, practice and attainment and therefore,

tearing himself off suddenly from the work of teaching, he went

back to Sri Satcidananda Sivabhinava Nrisimha Bharati Swami

at Sringeri. He spent the next eight years in the profoundest

study of the most advanced Vedanta Philosophy and practice of

the Brahmasadhana.

After several years in 1921 he was installed on the

pontifical throne of Sharada Peetha Sankaracharya and later in

1925 he became the pontifical head of Sri Govardhan Math Puri

where he served the remainder of his life spreading the holy

spiritual teachings of Sanatana Dharma.

In 1957, when he decided finally to undertake a tour of the

USA he rewrote from his memory the present volume of Vedic

Mathematics [51] giving an introductory account of the sixteen

formulae reconstructed by him. This is the only work on

mathematics that has been left behind by him.

Now we proceed on to give the 16 sutras (aphorisms or

formulae) and their corollaries [51]. As claimed by the editor,

the list of these main 16 sutras and of their sub-sutras or

corollaries is prefixed in the beginning of the text and the style

of language also points to their discovery by Sri Swamiji

himself. This is an open acknowledgement that they are not

from the Vedas. Further the editor feels that at any rate it is

needless to dwell longer on this point of origin since the vast

merit of these rules should be a matter of discovery for each

intelligent reader.

Now having known that even the 16 sutras are the

Jagadguru Sankaracharya’s invention we mention the name of

the sutras and the sub sutras or corollaries as given in the book

[51] pp. XVII to XVIII.

10

11.
Sixteen Sutras and their corollaries

Sutras Sub sutras or Corollaries

1. Ekādhikena Pūrvena

Ānurūpyena

(also a corollary)

2. Nikhilam

Śisyate Śesamjnah

Navataścaramam Daśatah

3. Ūrdhva - tiryagbhyām Ādyamādyenantyamantyena

4. Parāvartya Yojayet Kevalaih Saptakam Gunỹat

5. Sūnyam

Vestanam

Samyasamuccaye

6. (Ānurūpye) Śūnyamanyat Yāvadūnam Tāvadūnam

7. Sankalana - Yāvadūnam Tāvadūnīkrtya

vyavakalanābhyām Vargaňca Yojayet

8. Puranāpuranābhyām Antyayordasake’ pi

9. Calanā kalanābhyām Antyayoreva

10. Yāvadūnam Samuccayagunitah

11. Vyastisamastih Lopanasthāpanabhyām

12. Śesānyankena Caramena Vilokanam

13. Gunitasamuccayah

Sopantyadvayamantyam

Samuccayagunitah

14. Ekanyūnena Pūrvena

15. Gunitasamuccayah

16. Gunakasamuccayah

The editor further adds that the list of 16 slokas has been

complied from stray references in the text. Now we give

spectacular illustrations and a brief descriptions of the sutras.

The First Sutra: Ekādhikena Pūrvena

The relevant Sutra reads Ekādhikena Pūrvena which rendered

into English simply says “By one more than the previous one”.

Its application and “modus operandi” are as follows.

(1) The last digit of the denominator in this case being 1 and the

previous one being 1 “one more than the previous one”

11

Sutras Sub sutras or Corollaries

1. Ekādhikena Pūrvena

Ānurūpyena

(also a corollary)

2. Nikhilam

Śisyate Śesamjnah

Navataścaramam Daśatah

3. Ūrdhva - tiryagbhyām Ādyamādyenantyamantyena

4. Parāvartya Yojayet Kevalaih Saptakam Gunỹat

5. Sūnyam

Vestanam

Samyasamuccaye

6. (Ānurūpye) Śūnyamanyat Yāvadūnam Tāvadūnam

7. Sankalana - Yāvadūnam Tāvadūnīkrtya

vyavakalanābhyām Vargaňca Yojayet

8. Puranāpuranābhyām Antyayordasake’ pi

9. Calanā kalanābhyām Antyayoreva

10. Yāvadūnam Samuccayagunitah

11. Vyastisamastih Lopanasthāpanabhyām

12. Śesānyankena Caramena Vilokanam

13. Gunitasamuccayah

Sopantyadvayamantyam

Samuccayagunitah

14. Ekanyūnena Pūrvena

15. Gunitasamuccayah

16. Gunakasamuccayah

The editor further adds that the list of 16 slokas has been

complied from stray references in the text. Now we give

spectacular illustrations and a brief descriptions of the sutras.

The First Sutra: Ekādhikena Pūrvena

The relevant Sutra reads Ekādhikena Pūrvena which rendered

into English simply says “By one more than the previous one”.

Its application and “modus operandi” are as follows.

(1) The last digit of the denominator in this case being 1 and the

previous one being 1 “one more than the previous one”

11

12.
evidently means 2. Further the proposition ‘by’ (in the sutra)

indicates that the arithmetical operation prescribed is either

multiplication or division. We illustrate this example from pp. 1

to 3. [51]

Let us first deal with the case of a fraction say 1/19. 1/19

where denominator ends in 9.

By the Vedic one - line mental method.

A. First method

1 .0 5 2 6 31 5 7 8 9 4 7 3 6 8 4 2 i

=

19 1 1 111 1 1 11

B. Second Method

1 .0 5 2 6 3 1 5 7 8 / 9 4 7 3 6 8 4 2 i

=

19 1 1 11 1 1 1 1 1

This is the whole working. And the modus operandi is

explained below.

A. First Method

Modus operandi chart is as follows:

(i) We put down 1 as the right-hand most digit 1

(ii) We multiply that last digit 1 by 2 and put the 2

down as the immediately preceding digit. 21

(iii) We multiply that 2 by 2 and put 4 down as the

next previous digit. 421

(iv) We multiply that 4 by 2 and put it down thus 8421

(v) We multiply that 8 by 2 and get 16 as the

product. But this has two digits. We therefore

put the product. But this has two digits we

therefore put the 6 down immediately to the

left of the 8 and keep the 1 on hand to be

carried over to the left at the next step (as we

12

indicates that the arithmetical operation prescribed is either

multiplication or division. We illustrate this example from pp. 1

to 3. [51]

Let us first deal with the case of a fraction say 1/19. 1/19

where denominator ends in 9.

By the Vedic one - line mental method.

A. First method

1 .0 5 2 6 31 5 7 8 9 4 7 3 6 8 4 2 i

=

19 1 1 111 1 1 11

B. Second Method

1 .0 5 2 6 3 1 5 7 8 / 9 4 7 3 6 8 4 2 i

=

19 1 1 11 1 1 1 1 1

This is the whole working. And the modus operandi is

explained below.

A. First Method

Modus operandi chart is as follows:

(i) We put down 1 as the right-hand most digit 1

(ii) We multiply that last digit 1 by 2 and put the 2

down as the immediately preceding digit. 21

(iii) We multiply that 2 by 2 and put 4 down as the

next previous digit. 421

(iv) We multiply that 4 by 2 and put it down thus 8421

(v) We multiply that 8 by 2 and get 16 as the

product. But this has two digits. We therefore

put the product. But this has two digits we

therefore put the 6 down immediately to the

left of the 8 and keep the 1 on hand to be

carried over to the left at the next step (as we

12

13.
always do in all multiplication e.g. of 69 × 2 =

138 and so on). 68421

1

(vi) We now multiply 6 by 2 get 12 as product, add

thereto the 1 (kept to be carried over from the

right at the last step), get 13 as the

consolidated product, put the 3 down and keep

the 1 on hand for carrying over to the left at

the next step. 368421

1 1

(vii) We then multiply 3 by 2 add the one carried

over from the right one, get 7 as the

consolidated product. But as this is a single

digit number with nothing to carry over to

the left, we put it down as our next

multiplicand. 7368421

1 1

((viii) and xviii) we follow this procedure

continually until we reach the 18th digit

counting leftwards from the right, when we

find that the whole decimal has begun to

repeat itself. We therefore put up the usual

recurring marks (dots) on the first and the last

digit of the answer (from betokening that the

whole of it is a Recurring Decimal) and stop

the multiplication there.

Our chart now reads as follows:

1

= .052631578/94736842i.

1 1 1111/ 1 11

B. Second Method

The second method is the method of division (instead of

multiplication) by the self-same “Ekādhikena Pūrvena” namely

2. And as division is the exact opposite of multiplication it

13

138 and so on). 68421

1

(vi) We now multiply 6 by 2 get 12 as product, add

thereto the 1 (kept to be carried over from the

right at the last step), get 13 as the

consolidated product, put the 3 down and keep

the 1 on hand for carrying over to the left at

the next step. 368421

1 1

(vii) We then multiply 3 by 2 add the one carried

over from the right one, get 7 as the

consolidated product. But as this is a single

digit number with nothing to carry over to

the left, we put it down as our next

multiplicand. 7368421

1 1

((viii) and xviii) we follow this procedure

continually until we reach the 18th digit

counting leftwards from the right, when we

find that the whole decimal has begun to

repeat itself. We therefore put up the usual

recurring marks (dots) on the first and the last

digit of the answer (from betokening that the

whole of it is a Recurring Decimal) and stop

the multiplication there.

Our chart now reads as follows:

1

= .052631578/94736842i.

1 1 1111/ 1 11

B. Second Method

The second method is the method of division (instead of

multiplication) by the self-same “Ekādhikena Pūrvena” namely

2. And as division is the exact opposite of multiplication it

13

14.
stands to reason that the operation of division should proceed

not from right to left (as in the case of multiplication as

expounded here in before) but in the exactly opposite direction;

i.e. from left to right. And such is actually found to be the case.

Its application and modus operandi are as follows:

(i) Dividing 1 (The first digit of the dividend) by

2, we see the quotient is zero and the

remainder is 1. We therefore set 0 down as the

first digit of the quotient and prefix the

remainder 1 to that very digit of the quotient

(as a sort of reverse-procedure to the carrying

to the left process used in multiplication) and

thus obtain 10 as our next dividend. 0

1

(ii) Dividing this 10 by 2, we get 5 as the second

digit of the quotient, and as there is no

remainder to be prefixed thereto we take up

that digit 5 itself as our next dividend. .05

1

(iii) So, the next quotient – digit is 2, and the

remainder is 1. We therefore put 2 down as the

third digit of the quotient and prefix the

remainder 1 to that quotient digit 2 and thus

have 12 as our next dividend. .052

1 1

(iv) This gives us 6 as quotient digit and zero as

remainder. So we set 6 down as the fourth

digit of the quotient, and as there is no

remainder to be prefixed thereto we take 6

itself as our next digit for division which gives

the next quotient digit as 3. .052631

1 1 1

(v) That gives us 1 and 1 as quotient and

remainder respectively. We therefore put 1

down as the 6th quotient digit prefix the 1

thereto and have 11 as our next dividend. .0526315

1 1 11

14

not from right to left (as in the case of multiplication as

expounded here in before) but in the exactly opposite direction;

i.e. from left to right. And such is actually found to be the case.

Its application and modus operandi are as follows:

(i) Dividing 1 (The first digit of the dividend) by

2, we see the quotient is zero and the

remainder is 1. We therefore set 0 down as the

first digit of the quotient and prefix the

remainder 1 to that very digit of the quotient

(as a sort of reverse-procedure to the carrying

to the left process used in multiplication) and

thus obtain 10 as our next dividend. 0

1

(ii) Dividing this 10 by 2, we get 5 as the second

digit of the quotient, and as there is no

remainder to be prefixed thereto we take up

that digit 5 itself as our next dividend. .05

1

(iii) So, the next quotient – digit is 2, and the

remainder is 1. We therefore put 2 down as the

third digit of the quotient and prefix the

remainder 1 to that quotient digit 2 and thus

have 12 as our next dividend. .052

1 1

(iv) This gives us 6 as quotient digit and zero as

remainder. So we set 6 down as the fourth

digit of the quotient, and as there is no

remainder to be prefixed thereto we take 6

itself as our next digit for division which gives

the next quotient digit as 3. .052631

1 1 1

(v) That gives us 1 and 1 as quotient and

remainder respectively. We therefore put 1

down as the 6th quotient digit prefix the 1

thereto and have 11 as our next dividend. .0526315

1 1 11

14

15.
(vi to xvii) Carrying this process of straight continuous

division by 2 we get 2 as the 17th quotient digit and 0 as

(xviii) Dividing this 2 by 2 are get 1 as 18th

quotient digit and 0 as remainder. But this is . 0 5 2 6 3 1 5 7 8

exactly what we began with. This means that 1 1 1111

the decimal begins to repeat itself from here. 9 4 7 3 6 8 4 2 i

So we stop the mental division process and 1 11

put down the usual recurring symbols (dots)

st th

on the 1 and 18 digit to show that the

whole of it is a circulating decimal.

Now if we are interested to find 1/29 the student should

note down that the last digit of the denominator is 9, but the

penultimate one is 2 and one more than that means 3. Likewise

for 1/49 the last digit of the denominator is 9 but penultimate is

4 and one more than that is 5 so for each number the

observation must be memorized by the student and remembered.

The following are to be noted

1. Student should find out the procedure to be followed.

The technique must be memorized. They feel it is

difficult and cumbersome and wastes their time and

repels them from mathematics.

2. “This problem can be solved by a calculator in a time

less than a second. Who in this modernized world take

so much strain to work and waste time over such simple

calculation?” asked several of the students.

3. According to many students the long division method

was itself more interesting.

The Second Sutra: Nikhilam Navataścaramam Daśatah

Now we proceed on to the next sutra “Nikhilam sutra” The sutra

reads “Nikhilam Navataścaramam Daśatah”, which literally

translated means: all from 9 and the last from 10”. We shall

15

division by 2 we get 2 as the 17th quotient digit and 0 as

(xviii) Dividing this 2 by 2 are get 1 as 18th

quotient digit and 0 as remainder. But this is . 0 5 2 6 3 1 5 7 8

exactly what we began with. This means that 1 1 1111

the decimal begins to repeat itself from here. 9 4 7 3 6 8 4 2 i

So we stop the mental division process and 1 11

put down the usual recurring symbols (dots)

st th

on the 1 and 18 digit to show that the

whole of it is a circulating decimal.

Now if we are interested to find 1/29 the student should

note down that the last digit of the denominator is 9, but the

penultimate one is 2 and one more than that means 3. Likewise

for 1/49 the last digit of the denominator is 9 but penultimate is

4 and one more than that is 5 so for each number the

observation must be memorized by the student and remembered.

The following are to be noted

1. Student should find out the procedure to be followed.

The technique must be memorized. They feel it is

difficult and cumbersome and wastes their time and

repels them from mathematics.

2. “This problem can be solved by a calculator in a time

less than a second. Who in this modernized world take

so much strain to work and waste time over such simple

calculation?” asked several of the students.

3. According to many students the long division method

was itself more interesting.

The Second Sutra: Nikhilam Navataścaramam Daśatah

Now we proceed on to the next sutra “Nikhilam sutra” The sutra

reads “Nikhilam Navataścaramam Daśatah”, which literally

translated means: all from 9 and the last from 10”. We shall

15

16.
presently give the detailed explanation presently of the meaning

and applications of this cryptical-sounding formula [51] and

then give details about the three corollaries.

He has given a very simple multiplication.

Suppose we have to multiply 9 by 7.

1. We should take, as base for our calculations

that power of 10 which is nearest to the

numbers to be multiplied. In this case 10 itself

is that power. (10)

9–1

7–3

6/ 3

2. Put the numbers 9 and 7 above and below on the left hand

side (as shown in the working alongside here on the right

hand side margin);

3. Subtract each of them from the base (10) and write down the

remainders (1 and 3) on the right hand side with a

connecting minus sign (–) between them, to show that the

numbers to be multiplied are both of them less than 10.

4. The product will have two parts, one on the left side and one

on the right. A vertical dividing line may be drawn for the

purpose of demarcation of the two parts.

5. Now, the left hand side digit can be arrived at in one of the 4

ways

a) Subtract the base 10 from the sum of the

given numbers (9 and 7 i.e. 16). And put

(16 – 10) i.e. 6 as the left hand part of the

answer 9 + 7 – 10 = 6

or b) Subtract the sum of two deficiencies (1 +

3 = 4) from the base (10) you get the same

answer (6) again 10 – 1 – 3 = 6

or c) Cross subtract deficiency 3 on the second

row from the original number 9 in the first

row. And you find that you have got (9 –

3) i.e. 6 again 9–3=6

or d) Cross subtract in the converse way (i.e. 1

from 7), and you get 6 again as the left

hand side portion of the required answer 7 – 1 = 6.

16

and applications of this cryptical-sounding formula [51] and

then give details about the three corollaries.

He has given a very simple multiplication.

Suppose we have to multiply 9 by 7.

