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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.
VEDIC MATHEMATICS -
‘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
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
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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
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
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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. […]
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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
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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
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
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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.
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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”
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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
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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
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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
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
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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
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
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;
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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‘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).
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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
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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,
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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-
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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
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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
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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
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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
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“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.
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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
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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-
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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
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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
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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’,
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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.
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