1. We should take, as base for our calculations

that power of 10 which is nearest to the

numbers to be multiplied. In this case 10 itself

is that power. (10)

9–1

7–3

6/ 3

2. Put the numbers 9 and 7 above and below on the left hand

side (as shown in the working alongside here on the right

hand side margin);

3. Subtract each of them from the base (10) and write down the

remainders (1 and 3) on the right hand side with a

connecting minus sign (–) between them, to show that the

numbers to be multiplied are both of them less than 10.

4. The product will have two parts, one on the left side and one

on the right. A vertical dividing line may be drawn for the

purpose of demarcation of the two parts.

5. Now, the left hand side digit can be arrived at in one of the 4

ways

a) Subtract the base 10 from the sum of the

given numbers (9 and 7 i.e. 16). And put

(16 – 10) i.e. 6 as the left hand part of the

answer 9 + 7 – 10 = 6

or b) Subtract the sum of two deficiencies (1 +

3 = 4) from the base (10) you get the same

answer (6) again 10 – 1 – 3 = 6

or c) Cross subtract deficiency 3 on the second

row from the original number 9 in the first

row. And you find that you have got (9 –

3) i.e. 6 again 9–3=6

or d) Cross subtract in the converse way (i.e. 1

from 7), and you get 6 again as the left

hand side portion of the required answer 7 – 1 = 6.

16

17.
Note: This availability of the same result in several easy ways is

a very common feature of the Vedic system and is great

advantage and help to the student as it enables him to test and

verify the correctness of his answer step by step.

6. Now vertically multiply the two deficit figures (1 and 3).

The product is 3. And this is the right hand side portion

of the answer (10) 9 – 1

7. Thus 9 × 7 = 63. 7–3

6/3

This method holds good in all cases and is therefore capable

of infinite application. Now we proceed on to give the

interpretation and working of the ‘Nikhilam’ sutra and its three

The First Corollary

The first corollary naturally arising out of the Nikhilam Sutra

reads in English “whatever the extent of its deficiency lessen it

still further to that very extent, and also set up the square of that

This evidently deals with the squaring of the numbers. A few

elementary examples will suffice to make its meaning and

application clear:

Suppose one wants to square 9, the following are the

successive stages in our mental working.

(i) We would take up the nearest power of 10, i.e. 10 itself as

our base.

(ii) As 9 is 1 less than 10 we should decrease it still further by 1

and set 8 down as our left side portion of the answer

8/

(iii) And on the right hand we put down the square

of that deficiency 12 8/1.

(iv) Thus 92 = 81 9–1

9–1

8/1

17

a very common feature of the Vedic system and is great

advantage and help to the student as it enables him to test and

verify the correctness of his answer step by step.

6. Now vertically multiply the two deficit figures (1 and 3).

The product is 3. And this is the right hand side portion

of the answer (10) 9 – 1

7. Thus 9 × 7 = 63. 7–3

6/3

This method holds good in all cases and is therefore capable

of infinite application. Now we proceed on to give the

interpretation and working of the ‘Nikhilam’ sutra and its three

The First Corollary

The first corollary naturally arising out of the Nikhilam Sutra

reads in English “whatever the extent of its deficiency lessen it

still further to that very extent, and also set up the square of that

This evidently deals with the squaring of the numbers. A few

elementary examples will suffice to make its meaning and

application clear:

Suppose one wants to square 9, the following are the

successive stages in our mental working.

(i) We would take up the nearest power of 10, i.e. 10 itself as

our base.

(ii) As 9 is 1 less than 10 we should decrease it still further by 1

and set 8 down as our left side portion of the answer

8/

(iii) And on the right hand we put down the square

of that deficiency 12 8/1.

(iv) Thus 92 = 81 9–1

9–1

8/1

17

18.
Now we proceed on to give second corollary from (p.27, [51]).

The Second Corollary

The second corollary in applicable only to a special case under

the first corollary i.e. the squaring of numbers ending in 5 and

other cognate numbers. Its wording is exactly the same as that

of the sutra which we used at the outset for the conversion of

‘vulgar’ fractions into their recurring decimal equivalents. The

sutra now takes a totally different meaning and in fact relates to

a wholly different setup and context.

Its literal meaning is the same as before (i.e. by one more

than the previous one”) but it now relates to the squaring of

numbers ending in 5. For example we want to multiply 15. Here

the last digit is 5 and the “previous” one is 1. So one more than

that is 2.

Now sutra in this context tells us to multiply the previous

digit by one more than itself i.e. by 2. So the left hand side digit

is 1 × 2 and the right hand side is the vertical multiplication

product i.e. 25 as usual. 1 /5

2 / 25

Thus 152 = 1 × 2 / 25 = 2 / 25.

Now we proceed on to give the third corollary.

The Third Corollary

Then comes the third corollary to the Nikhilam sutra which

relates to a very special type of multiplication and which is not

frequently in requisition elsewhere but is often required in

mathematical astronomy etc. It relates to and provides for

multiplications where the multiplier digits consists entirely of

The procedure applicable in this case is therefore evidently

as follows:

i) Divide the multiplicand off by a vertical line into a right

hand portion consisting of as many digits as the multiplier;

18

The Second Corollary

The second corollary in applicable only to a special case under

the first corollary i.e. the squaring of numbers ending in 5 and

other cognate numbers. Its wording is exactly the same as that

of the sutra which we used at the outset for the conversion of

‘vulgar’ fractions into their recurring decimal equivalents. The

sutra now takes a totally different meaning and in fact relates to

a wholly different setup and context.

Its literal meaning is the same as before (i.e. by one more

than the previous one”) but it now relates to the squaring of

numbers ending in 5. For example we want to multiply 15. Here

the last digit is 5 and the “previous” one is 1. So one more than

that is 2.

Now sutra in this context tells us to multiply the previous

digit by one more than itself i.e. by 2. So the left hand side digit

is 1 × 2 and the right hand side is the vertical multiplication

product i.e. 25 as usual. 1 /5

2 / 25

Thus 152 = 1 × 2 / 25 = 2 / 25.

Now we proceed on to give the third corollary.

The Third Corollary

Then comes the third corollary to the Nikhilam sutra which

relates to a very special type of multiplication and which is not

frequently in requisition elsewhere but is often required in

mathematical astronomy etc. It relates to and provides for

multiplications where the multiplier digits consists entirely of

The procedure applicable in this case is therefore evidently

as follows:

i) Divide the multiplicand off by a vertical line into a right

hand portion consisting of as many digits as the multiplier;

18

19.
and subtract from the multiplicand one more than the whole

excess portion on the left. This gives us the left hand side

portion of the product;

or take the Ekanyuna and subtract therefrom the previous i.e.

the excess portion on the left; and

ii) Subtract the right hand side part of the multiplicand by the

Nikhilam rule. This will give you the right hand side of the

product.

The following example will make it clear:

43 × 9

4 : 3 :

:–5 : 3

3 : 8 :7

The Third Sutra: Ūrdhva Tiryagbhyām

Ūrdhva Tiryagbhyām sutra which is the General Formula

applicable to all cases of multiplication and will also be found

very useful later on in the division of a large number by another

large number.

The formula itself is very short and terse, consisting of only one

compound word and means “vertically and cross-wise.” The

applications of this brief and terse sutra are manifold.

A simple example will suffice to clarify the modus operandi

thereof. Suppose we have to multiply 12 by 13.

(i) We multiply the left hand most digit 1 of the 12

multiplicand vertically by the left hand most 13 .

digit 1 of the multiplier get their product 1 1:3 + 2:6 = 156

and set down as the left hand most part of

the answer;

(ii) We then multiply 1 and 3 and 1 and 2 crosswise add the two

get 5 as the sum and set it down as the middle part of the

answer; and

19

excess portion on the left. This gives us the left hand side

portion of the product;

or take the Ekanyuna and subtract therefrom the previous i.e.

the excess portion on the left; and

ii) Subtract the right hand side part of the multiplicand by the

Nikhilam rule. This will give you the right hand side of the

product.

The following example will make it clear:

43 × 9

4 : 3 :

:–5 : 3

3 : 8 :7

The Third Sutra: Ūrdhva Tiryagbhyām

Ūrdhva Tiryagbhyām sutra which is the General Formula

applicable to all cases of multiplication and will also be found

very useful later on in the division of a large number by another

large number.

The formula itself is very short and terse, consisting of only one

compound word and means “vertically and cross-wise.” The

applications of this brief and terse sutra are manifold.

A simple example will suffice to clarify the modus operandi

thereof. Suppose we have to multiply 12 by 13.

(i) We multiply the left hand most digit 1 of the 12

multiplicand vertically by the left hand most 13 .

digit 1 of the multiplier get their product 1 1:3 + 2:6 = 156

and set down as the left hand most part of

the answer;

(ii) We then multiply 1 and 3 and 1 and 2 crosswise add the two

get 5 as the sum and set it down as the middle part of the

answer; and

19

20.
(iii) We multiply 2 and 3 vertically get 6 as their product and put

it down as the last the right hand most part of the answer.

Thus 12 × 13 = 156.

The Fourth Sutra: Parāvartya Yojayet

The term Parāvartya Yojayet which means “Transpose and

Apply.” Here he claims that the Vedic system gave a number is

applications one of which is discussed here. The very

acceptance of the existence of polynomials and the consequent

remainder theorem during the Vedic times is a big question so

we don’t wish to give this application to those polynomials.

However the four steps given by them in the polynomial

division are given below: Divide x3 + 7x2 + 6x + 5 by x – 2.

i. x3 divided by x gives us x2 which is therefore the first term

of the quotient

x 3 + 7x 2 + 6x + 5

∴Q = x2 + ….

x−2

ii. x2 × –2 = –2x2 but we have 7x2 in the divident. This means

that we have to get 9x2 more. This must result from the

multiplication of x by 9x. Hence the 2nd term of the divisor

must be 9x

x 3 + 7x 2 + 6x + 5

∴ Q = x2 + 9x +….

x−2

iii. As for the third term we already have –2 × 9x = –18x. But

we have 6x in the dividend. We must therefore get an

additional 24x. Thus can only come in by the multiplication

of x by 24. This is the third term of the quotient.

∴ Q = x2 + 9x + 24

iv. Now the last term of the quotient multiplied by – 2 gives us

– 48. But the absolute term in the dividend is 5. We have

therefore to get an additional 53 from some where. But

there is no further term left in the dividend. This means that

the 53 will remain as the remainder ∴ Q = x2 + 9x + 24 and

R = 53.

20

it down as the last the right hand most part of the answer.

Thus 12 × 13 = 156.

The Fourth Sutra: Parāvartya Yojayet

The term Parāvartya Yojayet which means “Transpose and

Apply.” Here he claims that the Vedic system gave a number is

applications one of which is discussed here. The very

acceptance of the existence of polynomials and the consequent

remainder theorem during the Vedic times is a big question so

we don’t wish to give this application to those polynomials.

However the four steps given by them in the polynomial

division are given below: Divide x3 + 7x2 + 6x + 5 by x – 2.

i. x3 divided by x gives us x2 which is therefore the first term

of the quotient

x 3 + 7x 2 + 6x + 5

∴Q = x2 + ….

x−2

ii. x2 × –2 = –2x2 but we have 7x2 in the divident. This means

that we have to get 9x2 more. This must result from the

multiplication of x by 9x. Hence the 2nd term of the divisor

must be 9x

x 3 + 7x 2 + 6x + 5

∴ Q = x2 + 9x +….

x−2

iii. As for the third term we already have –2 × 9x = –18x. But

we have 6x in the dividend. We must therefore get an

additional 24x. Thus can only come in by the multiplication

of x by 24. This is the third term of the quotient.

∴ Q = x2 + 9x + 24

iv. Now the last term of the quotient multiplied by – 2 gives us

– 48. But the absolute term in the dividend is 5. We have

therefore to get an additional 53 from some where. But

there is no further term left in the dividend. This means that

the 53 will remain as the remainder ∴ Q = x2 + 9x + 24 and

R = 53.

20

21.
This method for a general degree is not given. However this

does not involve anything new. Further is it even possible that

the concept of polynomials existed during the period of Vedas

Now we give the 5th sutra.

The Fifth Sutra: Sūnyam Samyasamuccaye

We begin this section with an exposition of several special types

of equations which can be practically solved at sight with the

aid of a beautiful special sutra which reads Sūnyam

Samyasamuccaye and which in cryptic language which renders

its applicable to a large number of different cases. It merely says

“when the Samuccaya is the same that Samuccaya is zero i.e. it

should be equated to zero.”

Samuccaya is a technical term which has several meanings

in different contexts which we shall explain one at a time.

Samuccaya firstly means a term which occurs as a common

factor in all the terms concerned.

Samuccaya secondly means the product of independent

Samuccaya thirdly means the sum of the denominators of

two fractions having same numerical numerator.

Fourthly Samuccaya means combination or total.

Fifth meaning: With the same meaning i.e. total of the word

(Samuccaya) there is a fifth kind of application possible with

quadratic equations.

Sixth meaning – With the same sense (total of the word –

Samuccaya) but in a different application it comes in handy to

solve harder equations equated to zero.

Thus one has to imagine how the six shades of meanings

have been perceived by the Jagadguru Sankaracharya that too

from the Vedas when such types of equations had not even been

invented in the world at that point of time. However the

immediate application of the subsutra Vestnam is not given but

extensions of this sutra are discussed.

So we next go to the sixth sutra given by His Holiness

21

does not involve anything new. Further is it even possible that

the concept of polynomials existed during the period of Vedas

Now we give the 5th sutra.

The Fifth Sutra: Sūnyam Samyasamuccaye

We begin this section with an exposition of several special types

of equations which can be practically solved at sight with the

aid of a beautiful special sutra which reads Sūnyam

Samyasamuccaye and which in cryptic language which renders

its applicable to a large number of different cases. It merely says

“when the Samuccaya is the same that Samuccaya is zero i.e. it

should be equated to zero.”

Samuccaya is a technical term which has several meanings

in different contexts which we shall explain one at a time.

Samuccaya firstly means a term which occurs as a common

factor in all the terms concerned.

Samuccaya secondly means the product of independent

Samuccaya thirdly means the sum of the denominators of

two fractions having same numerical numerator.

Fourthly Samuccaya means combination or total.

Fifth meaning: With the same meaning i.e. total of the word

(Samuccaya) there is a fifth kind of application possible with

quadratic equations.

Sixth meaning – With the same sense (total of the word –

Samuccaya) but in a different application it comes in handy to

solve harder equations equated to zero.

Thus one has to imagine how the six shades of meanings

have been perceived by the Jagadguru Sankaracharya that too

from the Vedas when such types of equations had not even been

invented in the world at that point of time. However the

immediate application of the subsutra Vestnam is not given but

extensions of this sutra are discussed.

So we next go to the sixth sutra given by His Holiness

21

22.
The Sixth Sutra: Ānurūpye Śūnyamanyat

As said by Dani [32] we see the 6th sutra happens to be the

subsutra of the first sutra. Its mention is made in {pp. 51, 74,

249 and 286 of [51]}. The two small subsutras (i) Anurpyena

and (ii) Adayamadyenantyamantyena of the sutras 1 and 3

which mean “proportionately” and “the first by the first and the

last by the last”.

Here the later subsutra acquires a new and beautiful double

application and significance. It works out as follows:

i. Split the middle coefficient into two such parts so that the

ratio of the first coefficient to the first part is the same as the

ratio of that second part to the last coefficient. Thus in the

quadratic 2x2 + 5x + 2 the middle term 5 is split into two

such parts 4 and 1 so that the ratio of the first coefficient to

the first part of the middle coefficient i.e. 2 : 4 and the ratio

of the second part to the last coefficient i.e. 1 : 2 are the

same. Now this ratio i.e. x + 2 is one factor.

ii. And the second factor is obtained by dividing the first

coefficient of the quadratic by the first coefficient of the

factor already found and the last coefficient of the quadratic

by the last coefficient of that factor. In other words the

second binomial factor is obtained thus

2x 2 2

+ = 2x + 1.

x 2

Thus 2x2 + 5x + 2 = (x + 2) (2x + 1). This sutra has

Yavadunam Tavadunam to be its subsutra which the book

claims to have been used.

The Seventh Sutra: Sankalana Vyavakalanābhyām

Sankalana Vyavakalan process and the Adyamadya rule

together from the seventh sutra. The procedure adopted is one of

alternate destruction of the highest and the lowest powers by a

suitable multiplication of the coefficients and the addition or

subtraction of the multiples.

A concrete example will elucidate the process.

22

As said by Dani [32] we see the 6th sutra happens to be the

subsutra of the first sutra. Its mention is made in {pp. 51, 74,

249 and 286 of [51]}. The two small subsutras (i) Anurpyena

and (ii) Adayamadyenantyamantyena of the sutras 1 and 3

which mean “proportionately” and “the first by the first and the

last by the last”.

Here the later subsutra acquires a new and beautiful double

application and significance. It works out as follows:

i. Split the middle coefficient into two such parts so that the

ratio of the first coefficient to the first part is the same as the

ratio of that second part to the last coefficient. Thus in the

quadratic 2x2 + 5x + 2 the middle term 5 is split into two

such parts 4 and 1 so that the ratio of the first coefficient to

the first part of the middle coefficient i.e. 2 : 4 and the ratio

of the second part to the last coefficient i.e. 1 : 2 are the

same. Now this ratio i.e. x + 2 is one factor.

ii. And the second factor is obtained by dividing the first

coefficient of the quadratic by the first coefficient of the

factor already found and the last coefficient of the quadratic

by the last coefficient of that factor. In other words the

second binomial factor is obtained thus

2x 2 2

+ = 2x + 1.

x 2

Thus 2x2 + 5x + 2 = (x + 2) (2x + 1). This sutra has

Yavadunam Tavadunam to be its subsutra which the book

claims to have been used.

The Seventh Sutra: Sankalana Vyavakalanābhyām

Sankalana Vyavakalan process and the Adyamadya rule

together from the seventh sutra. The procedure adopted is one of

alternate destruction of the highest and the lowest powers by a

suitable multiplication of the coefficients and the addition or

subtraction of the multiples.

A concrete example will elucidate the process.

22

23.
Suppose we have to find the HCF (Highest Common factor)

of (x2 + 7x + 6) and x2 – 5x – 6.

x2 + 7x + 6 = (x + 1) (x + 6) and

x2 – 5x – 6 = (x + 1) ( x – 6)

∴ the HCF is x + 1

but where the sutra is deployed is not clear.

This has a subsutra Yavadunam Tavadunikrtya. However it

is not mentioned in chapter 10 of Vedic Mathematics [51].

The Eight Sutra: Puranāpuranābhyām

Puranāpuranābhyām means “by the completion or not

completion” of the square or the cube or forth power etc. But

when the very existence of polynomials, quadratic equations

etc. was not defined it is a miracle the Jagadguru could

contemplate of the completion of squares (quadratic) cubic and

forth degree equation. This has a subsutra Antyayor dasake’pi

use of which is not mentioned in that section.

The Ninth Sutra: Calanā kalanābhyām

The term (Calanā kalanābhyām) means differential calculus

according to Jagadguru Sankaracharya. It is mentioned in page

178 [51] that this topic will be dealt with later on. We have not

dealt with it as differential calculus not pertaining to our

analysis as it means only differential calculus and has no

mathematical formula or sutra value.

The Tenth Sutra: Yāvadūnam

Yāvadūnam Sutra (for cubing) is the tenth sutra. However no

modus operandi for elementary squaring and cubing is given in

this book [51]. It has a subsutra called Samuccayagunitah.

The Eleventh Sutra: Vyastisamastih Sutra

Vyastisamastih sutra teaches one how to use the average or

exact middle binomial for breaking the biquadratic down into a

23

of (x2 + 7x + 6) and x2 – 5x – 6.

x2 + 7x + 6 = (x + 1) (x + 6) and

x2 – 5x – 6 = (x + 1) ( x – 6)

∴ the HCF is x + 1

but where the sutra is deployed is not clear.

This has a subsutra Yavadunam Tavadunikrtya. However it

is not mentioned in chapter 10 of Vedic Mathematics [51].

The Eight Sutra: Puranāpuranābhyām

Puranāpuranābhyām means “by the completion or not

completion” of the square or the cube or forth power etc. But

when the very existence of polynomials, quadratic equations

etc. was not defined it is a miracle the Jagadguru could

contemplate of the completion of squares (quadratic) cubic and

forth degree equation. This has a subsutra Antyayor dasake’pi

use of which is not mentioned in that section.

The Ninth Sutra: Calanā kalanābhyām

The term (Calanā kalanābhyām) means differential calculus

according to Jagadguru Sankaracharya. It is mentioned in page

178 [51] that this topic will be dealt with later on. We have not

dealt with it as differential calculus not pertaining to our

analysis as it means only differential calculus and has no

mathematical formula or sutra value.

The Tenth Sutra: Yāvadūnam

Yāvadūnam Sutra (for cubing) is the tenth sutra. However no

modus operandi for elementary squaring and cubing is given in

this book [51]. It has a subsutra called Samuccayagunitah.

The Eleventh Sutra: Vyastisamastih Sutra

Vyastisamastih sutra teaches one how to use the average or

exact middle binomial for breaking the biquadratic down into a

23

24.
simple quadratic by the easy device of mutual cancellations of

the odd powers. However the modus operandi is missing.

The Twelfth Sutra: Śesānyankena Caramena

The sutra Śesānyankena Caramena means “The remainders by

the last digit”. For instance if one wants to find decimal value of

1/7. The remainders are 3, 2, 6, 4, 5 and 1. Multiplied by 7 these

remainders give successively 21, 14, 42, 28, 35 and 7. Ignoring

the left hand side digits we simply put down the last digit of

each product and we get 1/7 = .14 28 57!

Now this 12th sutra has a subsutra Vilokanam. Vilokanam

means “mere observation” He has given a few trivial examples

for the same.

Next we proceed on to study the 13th sutra

The Thirteen Sutra: Sopantyadvayamantyam

The sutra Sopantyadvayamantyam means “the ultimate and

twice the penultimate” which gives the answer immediately. No

mention is made about the immediate subsutra.

The illustration given by them.

1 1 1 1

+ = + .

(x + 2)(x + 3) (x + 2)(x + 4) (x + 2)(x + 5) (x + 3)(x + 4)

Here according to this sutra L + 2P (the last + twice the

= (x + 5) + 2 (x + 4) = 3x + 13 = 0

∴ x = −4 13 .

The proof of this is as follows.

1 1 1 1

+ = +

(x + 2)(x + 3) (x + 2)(x + 4) (x + 2)(x + 5) (x + 3)(x + 4)

1 1 1 1

∴ − = −

(x + 2)(x + 3) (x + 2)(x + 5) (x + 3)(x + 4) (x + 2)(x + 4)

1 ⎡ 2 ⎤ 1 ⎡ −1 ⎤

∴ =

(x + 2) ⎣ (x + 3)(x + 5) ⎦ (x + 4) ⎣ (x + 2)(x + 3) ⎥⎦

⎢ ⎥ ⎢

Removing the factors (x + 2) and (x + 3);

24

the odd powers. However the modus operandi is missing.

The Twelfth Sutra: Śesānyankena Caramena

The sutra Śesānyankena Caramena means “The remainders by

the last digit”. For instance if one wants to find decimal value of

1/7. The remainders are 3, 2, 6, 4, 5 and 1. Multiplied by 7 these

remainders give successively 21, 14, 42, 28, 35 and 7. Ignoring

the left hand side digits we simply put down the last digit of

each product and we get 1/7 = .14 28 57!

Now this 12th sutra has a subsutra Vilokanam. Vilokanam

means “mere observation” He has given a few trivial examples

for the same.

Next we proceed on to study the 13th sutra

The Thirteen Sutra: Sopantyadvayamantyam

The sutra Sopantyadvayamantyam means “the ultimate and

twice the penultimate” which gives the answer immediately. No

mention is made about the immediate subsutra.

The illustration given by them.

1 1 1 1

+ = + .

(x + 2)(x + 3) (x + 2)(x + 4) (x + 2)(x + 5) (x + 3)(x + 4)

Here according to this sutra L + 2P (the last + twice the

= (x + 5) + 2 (x + 4) = 3x + 13 = 0

∴ x = −4 13 .

The proof of this is as follows.

1 1 1 1

+ = +

(x + 2)(x + 3) (x + 2)(x + 4) (x + 2)(x + 5) (x + 3)(x + 4)

1 1 1 1

∴ − = −

(x + 2)(x + 3) (x + 2)(x + 5) (x + 3)(x + 4) (x + 2)(x + 4)

1 ⎡ 2 ⎤ 1 ⎡ −1 ⎤

∴ =

(x + 2) ⎣ (x + 3)(x + 5) ⎦ (x + 4) ⎣ (x + 2)(x + 3) ⎥⎦

⎢ ⎥ ⎢

Removing the factors (x + 2) and (x + 3);

24

25.
2 −1 2 −1

= i.e. =

x +5 x +4 L P

∴L + 2P = 0.

The General Algebraic Proof is as follows.

1 1 1 1

+ = +

AB AC AD BC

(where A, B, C and D are in A.P).

Let d be the common difference

1 1 1 1

+ = +

A(A + d) A(A + 2d) A(A + 3d) (A + d)(A + 2d)

1 1 1 1

∴ − = +

A(A + d) A(A + 3d) (A + d)(A + 2d) A(A + 2d)

1⎧ 2d ⎫ 1 ⎧ −d ⎫

∴ ⎨ ⎬= ⎨ ⎬.

A ⎩ (A + d)(A + 3d) ⎭ (A + 2d) ⎩ A(A + d) ⎭

Canceling the factors A (A + d) of the denominators and d of

the numerators:

2 −1

∴ = (p. 137)

A + 3d A + 2d

2 −1

In other words =

L P

∴ L + 2P = 0

It is a pity that all samples given by the book form a special

We now proceed on to present the 14th Sutra.

The Fourteenth Sutra: Ekanyūnena Pūrvena

The Ekanyūnena Pūrvena Sutra sounds as if it were the

converse of the Ekadhika Sutra. It actually relates and provides

for multiplications where the multiplier the digits consists

entirely of nines. The procedure applicable in this case is

therefore evidently as follows.

25

= i.e. =

x +5 x +4 L P

∴L + 2P = 0.

The General Algebraic Proof is as follows.

1 1 1 1

+ = +

AB AC AD BC

(where A, B, C and D are in A.P).

Let d be the common difference

1 1 1 1

+ = +

A(A + d) A(A + 2d) A(A + 3d) (A + d)(A + 2d)

1 1 1 1

∴ − = +

A(A + d) A(A + 3d) (A + d)(A + 2d) A(A + 2d)

1⎧ 2d ⎫ 1 ⎧ −d ⎫

∴ ⎨ ⎬= ⎨ ⎬.

A ⎩ (A + d)(A + 3d) ⎭ (A + 2d) ⎩ A(A + d) ⎭

Canceling the factors A (A + d) of the denominators and d of

the numerators:

2 −1

∴ = (p. 137)

A + 3d A + 2d

2 −1

In other words =

L P

∴ L + 2P = 0

It is a pity that all samples given by the book form a special

We now proceed on to present the 14th Sutra.

The Fourteenth Sutra: Ekanyūnena Pūrvena

The Ekanyūnena Pūrvena Sutra sounds as if it were the

converse of the Ekadhika Sutra. It actually relates and provides

for multiplications where the multiplier the digits consists

entirely of nines. The procedure applicable in this case is

therefore evidently as follows.

25

26.
For instance 43 × 9.

i. Divide the multiplicand off by a vertical line into a right

hand portion consisting of as many digits as the multiplier;

and subtract from the multiplicand one more than the whole

excess portion on the left. This gives us the left hand side

portion of the product or take the Ekanyuna and subtract it

from the previous i.e. the excess portion on the left and

ii. Subtract the right hand side part of the multiplicand by the

Nikhilam rule. This will give you the right hand side of the

product

43 × 9

4 : 3

:–5 : 3

3: 8 :7

This Ekanyuna Sutra can be utilized for the purpose of

postulating mental one-line answers to the question.

We now go to the 15th Sutra.

The Fifthteen Sutra: Gunitasamuccayah

Gunitasamuccayah rule i.e. the principle already explained with

regard to the Sc of the product being the same as the product of

the Sc of the factors.

Let us take a concrete example and see how this method

(p. 81) [51] can be made use of. Suppose we have to factorize x3

+ 6x2 + 11x + 6 and by some method, we know (x + 1) to be a

factor. We first use the corollary of the 3rd sutra viz.

Adayamadyena formula and thus mechanically put down x2 and

6 as the first and the last coefficients in the quotient; i.e. the

product of the remaining two binomial factors. But we know

already that the Sc of the given expression is 24 and as the Sc of

(x + 1) = 2 we therefore know that the Sc of the quotient must be

12. And as the first and the last digits thereof are already known

to be 1 and 6, their total is 7. And therefore the middle term

must be 12 – 7 = 5. So, the quotient x2 + 5x + 6.

This is a very simple and easy but absolutely certain and

effective process.

26

i. Divide the multiplicand off by a vertical line into a right

hand portion consisting of as many digits as the multiplier;

and subtract from the multiplicand one more than the whole

excess portion on the left. This gives us the left hand side

portion of the product or take the Ekanyuna and subtract it

from the previous i.e. the excess portion on the left and

ii. Subtract the right hand side part of the multiplicand by the

Nikhilam rule. This will give you the right hand side of the

product

43 × 9

4 : 3

:–5 : 3

3: 8 :7

This Ekanyuna Sutra can be utilized for the purpose of

postulating mental one-line answers to the question.

We now go to the 15th Sutra.

The Fifthteen Sutra: Gunitasamuccayah

Gunitasamuccayah rule i.e. the principle already explained with

regard to the Sc of the product being the same as the product of

the Sc of the factors.

Let us take a concrete example and see how this method

(p. 81) [51] can be made use of. Suppose we have to factorize x3

+ 6x2 + 11x + 6 and by some method, we know (x + 1) to be a

factor. We first use the corollary of the 3rd sutra viz.

Adayamadyena formula and thus mechanically put down x2 and

6 as the first and the last coefficients in the quotient; i.e. the

product of the remaining two binomial factors. But we know

already that the Sc of the given expression is 24 and as the Sc of

(x + 1) = 2 we therefore know that the Sc of the quotient must be

12. And as the first and the last digits thereof are already known

to be 1 and 6, their total is 7. And therefore the middle term

must be 12 – 7 = 5. So, the quotient x2 + 5x + 6.

This is a very simple and easy but absolutely certain and

effective process.

26

27.
As per pp. XVII to XVIII [51] of the book there is no

corollary to the 15th sutra i.e. to the sutra Gunitasamuccayah but

in p. 82 [51] of the same book they have given under the title

corollaries 8 methods of factorization which makes use of

mainly the Adyamadyena sutra. The interested reader can refer

pp. 82-85 of [51].

Now we proceed on to give the last sutra enlisted in page

XVIII of the book [51].

The Sixteen Sutra :Gunakasamuccayah.

“It means the product of the sum of the coefficients in the

factors is equal to the sum of the coefficients in the product”.

In symbols we may put this principle as follows:

Sc of the product = Product of the Sc (in factors).

For example

(x + 7) (x + 9) = x2 + 16 x + 63

and we observe

(1 + 7) (1 + 9) = 1 + 16 + 63 = 80.

Similarly in the case of cubics, biquadratics etc. the same rule

holds good.

For example

(x + 1) (x + 2) (x + 3) = x3 + 6x2 + 11 x + 6

2×3×4 = 1 + 6 + 11 + 6

= 24.

Thus if and when some factors are known this rule helps us to

fill in the gaps.

It will be found useful in the factorization of cubics,

biquadratics and will also be discussed in some other such

contexts later on.

In several places in the use of sutras the corollaries are

subsutras are dealt separately. One such instance is the subsutra

of the 11th sutra i.e., Vyastisamastih and its corollary viz.

Lapanasthapanabhyam finds its mention in page 77 [51] which

is cited verbatim here. The Lapana Sthapana subsutra however

removes the whole difficulty and makes the factorization of a

27

corollary to the 15th sutra i.e. to the sutra Gunitasamuccayah but

in p. 82 [51] of the same book they have given under the title

corollaries 8 methods of factorization which makes use of

mainly the Adyamadyena sutra. The interested reader can refer

pp. 82-85 of [51].

Now we proceed on to give the last sutra enlisted in page

XVIII of the book [51].

The Sixteen Sutra :Gunakasamuccayah.

“It means the product of the sum of the coefficients in the

factors is equal to the sum of the coefficients in the product”.

In symbols we may put this principle as follows:

Sc of the product = Product of the Sc (in factors).

For example

(x + 7) (x + 9) = x2 + 16 x + 63

and we observe

(1 + 7) (1 + 9) = 1 + 16 + 63 = 80.

Similarly in the case of cubics, biquadratics etc. the same rule

holds good.

For example

(x + 1) (x + 2) (x + 3) = x3 + 6x2 + 11 x + 6

2×3×4 = 1 + 6 + 11 + 6

= 24.

Thus if and when some factors are known this rule helps us to

fill in the gaps.

It will be found useful in the factorization of cubics,

biquadratics and will also be discussed in some other such

contexts later on.

In several places in the use of sutras the corollaries are

subsutras are dealt separately. One such instance is the subsutra

of the 11th sutra i.e., Vyastisamastih and its corollary viz.

Lapanasthapanabhyam finds its mention in page 77 [51] which

is cited verbatim here. The Lapana Sthapana subsutra however

removes the whole difficulty and makes the factorization of a

27

28.
quadratic of this type as easy and simple as that of the ordinary

quadratic already explained. The procedure is as follows:

Suppose we have to factorise the following long quadratic.

2x2 + 6y2 + 6z2 + 7xy + 11yz + 7zx

i. We first eliminate by putting z = 0 and retain only x and y

and factorise the resulting ordinary quadratic in x and y with

Adyam sutra which is only a corollary to the 3rd sutra viz.

Urdhva tryyagbhyam.

ii. We then similarly eliminate y and retain only x and z and

factorise the simple quadratic in x and z.

iii. With these two sets of factors before us we fill in the gaps

caused by our own deliberate elimination of z and y

respectively. And that gives us the real factors of the given

long expression. The procedure is an argumentative one and

is as follows:

If z = 0 then the given expression is 2x2 + 7xy + 6y2 = (x + 2y)

(2x + 3y). Similarly if y = 0 then 2x2 + 7xz + 3z2 = (x + 3z) (2x

+ z).

Filling in the gaps which we ourselves have created by leaving

out z and y, we get E = (x + 2y + 3z) (2x + 3y + z)

Note:

This Lopanasthapana method of alternate elimination and

retention will be found highly useful later on in finding HCF, in

solid geometry and in co-ordinate geometry of the straight line,

the hyperbola, the conjugate hyperbola, the asymptotes etc.

In the current system of mathematics we have two methods

which are used for finding the HCF of two or more given

expressions.

The first is by means of factorization which is not always

easy and the second is by a process of continuous division like

the method used in the G.C.M chapter of arithmetic. The latter

is a mechanical process and can therefore be applied in all

cases. But it is rather too mechanical and consequently long and

cumbrous.

28

quadratic already explained. The procedure is as follows:

Suppose we have to factorise the following long quadratic.

2x2 + 6y2 + 6z2 + 7xy + 11yz + 7zx

i. We first eliminate by putting z = 0 and retain only x and y

and factorise the resulting ordinary quadratic in x and y with

Adyam sutra which is only a corollary to the 3rd sutra viz.

Urdhva tryyagbhyam.

ii. We then similarly eliminate y and retain only x and z and

factorise the simple quadratic in x and z.

iii. With these two sets of factors before us we fill in the gaps

caused by our own deliberate elimination of z and y

respectively. And that gives us the real factors of the given

long expression. The procedure is an argumentative one and

is as follows:

If z = 0 then the given expression is 2x2 + 7xy + 6y2 = (x + 2y)

(2x + 3y). Similarly if y = 0 then 2x2 + 7xz + 3z2 = (x + 3z) (2x

+ z).

Filling in the gaps which we ourselves have created by leaving

out z and y, we get E = (x + 2y + 3z) (2x + 3y + z)

Note:

This Lopanasthapana method of alternate elimination and

retention will be found highly useful later on in finding HCF, in

solid geometry and in co-ordinate geometry of the straight line,

the hyperbola, the conjugate hyperbola, the asymptotes etc.

In the current system of mathematics we have two methods

which are used for finding the HCF of two or more given

expressions.

The first is by means of factorization which is not always

easy and the second is by a process of continuous division like

the method used in the G.C.M chapter of arithmetic. The latter

is a mechanical process and can therefore be applied in all

cases. But it is rather too mechanical and consequently long and

cumbrous.

28

29.
The Vedic methods provides a third method which is

applicable to all cases and is at the same time free from this

It is mainly an application of the subsutras or corollaries of

the 11th sutra viz. Vyastisamastih, the corollary Lapanasthapana

sutra the 7th sutra viz. Sankalana Vyavakalanabhyam process

and the subsutra of the 3rd sutra viz.

The procedure adopted is one of alternate destruction of the

highest and the lowest powers by a suitable multiplication of the

coefficients and the addition or subtraction of the multiples.

A concrete example will elucidate the process.

Suppose we have to find the H.C.F of x2 + 7x + 6 and x2 –

5x – 6

i. x2 + 7x + 6 = (x + 1) (x + 6) and x2 – 5x – 6 = (x + 1) (x –

6). HCF is (x + 1). This is the first method.

ii. The second method the GCM one is well-known and need

not be put down here.

iii. The third process of ‘Lopanasthapana’ i.e. of the

elimination and retention or alternate destruction of the

highest and the lowest powers is explained below.

Let E1 and E2 be the two expressions. Then for destroying the

highest power we should substract E2 from E1 and for

destroying the lowest one we should add the two. The chart is as

x 2 + 7x + 6 ⎫⎪ x 2 − 5x − 6 ⎫⎪

⎬ subtraction ⎬ addition

x 2 − 5x − 6 ⎪⎭ x 2 + 7x + 6 ⎪⎭

12x + 12 2x2 + 2x

12) 12x + 12 2x) 2x 2 + 2x

x +1 x +1

We then remove the common factor if any from each and we

find x + 1 staring us in the face i.e. x + 1 is the HCF. Two things

are to be noted importantly.

29

applicable to all cases and is at the same time free from this

It is mainly an application of the subsutras or corollaries of

the 11th sutra viz. Vyastisamastih, the corollary Lapanasthapana

sutra the 7th sutra viz. Sankalana Vyavakalanabhyam process

and the subsutra of the 3rd sutra viz.

The procedure adopted is one of alternate destruction of the

highest and the lowest powers by a suitable multiplication of the

coefficients and the addition or subtraction of the multiples.

A concrete example will elucidate the process.

Suppose we have to find the H.C.F of x2 + 7x + 6 and x2 –

5x – 6

i. x2 + 7x + 6 = (x + 1) (x + 6) and x2 – 5x – 6 = (x + 1) (x –

6). HCF is (x + 1). This is the first method.

ii. The second method the GCM one is well-known and need

not be put down here.

iii. The third process of ‘Lopanasthapana’ i.e. of the

elimination and retention or alternate destruction of the

highest and the lowest powers is explained below.

Let E1 and E2 be the two expressions. Then for destroying the

highest power we should substract E2 from E1 and for

destroying the lowest one we should add the two. The chart is as

x 2 + 7x + 6 ⎫⎪ x 2 − 5x − 6 ⎫⎪

⎬ subtraction ⎬ addition

x 2 − 5x − 6 ⎪⎭ x 2 + 7x + 6 ⎪⎭

12x + 12 2x2 + 2x

12) 12x + 12 2x) 2x 2 + 2x

x +1 x +1

We then remove the common factor if any from each and we

find x + 1 staring us in the face i.e. x + 1 is the HCF. Two things

are to be noted importantly.

29

30.
(1) We see that often the subsutras are not used under the main

sutra for which it is the subsutra or the corollary. This is the

main deviation from the usual mathematical principles of

theorem (sutra) and corollaries (subsutra).

(2) It cannot be easily compromised that a single sutra (a

Sanskrit word) can be mathematically interpreted in this

manner even by a stalwart in Sanskrit except the Jagadguru

Puri Sankaracharya.

We wind up the material from the book of Vedic Mathematics

and proceed on to give the opinion/views of great personalities

on Vedic Mathematics given by Jagadguru.

Since the notion of integral and differential calculus was not

in vogue in Vedic times, here we do not discuss about the

authenticated inventor, further we have not given the adaptation

of certain sutras in these fields. Further as most of the educated

experts felt that since the Jagadguru had obtained his degree

with mathematics as one of the subjects, most of the results

given in book on Vedic Mathematics were manipulated by His

30

sutra for which it is the subsutra or the corollary. This is the

main deviation from the usual mathematical principles of

theorem (sutra) and corollaries (subsutra).

(2) It cannot be easily compromised that a single sutra (a

Sanskrit word) can be mathematically interpreted in this

manner even by a stalwart in Sanskrit except the Jagadguru

Puri Sankaracharya.

We wind up the material from the book of Vedic Mathematics

and proceed on to give the opinion/views of great personalities

on Vedic Mathematics given by Jagadguru.

Since the notion of integral and differential calculus was not

in vogue in Vedic times, here we do not discuss about the

authenticated inventor, further we have not given the adaptation

of certain sutras in these fields. Further as most of the educated

experts felt that since the Jagadguru had obtained his degree

with mathematics as one of the subjects, most of the results

given in book on Vedic Mathematics were manipulated by His

30

31.
Chapter Two

ANALYSIS OF VEDIC MATHEMATICS BY

MATHEMATICIANS AND OTHERS

In this chapter we give the verbatim opinion of mathematicians

and experts about Vedic Mathematics in their articles, that have

appeared in the print media. The article of Prof. S.G. Dani,

School of Mathematics, Tata Institute of Fundamental Research

happen to give a complete analysis of Vedic Mathematics.

We have given his second article verbatim because we do

not want any bias or our opinion to play any role in our analysis

However we do not promise to discuss all the articles. Only

articles which show “How Vedic is Vedic Mathematics?” is

given for the perusal of the reader. We thank them for their

articles and quote them verbatim. The book on Vedic

Mathematics by Jagadguru Sankaracharya of Puri has been

translated into Tamil by Dr. V.S. Narasimhan, a Retired

Professor of an arts college and C. Mailvanan, M.Sc

Mathematics (Vidya Barathi state-level Vedic Mathematics

expert) in two volumes. The first edition appeared in 1998 and

the corrected second edition in 2003.

In Volume I of the Tamil book the introduction is as

follows: “Why was the name Vedic Mathematics given? On the

title “a trick in the name of Vedic Mathematics” though

professors in mathematics praise the sutras, they argue that the

title Vedic Mathematics is not well suited. According to them

31

ANALYSIS OF VEDIC MATHEMATICS BY

MATHEMATICIANS AND OTHERS

In this chapter we give the verbatim opinion of mathematicians

and experts about Vedic Mathematics in their articles, that have

appeared in the print media. The article of Prof. S.G. Dani,

School of Mathematics, Tata Institute of Fundamental Research

happen to give a complete analysis of Vedic Mathematics.

We have given his second article verbatim because we do

not want any bias or our opinion to play any role in our analysis

However we do not promise to discuss all the articles. Only

articles which show “How Vedic is Vedic Mathematics?” is

given for the perusal of the reader. We thank them for their

articles and quote them verbatim. The book on Vedic

Mathematics by Jagadguru Sankaracharya of Puri has been

translated into Tamil by Dr. V.S. Narasimhan, a Retired

Professor of an arts college and C. Mailvanan, M.Sc

Mathematics (Vidya Barathi state-level Vedic Mathematics

expert) in two volumes. The first edition appeared in 1998 and

the corrected second edition in 2003.

In Volume I of the Tamil book the introduction is as

follows: “Why was the name Vedic Mathematics given? On the

title “a trick in the name of Vedic Mathematics” though

professors in mathematics praise the sutras, they argue that the

title Vedic Mathematics is not well suited. According to them

31

32.
the sutras published by the Swamiji are not found anywhere in

the Vedas. Further the branches of mathematics like algebra and

calculus which he mentions, did not exist in the Vedic times. It

may help school students but only in certain problems where

shortcut methods can be used. The Exaggeration that, it can be

used in all branches of mathematics cannot be accepted.

Because it gives answers very fast it can be called “speed

maths”. He has welcomed suggestions and opinions of one and

It has also become pertinent to mention here that Jagadguru

Puri Sankaracharya for the first time visited the west in 1958.

He had been to America at the invitation of the Self Realization

Fellowship Los Angeles, to spread the message of Vedanta. The

book Vedic Metaphysics is a compilation of some of his

discourses delivered there. On 19 February 1958, he has given a

talk and demonstration to a small group of student

mathematicians at the California Institute of Technology,

Pasadena, California.

This talk finds its place in chapter XII of the book Vedic

Metaphysics pp. 156-196 [52] most of which has appeared later

on, in his book on Vedic Mathematics [51]. However some

experts were of the opinion, that if Swamiji would have

remained as Swamiji ‘or’ as a ‘mathematician’ it would have

been better. His intermingling and trying to look like both has

only brought him less recognition in both Mathematics and on

Vedanta. The views of Wing Commander Vishva Mohan

Tiwari, under the titles conventional to unconventionally

original speaks of Vedic Mathematics as follows:

“Vedic Mathematics mainly deals with various Vedic

mathematical formulas and their applications of carrying out

tedious and cumbersome arithmetical operations, and to a very

large extent executing them mentally. He feels that in this field

of mental arithmetical operations the works of the famous

mathematicians Trachtenberg and Lester Meyers (High speed

mathematics) are elementary compared to that of Jagadguruji …

An attempt has been made in this note to explain the

unconventional aspects of the methods. He then gives a very

brief sketch of first four chapters of Vedic Mathematics”.

32

the Vedas. Further the branches of mathematics like algebra and

calculus which he mentions, did not exist in the Vedic times. It

may help school students but only in certain problems where

shortcut methods can be used. The Exaggeration that, it can be

used in all branches of mathematics cannot be accepted.

Because it gives answers very fast it can be called “speed

maths”. He has welcomed suggestions and opinions of one and

It has also become pertinent to mention here that Jagadguru

Puri Sankaracharya for the first time visited the west in 1958.

He had been to America at the invitation of the Self Realization

Fellowship Los Angeles, to spread the message of Vedanta. The

book Vedic Metaphysics is a compilation of some of his

discourses delivered there. On 19 February 1958, he has given a

talk and demonstration to a small group of student

mathematicians at the California Institute of Technology,

Pasadena, California.

This talk finds its place in chapter XII of the book Vedic

Metaphysics pp. 156-196 [52] most of which has appeared later

on, in his book on Vedic Mathematics [51]. However some

experts were of the opinion, that if Swamiji would have

remained as Swamiji ‘or’ as a ‘mathematician’ it would have

been better. His intermingling and trying to look like both has

only brought him less recognition in both Mathematics and on

Vedanta. The views of Wing Commander Vishva Mohan

Tiwari, under the titles conventional to unconventionally

original speaks of Vedic Mathematics as follows:

“Vedic Mathematics mainly deals with various Vedic

mathematical formulas and their applications of carrying out

tedious and cumbersome arithmetical operations, and to a very

large extent executing them mentally. He feels that in this field

of mental arithmetical operations the works of the famous

mathematicians Trachtenberg and Lester Meyers (High speed

mathematics) are elementary compared to that of Jagadguruji …

An attempt has been made in this note to explain the

unconventional aspects of the methods. He then gives a very

brief sketch of first four chapters of Vedic Mathematics”.

32

33.
This chapter has seven sections; Section one gives the

verbatim analysis of Vedic Mathematics given by Prof. Dani in

his article in Frontline [31].

A list of eminent signatories asking people to stop this fraud

on our children is given verbatim in section two. Some views

given about the book both in favour of and against is given in

section three.

Section four gives the essay Vedas: Repositories of ancient

lore. “A rational approach to study ancient literature” an article

found in Current Science, volume 87, August 2004 is given in

Section five. Section Six gives the “Shanghai Rankings and

Indian Universities.” The final section gives conclusion derived

on Vedic Mathematics and calculation of Guru Tirthaji.

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

from Frontline

Views of Prof. S.G.Dani gave the authors a greater technical

insight into Vedic Mathematics because he has written 2 articles

in Frontline in 1993. He has analyzed the book extremely well

and we deeply acknowledge the services of professor S.G.Dani

to the educated community in general and school students in

particular. This section contains the verbatim views of Prof.

Dani that appeared in Frontline magazine. He has given a

marvelous analysis of the book Vedic Mathematics and has

daringly concluded.

“One would hardly have imagine that a book which is

transparently not from any ancient source or of any great

mathematical significance would one day be passed off as a

storehouse of some ancient mathematical treasure. It is high

time saner elements joined hands to educate people on the truth

of this so-called Vedic Mathematics and prevent the use of

public money and energy on its propagation, beyond the limited

extent that may be deserved, lest the intellectual and educational

life in the country should get vitiated further and result in wrong

attitudes to both history and mathematics, especially in the

coming generation.”

33

verbatim analysis of Vedic Mathematics given by Prof. Dani in

his article in Frontline [31].

A list of eminent signatories asking people to stop this fraud

on our children is given verbatim in section two. Some views

given about the book both in favour of and against is given in

section three.

Section four gives the essay Vedas: Repositories of ancient

lore. “A rational approach to study ancient literature” an article

found in Current Science, volume 87, August 2004 is given in

Section five. Section Six gives the “Shanghai Rankings and

Indian Universities.” The final section gives conclusion derived

on Vedic Mathematics and calculation of Guru Tirthaji.

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

from Frontline

Views of Prof. S.G.Dani gave the authors a greater technical

insight into Vedic Mathematics because he has written 2 articles

in Frontline in 1993. He has analyzed the book extremely well

and we deeply acknowledge the services of professor S.G.Dani

to the educated community in general and school students in

particular. This section contains the verbatim views of Prof.

Dani that appeared in Frontline magazine. He has given a

marvelous analysis of the book Vedic Mathematics and has

daringly concluded.

“One would hardly have imagine that a book which is

transparently not from any ancient source or of any great

mathematical significance would one day be passed off as a

storehouse of some ancient mathematical treasure. It is high

time saner elements joined hands to educate people on the truth

of this so-called Vedic Mathematics and prevent the use of

public money and energy on its propagation, beyond the limited

extent that may be deserved, lest the intellectual and educational

life in the country should get vitiated further and result in wrong

attitudes to both history and mathematics, especially in the

coming generation.”

33

34.
Myths and Reality: On ‘Vedic Mathematics’

S.G. Dani, School of Mathematics,

Tata Institute of Fundamental Research

An updated version of the 2-part article in Frontline, 22 Oct. and 5 Nov. 1993

We in India have good reasons to be proud of a rich heritage

in science, philosophy and culture in general, coming to us

down the ages. In mathematics, which is my own area of

specialization, the ancient Indians not only took great strides

long before the Greek advent, which is a standard reference

point in the Western historical perspective, but also enriched it

for a long period making in particular some very fundamental

contributions such as the place-value system for writing

numbers as we have today, introduction of zero and so on.

Further, the sustained development of mathematics in India in

the post-Greek period was indirectly instrumental in the revival

in Europe after “its dark ages”.

Notwithstanding the enviable background, lack of adequate

attention to academic pursuits over a prolonged period,

occasioned by several factors, together with about two centuries

of Macaulayan educational system, has unfortunately resulted,

on the one hand, in a lack of awareness of our historical role in

actual terms and, on the other, an empty sense of pride which is

more of an emotional reaction to the colonial domination rather

than an intellectual challenge. Together they provide a

convenient ground for extremist and misguided elements in

society to “reconstruct history” from nonexistent or concocted

source material to whip up popular euphoria.

That this anti-intellectual endeavour is counter-productive

in the long run and, more important, harmful to our image as a

mature society, is either not recognized or ignored in favour of

short-term considerations. Along with the obvious need to

accelerate the process of creating an awareness of our past

achievements, on the strength of authentic information, a more

urgent need has also arisen to confront and expose such baseless

constructs before it is too late. This is not merely a question of

setting the record straight. The motivated versions have a way

of corrupting the intellectual processes in society and

weakening their very foundations in the long run, which needs

to be prevented at all costs. The so-called “Vedic Mathematics”

34

S.G. Dani, School of Mathematics,

Tata Institute of Fundamental Research

An updated version of the 2-part article in Frontline, 22 Oct. and 5 Nov. 1993

We in India have good reasons to be proud of a rich heritage

in science, philosophy and culture in general, coming to us

down the ages. In mathematics, which is my own area of

specialization, the ancient Indians not only took great strides

long before the Greek advent, which is a standard reference

point in the Western historical perspective, but also enriched it

for a long period making in particular some very fundamental

contributions such as the place-value system for writing

numbers as we have today, introduction of zero and so on.

Further, the sustained development of mathematics in India in

the post-Greek period was indirectly instrumental in the revival

in Europe after “its dark ages”.

Notwithstanding the enviable background, lack of adequate

attention to academic pursuits over a prolonged period,

occasioned by several factors, together with about two centuries

of Macaulayan educational system, has unfortunately resulted,

on the one hand, in a lack of awareness of our historical role in

actual terms and, on the other, an empty sense of pride which is

more of an emotional reaction to the colonial domination rather

than an intellectual challenge. Together they provide a

convenient ground for extremist and misguided elements in

society to “reconstruct history” from nonexistent or concocted

source material to whip up popular euphoria.

That this anti-intellectual endeavour is counter-productive

in the long run and, more important, harmful to our image as a

mature society, is either not recognized or ignored in favour of

short-term considerations. Along with the obvious need to

accelerate the process of creating an awareness of our past

achievements, on the strength of authentic information, a more

urgent need has also arisen to confront and expose such baseless

constructs before it is too late. This is not merely a question of

setting the record straight. The motivated versions have a way

of corrupting the intellectual processes in society and

weakening their very foundations in the long run, which needs

to be prevented at all costs. The so-called “Vedic Mathematics”

34

35.
is a case in point. A book by that name written by Jagadguru

Swami Shri Bharati Krishna Tirthaji Maharaja (Tirthaji, 1965)

is at the centre of this pursuit, which has now acquired wide

following; Tirthaji was the Shankaracharya of Govardhan Math,

Puri, from 1925 until he passed away in 1960. The book was

published posthumously, but he had been carrying out a

campaign on the theme for a long time, apparently for several

decades, by means of lectures, blackboard demonstrations,

classes and so on. It has been known from the beginning that

there is no evidence of the contents of the book being of Vedic

origin; the Foreword to the book by the General Editor, Dr.

A.S.Agrawala, and an account of the genesis of the work written

by Manjula Trivedi, a disciple of the swamiji, make this clear

even before one gets to the text of the book. No one has come

up with any positive evidence subsequently either.

There has, however, been a persistent propaganda that the

material is from the Vedas. In the face of a false sense of

national pride associated with it and the neglect, on the part of

the knowledgeable, in countering the propaganda, even

educated and well meaning people have tended to accept it

uncritically. The vested interests have also involved politicians

in the propaganda process to gain state support. Several leaders

have lent support to the “Vedic Mathematics” over the years,

evidently in the belief of its being from ancient scriptures. In the

current environment, when a label as ancient seems to carry

considerable premium irrespective of its authenticity or merit,

the purveyors would have it going easy.

Large sums have been spent both by the Government and

several private agencies to support this “Vedic Mathematics”,

while authentic Vedic studies continue to be neglected. People,

especially children, are encouraged to learn and spread the

contents of the book, largely on the baseless premise of their

being from the Vedas. With missionary zeal several “devotees”

of this cause have striven to take the “message” around the

world; not surprisingly, they have even met with some success

in the West, not unlike some of the gurus and yogis peddling

their own versions of “Indian philosophy”. Several people are

also engaged in “research” in the new “Vedic Mathematics.”

35

Swami Shri Bharati Krishna Tirthaji Maharaja (Tirthaji, 1965)

is at the centre of this pursuit, which has now acquired wide

following; Tirthaji was the Shankaracharya of Govardhan Math,

Puri, from 1925 until he passed away in 1960. The book was

published posthumously, but he had been carrying out a

campaign on the theme for a long time, apparently for several

decades, by means of lectures, blackboard demonstrations,

classes and so on. It has been known from the beginning that

there is no evidence of the contents of the book being of Vedic

origin; the Foreword to the book by the General Editor, Dr.

A.S.Agrawala, and an account of the genesis of the work written

by Manjula Trivedi, a disciple of the swamiji, make this clear

even before one gets to the text of the book. No one has come

up with any positive evidence subsequently either.

There has, however, been a persistent propaganda that the

material is from the Vedas. In the face of a false sense of

national pride associated with it and the neglect, on the part of

the knowledgeable, in countering the propaganda, even

educated and well meaning people have tended to accept it

uncritically. The vested interests have also involved politicians

in the propaganda process to gain state support. Several leaders

have lent support to the “Vedic Mathematics” over the years,

evidently in the belief of its being from ancient scriptures. In the

current environment, when a label as ancient seems to carry

considerable premium irrespective of its authenticity or merit,

the purveyors would have it going easy.

Large sums have been spent both by the Government and

several private agencies to support this “Vedic Mathematics”,

while authentic Vedic studies continue to be neglected. People,

especially children, are encouraged to learn and spread the

contents of the book, largely on the baseless premise of their

being from the Vedas. With missionary zeal several “devotees”

of this cause have striven to take the “message” around the

world; not surprisingly, they have even met with some success

in the West, not unlike some of the gurus and yogis peddling

their own versions of “Indian philosophy”. Several people are

also engaged in “research” in the new “Vedic Mathematics.”

35

36.
To top it all, when in the early nineties the Uttar Pradesh

Government introduced “Vedic Mathematics” in school text

books, the contents of the swamiji’s book were treated as if they

were genuinely from the Vedas; this also naturally seems to

have led them to include a list of the swamiji’s sutras on one of

the opening pages (presumably for the students to learn them by

heart and recite!) and to accord the swamiji a place of honour in

the “brief history of Indian mathematics” described in the

beginning of the textbook, together with a chart, which cu-

riously has Srinivasa Ramanujan’s as the only other name from

the twentieth century!

For all their concern to inculcate a sense of national pride in

children, those responsible for this have not cared for the simple

fact that modern India has also produced several notable

mathematicians and built a worthwhile edifice in mathematics

(as also in many other areas). Harish Chandra’s work is held in

great esteem all over the world and several leading seats of

learning of our times pride themselves in having members

pursuing his ideas; (see, for instance, Langlands, 1993). Even

among those based in India, several like Syamdas

Mukhopadhyay, Ganesh Prasad, B.N.Prasad, K.Anand Rau,

T.Vijayaraghavan, S.S.Pillai, S.Minakshisundaram, Hansraj

Gupta, K.G.Ramanathan, B.S.Madhava Rao, V.V.Narlikar,

P.L.Bhatnagar and so on and also many living Indian

mathematicians have carved a niche for themselves on the

international mathematical scene (see Narasimhan, 1991).

Ignoring all this while introducing the swamiji’s name in the

“brief history” would inevitably create a warped perspective in

children’s minds, favouring gimmickry rather than professional

work. What does the swamiji’s “Vedic Mathematics” seek to do

and what does it achieve? In his preface of the book, grandly

titled” A Descriptive Prefatory Note on the astounding Wonders

of Ancient Indian Vedic Mathematics,” the swamiji tells us that

he strove from his childhood to study the Vedas critically “to

prove to ourselves (and to others) the correctness (or

otherwise)”of the “derivational meaning” of “Veda” that the”

Vedas should contain within themselves all the knowledge

needed by the mankind relating not only to spiritual matters but

also those usually described as purely ‘secular’, ‘temporal’ or

36

Government introduced “Vedic Mathematics” in school text

books, the contents of the swamiji’s book were treated as if they

were genuinely from the Vedas; this also naturally seems to

have led them to include a list of the swamiji’s sutras on one of

the opening pages (presumably for the students to learn them by

heart and recite!) and to accord the swamiji a place of honour in

the “brief history of Indian mathematics” described in the

beginning of the textbook, together with a chart, which cu-

riously has Srinivasa Ramanujan’s as the only other name from

the twentieth century!

For all their concern to inculcate a sense of national pride in

children, those responsible for this have not cared for the simple

fact that modern India has also produced several notable

mathematicians and built a worthwhile edifice in mathematics

(as also in many other areas). Harish Chandra’s work is held in

great esteem all over the world and several leading seats of

learning of our times pride themselves in having members

pursuing his ideas; (see, for instance, Langlands, 1993). Even

among those based in India, several like Syamdas

Mukhopadhyay, Ganesh Prasad, B.N.Prasad, K.Anand Rau,

T.Vijayaraghavan, S.S.Pillai, S.Minakshisundaram, Hansraj

Gupta, K.G.Ramanathan, B.S.Madhava Rao, V.V.Narlikar,

P.L.Bhatnagar and so on and also many living Indian

mathematicians have carved a niche for themselves on the

international mathematical scene (see Narasimhan, 1991).

Ignoring all this while introducing the swamiji’s name in the

“brief history” would inevitably create a warped perspective in

children’s minds, favouring gimmickry rather than professional

work. What does the swamiji’s “Vedic Mathematics” seek to do

and what does it achieve? In his preface of the book, grandly

titled” A Descriptive Prefatory Note on the astounding Wonders

of Ancient Indian Vedic Mathematics,” the swamiji tells us that

he strove from his childhood to study the Vedas critically “to

prove to ourselves (and to others) the correctness (or

otherwise)”of the “derivational meaning” of “Veda” that the”

Vedas should contain within themselves all the knowledge

needed by the mankind relating not only to spiritual matters but

also those usually described as purely ‘secular’, ‘temporal’ or

36

37.
‘worldly’; in other words, simply because of the meaning of the

word ‘Veda’, everything that is worth knowing is expected to be

contained in the vedas and the swamiji seeks to prove it to be

the case!

It may be worthwhile to point out here that there would be

room for starting such an enterprise with the word ‘science’! He

also describes how the “contemptuous or at best patronising ”

attitude of Orientalists, Indologists and so on strengthened his

determination to unravel the too-long-hidden mysteries of

philosophy and science contained in ancient India’s Vedic lore,

with the consequence that, “after eight years of concentrated

contemplation in forest solitude, we were at long last able to

recover the long lost keys which alone could unlock the portals

The mindset revealed in this can hardly be said to be

suitable in scientific and objective inquiry or pursuit of

knowledge, but perhaps one should not grudge it in someone

from a totally different milieu, if the outcome is positive. One

would have thought that with all the commitment and grit the

author would have come up with at least a few new things

which can be attributed to the Vedas, with solid evidence. This

would have made a worthwhile contribution to our

understanding of our heritage. Instead, all said and done there is

only the author’s certificate that “we were agreeably astonished

and intensely gratified to find that exceedingly though

mathematical problems can be easily and readily solved with the

help of these ultra-easy Vedic sutras (or mathematical

aphorisms) contained in the Parishishta (the appendix portion)

of the Atharva Veda in a few simple steps and by methods

which can be conscientiously described as mere ‘mental

arithmetic’ ”(paragraph 9 in the preface). That passing reference

to the Atharva Veda is all that is ever said by way of source

material for the contents. The sutras, incidentally, which

appeared later scattered in the book, are short phrases of just

about two to four words in Sanskrit, such as Ekadhikena

Purvena or Anurupye Shunyam Anyat. (There are 16 of them

and in addition there are 13 of what are called sub-sutras,

similar in nature to the sutras).

37

word ‘Veda’, everything that is worth knowing is expected to be

contained in the vedas and the swamiji seeks to prove it to be

the case!

It may be worthwhile to point out here that there would be

room for starting such an enterprise with the word ‘science’! He

also describes how the “contemptuous or at best patronising ”

attitude of Orientalists, Indologists and so on strengthened his

determination to unravel the too-long-hidden mysteries of

philosophy and science contained in ancient India’s Vedic lore,

with the consequence that, “after eight years of concentrated

contemplation in forest solitude, we were at long last able to

recover the long lost keys which alone could unlock the portals

The mindset revealed in this can hardly be said to be

suitable in scientific and objective inquiry or pursuit of

knowledge, but perhaps one should not grudge it in someone

from a totally different milieu, if the outcome is positive. One

would have thought that with all the commitment and grit the

author would have come up with at least a few new things

which can be attributed to the Vedas, with solid evidence. This

would have made a worthwhile contribution to our

understanding of our heritage. Instead, all said and done there is

only the author’s certificate that “we were agreeably astonished

and intensely gratified to find that exceedingly though

mathematical problems can be easily and readily solved with the

help of these ultra-easy Vedic sutras (or mathematical

aphorisms) contained in the Parishishta (the appendix portion)

of the Atharva Veda in a few simple steps and by methods

which can be conscientiously described as mere ‘mental

arithmetic’ ”(paragraph 9 in the preface). That passing reference

to the Atharva Veda is all that is ever said by way of source

material for the contents. The sutras, incidentally, which

appeared later scattered in the book, are short phrases of just

about two to four words in Sanskrit, such as Ekadhikena

Purvena or Anurupye Shunyam Anyat. (There are 16 of them

and in addition there are 13 of what are called sub-sutras,

similar in nature to the sutras).

37

38.
The first key question, which would occur to anyone, is

where are these sutras to be found in the Atharva Veda. One

does not mean this as a rhetorical question. Considering that at

the outset the author seemed set to send all doubting Thomases

packing, the least one would expect is that he would point out

where the sutras are, say in which part, stanza, page and so on,

especially since it is not a small article that is being referred to.

Not only has the author not cared to do so, but when

Prof.K.S.Shukla, a renowned scholar of ancient Indian

mathematics, met him in 1950, when the swamiji visited

Lucknow to give a blackboard demonstration of his “Vedic

Mathematics”, and requested him to point out the sutras in

question in the Parishishta of the Atharva Veda, of which he

even carried a copy (the standard version edited by G.M.Bolling

and J.Von Negelein), the swamiji is said to have told him that

the 16 sutra demonstrated by him were not in those Parishishtas

and that “they occurred in his own Parishishta and not any

other” (Shukla, 1980, or Shukla, 1991). What justification the

swamiji thought he had for introducing an appendix in the

Atharva Veda, the contents of which are nevertheless to be

viewed as from the Veda, is anybody’s guess. In any case, even

such a Parishishta, written by the swamiji, does not exist in the

form of a Sanskrit text.

Let us suppose for a moment that the author indeed found

the sutras in some manuscript of the Atharva Veda, which he

came across. Would he not then have preserved the manuscript?

Would he not have shown at least to some people where the

sutras are in the manuscript? Would he not have revealed to

some cherished students how to look for sutras with such

profound mathematical implications as he attributes to the sutras

in question, in that or other manuscripts that may be found?

While there is a specific mention in the write-up of Manjula

Trivedi, in the beginning of the book, about some 16volume

manuscript written by the swamiji having been lost in 1956,

there is no mention whatever (let alone any lamentation that

would be due in such an event) either in her write-up nor in the

swamiji’s preface about any original manuscript having been

lost. No one certainly has come forward with any information

received from the swamiji with regard to the other questions

38

where are these sutras to be found in the Atharva Veda. One

does not mean this as a rhetorical question. Considering that at

the outset the author seemed set to send all doubting Thomases

packing, the least one would expect is that he would point out

where the sutras are, say in which part, stanza, page and so on,

especially since it is not a small article that is being referred to.

Not only has the author not cared to do so, but when

Prof.K.S.Shukla, a renowned scholar of ancient Indian

mathematics, met him in 1950, when the swamiji visited

Lucknow to give a blackboard demonstration of his “Vedic

Mathematics”, and requested him to point out the sutras in

question in the Parishishta of the Atharva Veda, of which he

even carried a copy (the standard version edited by G.M.Bolling

and J.Von Negelein), the swamiji is said to have told him that

the 16 sutra demonstrated by him were not in those Parishishtas

and that “they occurred in his own Parishishta and not any

other” (Shukla, 1980, or Shukla, 1991). What justification the

swamiji thought he had for introducing an appendix in the

Atharva Veda, the contents of which are nevertheless to be

viewed as from the Veda, is anybody’s guess. In any case, even

such a Parishishta, written by the swamiji, does not exist in the

form of a Sanskrit text.

Let us suppose for a moment that the author indeed found

the sutras in some manuscript of the Atharva Veda, which he

came across. Would he not then have preserved the manuscript?

Would he not have shown at least to some people where the

sutras are in the manuscript? Would he not have revealed to

some cherished students how to look for sutras with such

profound mathematical implications as he attributes to the sutras

in question, in that or other manuscripts that may be found?

While there is a specific mention in the write-up of Manjula

Trivedi, in the beginning of the book, about some 16volume

manuscript written by the swamiji having been lost in 1956,

there is no mention whatever (let alone any lamentation that

would be due in such an event) either in her write-up nor in the

swamiji’s preface about any original manuscript having been

lost. No one certainly has come forward with any information

received from the swamiji with regard to the other questions

38

39.
above. It is to be noted that want of time could not be a factor in

any of this, since the swamiji kindly informs us in the preface

that “Ever since (i.e. since several decades ago), we have been

carrying on an incessant and strenuous campaign for the India-

wide diffusion of all this scientific knowledge”.

The only natural explanation is that there was no such

manuscript. It has in fact been mentioned by Agrawala in his

general editor’s foreword to the book, and also by Manjula

Trivedi in the short account of the genesis of the work, included

in the book together with a biographical sketch of the swamiji,

that the sutras do not appear in hitherto known Parishishtas. The

general editor also notes that the style of language of the sutras

“point to their discovery by Shri Swamiji himself ” (emphasis

added); the language style being contemporary can be

confirmed independently from other Sanskrit scholars as well.

The question why then the contents should be considered

‘Vedic’ apparently did not bother the general editor, as he

agreed with the author that “by definition” the Vedas should

contain all knowledge (never mind whether found in the 20th

century, or perhaps even later)! Manjula Trivedi, the disciple

has of course no problem with the sutras not being found in the

Vedas as she in fact says that they were actually reconstructed

by her beloved “Gurudeva,” on the basis of intuitive revelation

from material scattered here and there in the Atharva Veda, after

“assiduous research” and ‘Tapas’ for about eight years in the

forests surrounding Shringeri.” Isn’t that adequate to consider

them to be “Vedic”? Well, one can hardly argue with the

devout! There is a little problem as to why the Gurudeva him-

self did not say so (that the sutras were reconstructed) rather

than referring to them as sutras contained in the Parishishta of

the Atharva Veda, but we will have to let it pass. Anyway the

fact remains that she was aware that they could not actually be

located in what we lesser mortals consider to be the Atharva

Veda. The question of the source of the sutras is merely the first

that would come to mind, and already on that there is such a

muddle. Actually, even if the sutras were to be found, say in the

Atharva Veda or some other ancient text, that still leaves open

another fundamental question as to whether they mean or yield,

in some cognisable way, what the author claims; in other words,

39

any of this, since the swamiji kindly informs us in the preface

that “Ever since (i.e. since several decades ago), we have been

carrying on an incessant and strenuous campaign for the India-

wide diffusion of all this scientific knowledge”.

The only natural explanation is that there was no such

manuscript. It has in fact been mentioned by Agrawala in his

general editor’s foreword to the book, and also by Manjula

Trivedi in the short account of the genesis of the work, included

in the book together with a biographical sketch of the swamiji,

that the sutras do not appear in hitherto known Parishishtas. The

general editor also notes that the style of language of the sutras

“point to their discovery by Shri Swamiji himself ” (emphasis

added); the language style being contemporary can be

confirmed independently from other Sanskrit scholars as well.

The question why then the contents should be considered

‘Vedic’ apparently did not bother the general editor, as he

agreed with the author that “by definition” the Vedas should

contain all knowledge (never mind whether found in the 20th

century, or perhaps even later)! Manjula Trivedi, the disciple

has of course no problem with the sutras not being found in the

Vedas as she in fact says that they were actually reconstructed

by her beloved “Gurudeva,” on the basis of intuitive revelation

from material scattered here and there in the Atharva Veda, after

“assiduous research” and ‘Tapas’ for about eight years in the

forests surrounding Shringeri.” Isn’t that adequate to consider

them to be “Vedic”? Well, one can hardly argue with the

devout! There is a little problem as to why the Gurudeva him-

self did not say so (that the sutras were reconstructed) rather

than referring to them as sutras contained in the Parishishta of

the Atharva Veda, but we will have to let it pass. Anyway the

fact remains that she was aware that they could not actually be

located in what we lesser mortals consider to be the Atharva

Veda. The question of the source of the sutras is merely the first

that would come to mind, and already on that there is such a

muddle. Actually, even if the sutras were to be found, say in the

Atharva Veda or some other ancient text, that still leaves open

another fundamental question as to whether they mean or yield,

in some cognisable way, what the author claims; in other words,

39

40.
we would still need to know whether such a source really

contains the mathematics the swamiji deals with or merely the

phrases, may be in some quite different context. It is interesting

to consider the swamiji’s sutras in this light. One of them, for

instance, is Ekadhikena Purvena which literally just means “by

one more than the previous one.” In chapter I, the swamiji tells

us that it is a sutra for finding the digits in the decimal

expansion of numbers such as 1/19, and 1/29, where the

denominator is a number with 9 in the unit’s place; he goes on

to give a page-long description of the procedure to be followed,

whose only connection with the sutra is that it involves, in

particular, repeatedly multiplying by one more than the previous

one, namely 2, 3 and so on, respectively, the “previous one”

being the number before the unit’s place; the full procedure

involves a lot more by way of arranging the digits which can in

no way be read off from the phrase.

In Chapter II, we are told that the same sutra also means

that to find the square of a number like 25 and 35, (with five in

unit’s place) multiply the number of tens by one more than itself

and write 25 ahead of that; like 625, 1,225 and so on. The

phrase Ekanyunena Purvena which means “by one less than the

previous one” is however given to mean something which has

neither to do with decimal expansions nor with squaring of

numbers but concerns multiplying together two numbers, one of

which has 9 in all places (like 99,999, so on.)!

Allowing oneself such unlimited freedom of interpretation,

one can also interpret the same three-word phrase to mean also

many other things not only in mathematics but also in many

other subjects such as physics, chemistry, biology, economics,

sociology and politics. Consider, for instance, the following

“meaning”: the family size may be allowed to grow, at most, by

one more than the previous one. In this we have the family-

planning message of the 1960s; the “previous one” being the

couple, the prescription is that they should have no more than

three children. Thus the lal trikon (red triangle) formula may be

seen to be “from the Atharva Veda,” thanks to the swamiji’s

novel technique (with just a bit of credit to yours faithfully). If

you think the three children norm now outdated, there is no

need to despair. One can get the two-children or even the one-

40

contains the mathematics the swamiji deals with or merely the

phrases, may be in some quite different context. It is interesting

to consider the swamiji’s sutras in this light. One of them, for

instance, is Ekadhikena Purvena which literally just means “by

one more than the previous one.” In chapter I, the swamiji tells

us that it is a sutra for finding the digits in the decimal

expansion of numbers such as 1/19, and 1/29, where the

denominator is a number with 9 in the unit’s place; he goes on

to give a page-long description of the procedure to be followed,

whose only connection with the sutra is that it involves, in

particular, repeatedly multiplying by one more than the previous

one, namely 2, 3 and so on, respectively, the “previous one”

being the number before the unit’s place; the full procedure

involves a lot more by way of arranging the digits which can in

no way be read off from the phrase.

In Chapter II, we are told that the same sutra also means

that to find the square of a number like 25 and 35, (with five in

unit’s place) multiply the number of tens by one more than itself

and write 25 ahead of that; like 625, 1,225 and so on. The

phrase Ekanyunena Purvena which means “by one less than the

previous one” is however given to mean something which has

neither to do with decimal expansions nor with squaring of

numbers but concerns multiplying together two numbers, one of

which has 9 in all places (like 99,999, so on.)!

Allowing oneself such unlimited freedom of interpretation,

one can also interpret the same three-word phrase to mean also

many other things not only in mathematics but also in many

other subjects such as physics, chemistry, biology, economics,

sociology and politics. Consider, for instance, the following

“meaning”: the family size may be allowed to grow, at most, by

one more than the previous one. In this we have the family-

planning message of the 1960s; the “previous one” being the

couple, the prescription is that they should have no more than

three children. Thus the lal trikon (red triangle) formula may be

seen to be “from the Atharva Veda,” thanks to the swamiji’s

novel technique (with just a bit of credit to yours faithfully). If

you think the three children norm now outdated, there is no

need to despair. One can get the two-children or even the one-

40

41.
child formula also from the same sutra; count only the man as

the “previous one” (the woman is an outsider joining in

marriage, isn’t she) and in the growth of the family either count

only the children or include also the wife, depending on what

suits the desired formula!

Another sutra is Yavadunam, which means “as much less;”

a lifetime may not suffice to write down all the things such a

phrase could “mean,” in the spirit as above. There is even a sub-

sutra, Vilokanam (observation) and that is supposed to mean

various mathematical steps involving observation! In the same

vein one can actually suggest a single sutra adequate not only

for all of mathematics but many many subjects: Chintanam

It may be argued that there are, after all, ciphers which

convey more information than meets the eye. But the meaning

in those cases is either arrived at from the knowledge of the

deciphering code or deduced in one or other way using various

kinds of contexual information. Neither applies in the present

case. The sutras in the swamiji’s book are in reality mere names

for various steps to be followed in various contexts; the steps

themselves had to be known independently. In other words, the

mathematical step is not arrived at by understanding or

interpreting what are given as sutras; rather, sutras somewhat

suggestive of the meaning of the steps are attached to them like

names. It is like associating the ‘sutra’ VIBGYOR to the

sequence of colours in rainbow (which make up the white light).

Usage of words in Sanskrit, a language which the popular mind

unquestioningly associates with the distant past(!), lend the

contents a bit of antique finish!

An analysis of the mathematical contents of Tirthaji’s book

also shows that they cannot be from the Vedas. Though

unfortunately there is considerable ignorance about the subject,

mathematics from the Vedas is far from being an unexplored

area. Painstaking efforts have been made for well over a century

to study the original ancient texts from the point of view of

understanding the extent of mathematical knowledge in ancient

times. For instance, from the study of Vedic Samhitas and

Brahamanas it has been noted that they had the system of

counting progressing in multiples of 10 as we have today and

41

the “previous one” (the woman is an outsider joining in

marriage, isn’t she) and in the growth of the family either count

only the children or include also the wife, depending on what

suits the desired formula!

Another sutra is Yavadunam, which means “as much less;”

a lifetime may not suffice to write down all the things such a

phrase could “mean,” in the spirit as above. There is even a sub-

sutra, Vilokanam (observation) and that is supposed to mean

various mathematical steps involving observation! In the same

vein one can actually suggest a single sutra adequate not only

for all of mathematics but many many subjects: Chintanam

It may be argued that there are, after all, ciphers which

convey more information than meets the eye. But the meaning

in those cases is either arrived at from the knowledge of the

deciphering code or deduced in one or other way using various

kinds of contexual information. Neither applies in the present

case. The sutras in the swamiji’s book are in reality mere names

for various steps to be followed in various contexts; the steps

themselves had to be known independently. In other words, the

mathematical step is not arrived at by understanding or

interpreting what are given as sutras; rather, sutras somewhat

suggestive of the meaning of the steps are attached to them like

names. It is like associating the ‘sutra’ VIBGYOR to the

sequence of colours in rainbow (which make up the white light).

Usage of words in Sanskrit, a language which the popular mind

unquestioningly associates with the distant past(!), lend the

contents a bit of antique finish!

An analysis of the mathematical contents of Tirthaji’s book

also shows that they cannot be from the Vedas. Though

unfortunately there is considerable ignorance about the subject,

mathematics from the Vedas is far from being an unexplored

area. Painstaking efforts have been made for well over a century

to study the original ancient texts from the point of view of

understanding the extent of mathematical knowledge in ancient

times. For instance, from the study of Vedic Samhitas and

Brahamanas it has been noted that they had the system of

counting progressing in multiples of 10 as we have today and

41

42.
that they considered remarkably large numbers, even up to 14

digits, unlike other civilizations of those times. From the

Vedanga period there is in fact available a significant body of

mathematical literature in the form of Shulvasutras, from the

period between 800 bc and 500 bc, or perhaps even earlier,

some of which contain expositions of various mathematical

principles involved in construction of sacrificial ‘vedi’s needed

in performing’ yajna’s (see, for instance, Sen and Bag 1983).

Baudhyana Shulvasutra, the earliest of the extant

Shulvasutras, already contains, for instance, what is currently

known as Pythagoras’ Theorem (Sen and Bag, 1983, page 78,

1.12). It is the earliest known explicit statement of the theorem

in the general form (anywhere in the world) and precedes

Pythagoras by at least a few hundred years. The texts also show

a remarkable familiarity with many other facts from the so-

called Euclidean Geometry and it is clear that considerable use

was made of these, long before the Greeks formulated them.

The work of George Thibaut in the last century and that of

A.Burk around the turn of the century brought to the attention of

the world the significance of the mathematics of the

Shulvasutras. It has been followed up in this century by both

foreign and Indian historians of mathematics. It is this kind of

authentic work, and not some mumbo-jumbo that would

highlight our rich heritage. I would strongly recommend to the

reader to peruse the monograph, The Sulbasutras by S.N.Sen

and A.K.Bag (Sen and Bag, 1983), containing the original

sutras, their translation and a detailed commentary, which

includes a survey of a number of earlier works on the subject.

There are also several books on ancient Indian mathematics

from the Vedic period.

The contents of the swamiji’s book have practically nothing

in common with what is known of the mathematics from the

Vedic period or even with the subsequent rich tradition of

mathematics in India until the advent of the modern era;

incidentally, the descriptions of mathematical principles or

procedures in ancient mathematical texts are quite explicit and

not in terms of cryptic sutras. The very first chapter of the book

(as also chapters XXVI to XXVIII) involves the notion of

decimal fractions in an essential way. If the contents are to be

42

digits, unlike other civilizations of those times. From the

Vedanga period there is in fact available a significant body of

mathematical literature in the form of Shulvasutras, from the

period between 800 bc and 500 bc, or perhaps even earlier,

some of which contain expositions of various mathematical

principles involved in construction of sacrificial ‘vedi’s needed

in performing’ yajna’s (see, for instance, Sen and Bag 1983).

Baudhyana Shulvasutra, the earliest of the extant

Shulvasutras, already contains, for instance, what is currently

known as Pythagoras’ Theorem (Sen and Bag, 1983, page 78,

1.12). It is the earliest known explicit statement of the theorem

in the general form (anywhere in the world) and precedes

Pythagoras by at least a few hundred years. The texts also show

a remarkable familiarity with many other facts from the so-

called Euclidean Geometry and it is clear that considerable use

was made of these, long before the Greeks formulated them.

The work of George Thibaut in the last century and that of

A.Burk around the turn of the century brought to the attention of

the world the significance of the mathematics of the

Shulvasutras. It has been followed up in this century by both

foreign and Indian historians of mathematics. It is this kind of

authentic work, and not some mumbo-jumbo that would

highlight our rich heritage. I would strongly recommend to the

reader to peruse the monograph, The Sulbasutras by S.N.Sen

and A.K.Bag (Sen and Bag, 1983), containing the original

sutras, their translation and a detailed commentary, which

includes a survey of a number of earlier works on the subject.

There are also several books on ancient Indian mathematics

from the Vedic period.

The contents of the swamiji’s book have practically nothing

in common with what is known of the mathematics from the

Vedic period or even with the subsequent rich tradition of

mathematics in India until the advent of the modern era;

incidentally, the descriptions of mathematical principles or

procedures in ancient mathematical texts are quite explicit and

not in terms of cryptic sutras. The very first chapter of the book

(as also chapters XXVI to XXVIII) involves the notion of

decimal fractions in an essential way. If the contents are to be

42

43.
Vedic, there would have had to be a good deal of familiarity

with decimal fractions, even involving several digits, at that

time. It turns out that while the Shulvasutras make extensive use

of fractions in the usual form, nowhere is there any indication of

fractions in decimal form. It is inconceivable that such an

important notion would be left out, had it been known, from

what are really like users manuals of those times, produced at

different times over a prolonged period. Not only the

Shulvasutras and the earlier Vedic works, but even the works of

mathematicians such as Aryabhata, Brahmagupta and Bhaskara,

are not found to contain any decimal fractions. Is it possible that

none of them had access to some Vedic source that the swamiji

could lay his hands on (and still not describe it specifically)?

How far do we have to stretch our credulity?

The fact is that the use of decimal fractions started only in

the 16th century, propagated to a large extent by Francois Viete;

the use of the decimal point (separating the integer and the

fractional parts) itself, as a notation for the decimal

representation, began only towards the end of the century and

acquired popularity in the 17th century following their use in

John Napier’s logarithm tables (see, for instance, Boyer, 1968,

page 334).

Similarly, in chapter XXII the swamiji claims to give

“sutras relevant to successive differentiation, covering the

theorems of Leibnitz, Maclaurin, Taylor, etc. and a lot of other

material which is yet to be studied and decided on by the great

mathematicians of the present-day Western world;” it should

perhaps be mentioned before we proceed that the chapter does

not really deal with anything of the sort that would even

remotely justify such a grandiloquent announcement, but rather

deals with differentiation as an operation on polynomials, which

is a very special case reducing it all to elementary algebra

devoid of the very soul of calculus, as taught even at the college

Given the context, we shall leave Leibnitz and company

alone, but consider the notions of derivative and successive

differentiation. Did the notions exist in the Vedic times? While

certain elements preliminary to calculus have been found in the

works of Bhaskara II from the 12th century and later Indian

43

with decimal fractions, even involving several digits, at that

time. It turns out that while the Shulvasutras make extensive use

of fractions in the usual form, nowhere is there any indication of

fractions in decimal form. It is inconceivable that such an

important notion would be left out, had it been known, from

what are really like users manuals of those times, produced at

different times over a prolonged period. Not only the

Shulvasutras and the earlier Vedic works, but even the works of

mathematicians such as Aryabhata, Brahmagupta and Bhaskara,

are not found to contain any decimal fractions. Is it possible that

none of them had access to some Vedic source that the swamiji

could lay his hands on (and still not describe it specifically)?

How far do we have to stretch our credulity?

The fact is that the use of decimal fractions started only in

the 16th century, propagated to a large extent by Francois Viete;

the use of the decimal point (separating the integer and the

fractional parts) itself, as a notation for the decimal

representation, began only towards the end of the century and

acquired popularity in the 17th century following their use in

John Napier’s logarithm tables (see, for instance, Boyer, 1968,

page 334).

Similarly, in chapter XXII the swamiji claims to give

“sutras relevant to successive differentiation, covering the

theorems of Leibnitz, Maclaurin, Taylor, etc. and a lot of other

material which is yet to be studied and decided on by the great

mathematicians of the present-day Western world;” it should

perhaps be mentioned before we proceed that the chapter does

not really deal with anything of the sort that would even

remotely justify such a grandiloquent announcement, but rather

deals with differentiation as an operation on polynomials, which

is a very special case reducing it all to elementary algebra

devoid of the very soul of calculus, as taught even at the college

Given the context, we shall leave Leibnitz and company

alone, but consider the notions of derivative and successive

differentiation. Did the notions exist in the Vedic times? While

certain elements preliminary to calculus have been found in the

works of Bhaskara II from the 12th century and later Indian

43

44.
mathematicians in the pre-calculus era in international

mathematics, such crystallised notions as the derivative or the

integral were not known. Though a case may be made that the

developments here would have led to the discovery of calculus

in India, no historians of Indian mathematics would dream of

proposing that they actually had such a notion as the derivative,

let alone successive differentiation; the question here is not

about performing the operation on polynomials, but of the con-

cept. A similar comment applies with regard to integration, in

chapter XXIV. It should also be borne in mind that if calculus

were to be known in India in the early times, it would have been

acquired by foreigners as well, long before it actually came to

be discovered, as there was enough interaction between India

and the outside world.

If this is not enough, in Chapter XXXIX we learn that

analytic conics has an “important and predominating place for

itself in the Vedic system of mathematics,” and in Chapter XL

we find a whole list of subjects such as dynamics, statics,

hydrostatics, pneumatics and applied mathematics listed

alongside such elementary things as subtractions, ratios,

proportions and such money matters as interest and annuities

(!), discounts (!) to which we are assured, without going into

details, that the Vedic sutras can be applied. Need we comment

any further on this? The remaining chapters are mostly

elementary in content, on account of which one does not see

such marked incongruities in their respect. It has, however, been

pointed out by Shukla that many of the topics considered in the

book are alien to the pursuits of ancient Indian mathematicians,

not only form the Vedic period but until much later (Shukla,

1989 or Shukla, 1991). These include many such topics as

factorisation of algebraic expressions, HCF (highest common

factor) of algebraic expressions and various types of

simultaneous equations. The contents of the book are akin to

much later mathematics, mostly of the kind that appeared in

school books of our times or those of the swamiji’s youth, and it

is unthinkable, in the absence of any pressing evidence, that

they go back to the Vedic lore. The book really consists of a

compilation of tricks in elementary arithmetic and algebra, to be

applied in computations with numbers and polynomials. By a

44

mathematics, such crystallised notions as the derivative or the

integral were not known. Though a case may be made that the

developments here would have led to the discovery of calculus

in India, no historians of Indian mathematics would dream of

proposing that they actually had such a notion as the derivative,

let alone successive differentiation; the question here is not

about performing the operation on polynomials, but of the con-

cept. A similar comment applies with regard to integration, in

chapter XXIV. It should also be borne in mind that if calculus

were to be known in India in the early times, it would have been

acquired by foreigners as well, long before it actually came to

be discovered, as there was enough interaction between India

and the outside world.

If this is not enough, in Chapter XXXIX we learn that

analytic conics has an “important and predominating place for

itself in the Vedic system of mathematics,” and in Chapter XL

we find a whole list of subjects such as dynamics, statics,

hydrostatics, pneumatics and applied mathematics listed

alongside such elementary things as subtractions, ratios,

proportions and such money matters as interest and annuities

(!), discounts (!) to which we are assured, without going into

details, that the Vedic sutras can be applied. Need we comment

any further on this? The remaining chapters are mostly

elementary in content, on account of which one does not see

such marked incongruities in their respect. It has, however, been

pointed out by Shukla that many of the topics considered in the

book are alien to the pursuits of ancient Indian mathematicians,

not only form the Vedic period but until much later (Shukla,

1989 or Shukla, 1991). These include many such topics as

factorisation of algebraic expressions, HCF (highest common

factor) of algebraic expressions and various types of

simultaneous equations. The contents of the book are akin to

much later mathematics, mostly of the kind that appeared in

school books of our times or those of the swamiji’s youth, and it

is unthinkable, in the absence of any pressing evidence, that

they go back to the Vedic lore. The book really consists of a

compilation of tricks in elementary arithmetic and algebra, to be

applied in computations with numbers and polynomials. By a

44

45.
“trick” I do not mean a sleight of hand or something like that; in

a general sense a trick is a method or procedure which involves

observing and exploring some special features of a situation,

which generally tend to be overlooked; for example, the trick

described for finding the square of numbers like 15 and 25 with

5 in the unit’s place makes crucial use of the fact of 5 being half

of 10, the latter being the base in which the numbers are written.

Some of the tricks given in the book are quite interesting and

admittedly yield quicker solutions than by standard methods

(though the comparison made in the book are facetious and

misleading). They are of the kind that an intelligent hobbyist ex-

perimenting with numbers might be expected to come up with.

The tricks are, however, based on well-understood mathematical

principles and there is no mystery about them.

Of course to produce such a body of tricks, even using the

well-known is still a non-trivial task and there is a serious

question of how this came to be accomplished. It is sometimes

suggested that Tirthaji himself might have invented the tricks.

The fact that he had a M.A.degree in mathematics is notable in

this context. It is also possible that he might have learnt some of

the tricks from some elders during an early period in his life and

developed on them during those “eight years of concentrated

contemplation in forest solitude:” this would mean that they do

involve a certain element of tradition, though not to the absurd

extent that is claimed. These can, however, be viewed only as

possibilities and it would not be easy to settle these details. But

it is quite clear that the choice is only between alternatives

involving only the recent times.

It may be recalled here that there have also been other

instances of exposition and propagation of such faster methods

of computation applicable in various special situations (without

claims of their coming from ancient sources). Trachtenberg’s

Speed System (see Arther and McShane, 1965) and Lester

Meyers’ book, High-Speed Mathematics (Meyers, 1947) are

some well-known examples of this. Trachtenberg had even set

up an Institute in Germany to provide training in high-speed

mathematics. While the swamiji’s methods are independent of

these, for the most part they are similar in spirit.

45

a general sense a trick is a method or procedure which involves

observing and exploring some special features of a situation,

which generally tend to be overlooked; for example, the trick

described for finding the square of numbers like 15 and 25 with

5 in the unit’s place makes crucial use of the fact of 5 being half

of 10, the latter being the base in which the numbers are written.

Some of the tricks given in the book are quite interesting and

admittedly yield quicker solutions than by standard methods

(though the comparison made in the book are facetious and

misleading). They are of the kind that an intelligent hobbyist ex-

perimenting with numbers might be expected to come up with.

The tricks are, however, based on well-understood mathematical

principles and there is no mystery about them.

Of course to produce such a body of tricks, even using the

well-known is still a non-trivial task and there is a serious

question of how this came to be accomplished. It is sometimes

suggested that Tirthaji himself might have invented the tricks.

The fact that he had a M.A.degree in mathematics is notable in

this context. It is also possible that he might have learnt some of

the tricks from some elders during an early period in his life and

developed on them during those “eight years of concentrated

contemplation in forest solitude:” this would mean that they do

involve a certain element of tradition, though not to the absurd

extent that is claimed. These can, however, be viewed only as

possibilities and it would not be easy to settle these details. But

it is quite clear that the choice is only between alternatives

involving only the recent times.

It may be recalled here that there have also been other

instances of exposition and propagation of such faster methods

of computation applicable in various special situations (without

claims of their coming from ancient sources). Trachtenberg’s

Speed System (see Arther and McShane, 1965) and Lester

Meyers’ book, High-Speed Mathematics (Meyers, 1947) are

some well-known examples of this. Trachtenberg had even set

up an Institute in Germany to provide training in high-speed

mathematics. While the swamiji’s methods are independent of

these, for the most part they are similar in spirit.

45

46.
One may wonder why such methods are not commonly

adopted for practical purposes. One main point is that they turn

out to be quicker only for certain special classes of examples.

For a general example the amount of effort involved (for

instance, the count of the individual operations needed to be

performed with digits, in arriving at the final answer) is about

the same as required by the standard methods; in the swamiji’s

book, this is often concealed by not writing some of the steps

involved, viewing it as “mental arithmetic.” Using such

methods of fast arithmetic involves the ability or practice to

recognize various patterns which would simplify the

calculations. Without that, one would actually spend more time,

in first trying to recognize patterns and then working by rote

anyway, since in most cases it is not easy to find useful patterns.

People who in the course of their work have to do

computations as they arise, rather than choose the figures

suitably as in the demonstrations, would hardly find it

convenient to carry them out by employing umpteen different

ways depending on the particular case, as the methods of fast

arithmetic involve. It is more convenient to follow the standard

method, in which one has only to follow a set procedure to find

the answer, even though in some cases this might take more

time. Besides, equipment such as calculators and computers

have made it unnecessary to tax one’s mind with arithmetical

computations. Incidentally, the suggestion that this “Vedic

Mathematics” of the Shankaracharya could lead to improvement

in computers is totally fallacious, since the underlying

mathematical principles involved in it were by no means

unfamiliar in professional circles.

One of the factors causing people not to pay due attention to

the obvious questions about “Vedic Mathematics” seems to be

that they are overwhelmed by a sense of wonderment by the

tricks. The swamiji tells us in the preface how “the

educationists, the cream of the English educated section of the

people including highest officials (e.g. the high court judges, the

ministers etc.) and the general public as such were all highly

impressed; nay thrilled, wonder-struck and flabbergasted!” at

his demonstrations of the “Vedic Mathematics.” Sometimes one

comes across reports about similar thrilling demonstrations by

46

adopted for practical purposes. One main point is that they turn

out to be quicker only for certain special classes of examples.

For a general example the amount of effort involved (for

instance, the count of the individual operations needed to be

performed with digits, in arriving at the final answer) is about

the same as required by the standard methods; in the swamiji’s

book, this is often concealed by not writing some of the steps

involved, viewing it as “mental arithmetic.” Using such

methods of fast arithmetic involves the ability or practice to

recognize various patterns which would simplify the

calculations. Without that, one would actually spend more time,

in first trying to recognize patterns and then working by rote

anyway, since in most cases it is not easy to find useful patterns.

People who in the course of their work have to do

computations as they arise, rather than choose the figures

suitably as in the demonstrations, would hardly find it

convenient to carry them out by employing umpteen different

ways depending on the particular case, as the methods of fast

arithmetic involve. It is more convenient to follow the standard

method, in which one has only to follow a set procedure to find

the answer, even though in some cases this might take more

time. Besides, equipment such as calculators and computers

have made it unnecessary to tax one’s mind with arithmetical

computations. Incidentally, the suggestion that this “Vedic

Mathematics” of the Shankaracharya could lead to improvement

in computers is totally fallacious, since the underlying

mathematical principles involved in it were by no means

unfamiliar in professional circles.

One of the factors causing people not to pay due attention to

the obvious questions about “Vedic Mathematics” seems to be

that they are overwhelmed by a sense of wonderment by the

tricks. The swamiji tells us in the preface how “the

educationists, the cream of the English educated section of the

people including highest officials (e.g. the high court judges, the

ministers etc.) and the general public as such were all highly

impressed; nay thrilled, wonder-struck and flabbergasted!” at

his demonstrations of the “Vedic Mathematics.” Sometimes one

comes across reports about similar thrilling demonstrations by

46

47.
some of the present-day expositors of the subject. Though

inevitably they have to be taken with a pinch of salt, I do not

entirely doubt the truth of such reports. Since most people have

had a difficult time with their arithmetic at school and even

those who might have been fairly good would have lost touch,

the very fact of someone doing some computations rather fast

can make an impressive sight. This effect may be enhanced with

well-chosen examples, where some quicker methods are

Even in the case of general examples where the method

employed is not really more efficient than the standard one, the

computations might appear to be fast, since the demonstrator

would have a lot more practice than the people in the audience.

An objective assessment of the methods from the point of view

of overall use can only be made by comparing how many

individual calculations are involved in working out various

general examples, on an average, and in this respect the

methods of fast arithmetic do not show any marked advantage

which would offset the inconvenience indicated earlier. In any

case, it would be irrational to let the element of surprise

interfere in judging the issue of origin of “Vedic Mathematics”

or create a dreamy and false picture of its providing solutions to

all kinds of problems.

It should also be borne in mind that the book really deals

only with some middle and high school level mathematics; this

is true despite what appear to be chapters dealing with some

notions in calculus and coordinate geometry and the mention of

a few, little more advanced topics, in the book. The swamiji’s

claim that “there is no part of mathematics, pure or applied,

which is beyond their jurisdiction” is ludicrous. Mathematics

actually means a lot more than arithmetic of numbers and

algebra of polynomials; in fact multiplying big numbers

together, which a lot of people take for mathematics, is hardly

something a mathematician of today needs to engage himself in.

The mathematics of today concerns a great variety of objects

beyond the high school level, involving various kinds of ab-

stract objects generalising numbers, shapes, geometries,

measures and so on and several combinations of such structures,

various kinds of operations, often involving infinitely many en-

47

inevitably they have to be taken with a pinch of salt, I do not

entirely doubt the truth of such reports. Since most people have

had a difficult time with their arithmetic at school and even

those who might have been fairly good would have lost touch,

the very fact of someone doing some computations rather fast

can make an impressive sight. This effect may be enhanced with

well-chosen examples, where some quicker methods are

Even in the case of general examples where the method

employed is not really more efficient than the standard one, the

computations might appear to be fast, since the demonstrator

would have a lot more practice than the people in the audience.

An objective assessment of the methods from the point of view

of overall use can only be made by comparing how many

individual calculations are involved in working out various

general examples, on an average, and in this respect the

methods of fast arithmetic do not show any marked advantage

which would offset the inconvenience indicated earlier. In any

case, it would be irrational to let the element of surprise

interfere in judging the issue of origin of “Vedic Mathematics”

or create a dreamy and false picture of its providing solutions to

all kinds of problems.

It should also be borne in mind that the book really deals

only with some middle and high school level mathematics; this

is true despite what appear to be chapters dealing with some

notions in calculus and coordinate geometry and the mention of

a few, little more advanced topics, in the book. The swamiji’s

claim that “there is no part of mathematics, pure or applied,

which is beyond their jurisdiction” is ludicrous. Mathematics

actually means a lot more than arithmetic of numbers and

algebra of polynomials; in fact multiplying big numbers

together, which a lot of people take for mathematics, is hardly

something a mathematician of today needs to engage himself in.

The mathematics of today concerns a great variety of objects

beyond the high school level, involving various kinds of ab-

stract objects generalising numbers, shapes, geometries,

measures and so on and several combinations of such structures,

various kinds of operations, often involving infinitely many en-

47

48.
tities; this is not the case only about the frontiers of mathematics

but a whole lot of it, including many topics applied in physics,

engineering, medicine, finance and various other subjects.

Despite all its pretentious verbiage page after page, the

swamiji’s book offers nothing worthwhile in advanced

mathematics whether concretely or by way of insight. Modern

mathematics with its multitude of disciplines (group theory,

topology, algebraic geometry, harmonic analysis, ergodic

theory, combinatorial mathematics-to name just a few) would be

a long way from the level of the swamiji’s book. There are

occasionally reports of some “researchers” applying the

swamiji’s “Vedic Mathematics” to advanced problems such as

Kepler’s problem, but such work involves nothing more than

tinkering superficially with the topic, in the manner of the

swamiji’s treatment of calculus, and offers nothing of interest to

professionals in the area.

Even at the school level “Vedic Mathematics” deals only

with a small part and, more importantly, there too it concerns

itself with only one particular aspect, that of faster computation.

One of the main aims of mathematics education even at the

elementary level consists of developing familiarity with a

variety of concepts and their significance. Not only does the

approach of “Vedic Mathematics” not contribute anything

towards this crucial objective, but in fact might work to its

detriment, because of the undue emphasis laid on faster

computation. The swamiji’s assertion “8 months (or 12 months)

at an average rate of 2 or 3 hours per day should suffice for

completing the whole course of mathematical studies on these

Vedic lines instead of 15 or 20 years required according to the

existing systems of the Indian and also foreign universities,” is

patently absurd and hopefully nobody takes it seriously, even

among the activists in the area. It would work as a cruel joke if

some people choose to make such a substitution in respect of

their children.

It is often claimed that “Vedic Mathematics” is well-

appreciated in other countries, and even taught in some schools

in UK etc.. In the normal course one would not have the means

to examine such claims, especially since few details are

generally supplied while making the claims. Thanks to certain

48

but a whole lot of it, including many topics applied in physics,

engineering, medicine, finance and various other subjects.

Despite all its pretentious verbiage page after page, the

swamiji’s book offers nothing worthwhile in advanced

mathematics whether concretely or by way of insight. Modern

mathematics with its multitude of disciplines (group theory,

topology, algebraic geometry, harmonic analysis, ergodic

theory, combinatorial mathematics-to name just a few) would be

a long way from the level of the swamiji’s book. There are

occasionally reports of some “researchers” applying the

swamiji’s “Vedic Mathematics” to advanced problems such as

Kepler’s problem, but such work involves nothing more than

tinkering superficially with the topic, in the manner of the

swamiji’s treatment of calculus, and offers nothing of interest to

professionals in the area.

Even at the school level “Vedic Mathematics” deals only

with a small part and, more importantly, there too it concerns

itself with only one particular aspect, that of faster computation.

One of the main aims of mathematics education even at the

elementary level consists of developing familiarity with a

variety of concepts and their significance. Not only does the

approach of “Vedic Mathematics” not contribute anything

towards this crucial objective, but in fact might work to its

detriment, because of the undue emphasis laid on faster

computation. The swamiji’s assertion “8 months (or 12 months)

at an average rate of 2 or 3 hours per day should suffice for

completing the whole course of mathematical studies on these

Vedic lines instead of 15 or 20 years required according to the

existing systems of the Indian and also foreign universities,” is

patently absurd and hopefully nobody takes it seriously, even

among the activists in the area. It would work as a cruel joke if

some people choose to make such a substitution in respect of

their children.

It is often claimed that “Vedic Mathematics” is well-

appreciated in other countries, and even taught in some schools

in UK etc.. In the normal course one would not have the means

to examine such claims, especially since few details are

generally supplied while making the claims. Thanks to certain

48

49.
special circumstances I came to know a few things about the St.

James Independent School, London which I had seen quoted in

this context. The School is run by the ‘School of Economic

Science’ which is, according to a letter to me from Mr. James

Glover, the Head of Mathematics at the School, “engaged in the

practical study of Advaita philosophy”. The people who run it

have had substantial involvement with religious groups in India

over a long period. Thus in essence their adopting “Vedic

Mathematics” is much like a school in India run by a religious

group adopting it; that school being in London is beside the

point. (It may be noted here that while privately run schools in

India have limited freedom in choosing their curricula, it is not

the case in England). It would be interesting to look into the

background and motivation of other institutions about which

similar claims are made. At any rate, adoption by institutions

abroad is another propaganda feature, like being from ancient

source, and should not sway us.

It is not the contention here that the contents of the book are

not of any value. Indeed, some of the observations could be

used in teaching in schools. They are entertaining and could to

some extent enable children to enjoy mathematics. It would,

however, be more appropriate to use them as aids in teaching

the related concepts, rather than like a series of tricks of magic.

Ultimately, it is the understanding that is more important than

the transient excitement, By and large, however, such

pedagogical application has limited scope and needs to be made

with adequate caution, without being carried away by motivated

It is shocking to see the extent to which vested interests and

persons driven by guided notions are able to exploit the urge for

cultural self-assertion felt by the Indian psyche. One would

hardly have imagined that a book which is transparently not

from any ancient source or of any great mathematical

significance would one day be passed off as a storehouse of

some ancient mathematical treasure. It is high time saner

elements joined hands to educate people on the truth of this so-

called Vedic Mathematics and prevent the use of public money

and energy on its propagation, beyond the limited extent that

may be deserved, lest the intellectual and educational life in the

49

James Independent School, London which I had seen quoted in

this context. The School is run by the ‘School of Economic

Science’ which is, according to a letter to me from Mr. James

Glover, the Head of Mathematics at the School, “engaged in the

practical study of Advaita philosophy”. The people who run it

have had substantial involvement with religious groups in India

over a long period. Thus in essence their adopting “Vedic

Mathematics” is much like a school in India run by a religious

group adopting it; that school being in London is beside the

point. (It may be noted here that while privately run schools in

India have limited freedom in choosing their curricula, it is not

the case in England). It would be interesting to look into the

background and motivation of other institutions about which

similar claims are made. At any rate, adoption by institutions

abroad is another propaganda feature, like being from ancient

source, and should not sway us.

It is not the contention here that the contents of the book are

not of any value. Indeed, some of the observations could be

used in teaching in schools. They are entertaining and could to

some extent enable children to enjoy mathematics. It would,

however, be more appropriate to use them as aids in teaching

the related concepts, rather than like a series of tricks of magic.

Ultimately, it is the understanding that is more important than

the transient excitement, By and large, however, such

pedagogical application has limited scope and needs to be made

with adequate caution, without being carried away by motivated

It is shocking to see the extent to which vested interests and

persons driven by guided notions are able to exploit the urge for

cultural self-assertion felt by the Indian psyche. One would

hardly have imagined that a book which is transparently not

from any ancient source or of any great mathematical

significance would one day be passed off as a storehouse of

some ancient mathematical treasure. It is high time saner

elements joined hands to educate people on the truth of this so-

called Vedic Mathematics and prevent the use of public money

and energy on its propagation, beyond the limited extent that

may be deserved, lest the intellectual and educational life in the

49

50.
country should get vitiated further and result in wrong attitudes

to both history and mathematics, especially in the coming

References

[1] Ann Arther and Rudolph McShane, The Trachtenberg

Speed System of Basic Mathematics (English edition), Asia

Publishing House, New Delhi, 1965.

[2] Carl B. Boyer, A History of Mathematics, John Wiley and

Sons, 1968.

[3] R.P. Langlands, Harish-Chandra (11 October 1923 -16

October 1983), Current Science, Vol. 65: No. 12, 1993.

[4] Lester Meyers, High-Speed Mathematics, Van Nostrand,

New York, 1947.

[5] Raghavan Narasimhan, The Coming of Age of Mathematics

in India, Miscellanea Mathematica, 235–258, Springer-

Verlag, 1991.

[6] S.N. Sen and A.K. Bag, The Sulbasutras, Indian National

Science Academy, New Delhi, 1983. .

[7] K.S. Shukla, Vedic Mathematics — the illusive title of

Swamiji’s book, Mathematical Education, Vol 5: No. 3,

January-March 1989.

[8] K.S. Shukla, Mathematics — The Deceptive Title of

Swamiji’s Book, in Issues in Vedic Mathematics, (ed:

H.C.Khare), Rashtriya Veda Vidya Prakashan and Motilal

Banarasidass Publ., 1991.

[9] Shri Bharati Krishna Tirthaji, Vedic Mathematics, Motilal

Banarasidass, New Delhi, 1965.

2.2 Neither Vedic Nor Mathematics

We, the undersigned, are deeply concerned by the continuing

attempts to thrust the so-called `Vedic Mathematics' on the

school curriculum by the NCERT (National Council of

Educational Research and Training).

As has been pointed out earlier on several occasions,

the so-called ‘Vedic Mathematics’ is neither ‘Vedic’ nor can it

be dignified by the name of mathematics. ‘Vedic Mathematics’,

50

to both history and mathematics, especially in the coming

References

[1] Ann Arther and Rudolph McShane, The Trachtenberg

Speed System of Basic Mathematics (English edition), Asia

Publishing House, New Delhi, 1965.

[2] Carl B. Boyer, A History of Mathematics, John Wiley and

Sons, 1968.

[3] R.P. Langlands, Harish-Chandra (11 October 1923 -16

October 1983), Current Science, Vol. 65: No. 12, 1993.

[4] Lester Meyers, High-Speed Mathematics, Van Nostrand,

New York, 1947.

[5] Raghavan Narasimhan, The Coming of Age of Mathematics

in India, Miscellanea Mathematica, 235–258, Springer-

Verlag, 1991.

[6] S.N. Sen and A.K. Bag, The Sulbasutras, Indian National

Science Academy, New Delhi, 1983. .

[7] K.S. Shukla, Vedic Mathematics — the illusive title of

Swamiji’s book, Mathematical Education, Vol 5: No. 3,

January-March 1989.

[8] K.S. Shukla, Mathematics — The Deceptive Title of

Swamiji’s Book, in Issues in Vedic Mathematics, (ed:

H.C.Khare), Rashtriya Veda Vidya Prakashan and Motilal

Banarasidass Publ., 1991.

[9] Shri Bharati Krishna Tirthaji, Vedic Mathematics, Motilal

Banarasidass, New Delhi, 1965.

2.2 Neither Vedic Nor Mathematics

We, the undersigned, are deeply concerned by the continuing

attempts to thrust the so-called `Vedic Mathematics' on the

school curriculum by the NCERT (National Council of

Educational Research and Training).

As has been pointed out earlier on several occasions,

the so-called ‘Vedic Mathematics’ is neither ‘Vedic’ nor can it

be dignified by the name of mathematics. ‘Vedic Mathematics’,

50

51.
as is well-known, originated with a book of the same name by a

former Sankaracharya of Puri (the late Jagadguru Swami Shri

Bharati Krishna Tirthaji Maharaj) published posthumously in

1965. The book assembled a set of tricks in elementary

arithmetic and algebra to be applied in performing computations

with numbers and polynomials. As is pointed out even in the

foreword to the book by the General Editor, Dr. A.S. Agarwala,

the aphorisms in Sanskrit to be found in the book have nothing

to do with the Vedas. Nor are these aphorisms to be found in the

genuine Vedic literature.

The term “Vedic Mathematics” is therefore entirely

misleading and factually incorrect. Further, it is clear from the

notation used in the arithmetical tricks in the book that the

methods used in this text have nothing to do with the

arithmetical techniques of antiquity. Many of the Sanskrit

aphorisms in the book are totally cryptic (ancient Indian

mathematical writing was anything but cryptic) and often so

generalize to be devoid of any specific mathematical meaning.

There are several authoritative texts on the mathematics of

Vedic times that could be used in part to teach an authoritative

and correct account of ancient Indian mathematics but this book

clearly cannot be used for any such purpose. The teaching of

mathematics involves both the teaching of the basic concepts of

the subject as well as methods of mathematical computation.

The so-called “Vedic Mathematics” is entirely inadequate to

this task considering that it is largely made up of tricks to do

some elementary arithmetic computations. Many of these can be

far more easily performed on a simple computer or even an

advanced calculator.

The book “Vedic Mathematics” essentially deals with

arithmetic of the middle and high-school level. Its claims that

“there is no part of mathematics, pure or applied, which is

beyond their jurisdiction” is simply ridiculous. In an era when

the content of mathematics teaching has to be carefully designed

to keep pace with the general explosion of knowledge and the

needs of other modern professions that use mathematical

techniques, the imposition of “Vedic Mathematics” will be

nothing short of calamitous.

51

former Sankaracharya of Puri (the late Jagadguru Swami Shri

Bharati Krishna Tirthaji Maharaj) published posthumously in

1965. The book assembled a set of tricks in elementary

arithmetic and algebra to be applied in performing computations

with numbers and polynomials. As is pointed out even in the

foreword to the book by the General Editor, Dr. A.S. Agarwala,

the aphorisms in Sanskrit to be found in the book have nothing

to do with the Vedas. Nor are these aphorisms to be found in the

genuine Vedic literature.

The term “Vedic Mathematics” is therefore entirely

misleading and factually incorrect. Further, it is clear from the

notation used in the arithmetical tricks in the book that the

methods used in this text have nothing to do with the

arithmetical techniques of antiquity. Many of the Sanskrit

aphorisms in the book are totally cryptic (ancient Indian

mathematical writing was anything but cryptic) and often so

generalize to be devoid of any specific mathematical meaning.

There are several authoritative texts on the mathematics of

Vedic times that could be used in part to teach an authoritative

and correct account of ancient Indian mathematics but this book

clearly cannot be used for any such purpose. The teaching of

mathematics involves both the teaching of the basic concepts of

the subject as well as methods of mathematical computation.

The so-called “Vedic Mathematics” is entirely inadequate to

this task considering that it is largely made up of tricks to do

some elementary arithmetic computations. Many of these can be

far more easily performed on a simple computer or even an

advanced calculator.

The book “Vedic Mathematics” essentially deals with

arithmetic of the middle and high-school level. Its claims that

“there is no part of mathematics, pure or applied, which is

beyond their jurisdiction” is simply ridiculous. In an era when

the content of mathematics teaching has to be carefully designed

to keep pace with the general explosion of knowledge and the

needs of other modern professions that use mathematical

techniques, the imposition of “Vedic Mathematics” will be

nothing short of calamitous.

51