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This PDf contains Sixteen simple Mathematical Formulae i.e. the Sixteen Sutras from the Vedas.
1.
VEDIC MATHEMATICS
OR
Sixteen simple Mathematical
Formulae from the Vedas
(For OneUne Answers to all Mathematid Problems)
JAGADGURU
SWAMI SRI BHARATI KRSNA TIRTHAJI MAHARAJA,
~ A N K A R ~ C A R Y AOF OOVARDHANA MATHA, PURI
General Editor
DR. V.S. A G R A W U
1 The Author I
Jagadguru S a i ~ k a r i c i r ~ a MOTILAL BANARSIDASS
Sri Bharati K ~ s n aTirtha MahBrSjtja
(18841960)
Delhi :: Varanasi :: Patna
I
2.
PUBLICATION ANNOUNCEMENT
I have great pleasure in associating myself with the publi
cation of the book Vedic Mathematics or 'Sixteen Simple Mathe
matical Formulae,' by Jagadguru Swami Bharati Krishna
Tirtha, Shankaracharya of Govardhana Pitha. I t was long
awaited by his disciples. Shrimati Manjula Devi, sole inheriter
of Swamiji's, right, entered into am agreement with the Banaras
Hindu University to publieh i t and the same is now being
done in the Nepal Endowment Hindu Vishvavidyalaya Sanskrit
Granthamala.
I feel gratefuk to all those who have worked for it. Shri
Arvind N. Mafatlal business magnate of Bombay and a devotee
of Swamiji has taken interest in the publication of the work.
He has taken the trouble of being personally present in this
ceremony of publication (Prakashana Utsava). He has given
expression to his deep devotion to Shri Shankaracharyaji by
consenting t o found a chair a t the Banaras Hindu University
by the name of Shri Jagadguru Bharati Krishna Tirtha Shan
karacharya Chair of Vedic Studies for which he is making a
magnificent endowment. As ViceChancellor of this University
I accept the donation and offer my heartfelt thanks to him
for his generosity.
QMOT1LA.L B A N A R S I D A S S N. H. BHAGWATI
Indologlcal Publ~shers& Booksellers
Head Office : 41U.A., Bungalow Road, Delhi110007 Date 27365 VimChancellor
Braaches : 1. Chowk, Varanasc1 (u.P.) Banaras Hindu University
2. Ashok Rajpath. Patna4 (snraa)
First Edition : Varonasi, 1965
Reprint : Delhi, 1970, 1971 1975, 1978,1981
:Rs. 55 (Cloth)
Rs. 40 (Paper)
Printed in India
By Shantilal Jain at Shri Jainendra Press
A45 Phase I adu us trial Area Naraina. 'New Delhi110 028
Pubished by hlarendra ~ r a k d hJain, for Motilal Banarsidass,
Bungalow Road, Jawahar Nagar. Delhi110007.
3.
GENERAL EDITOR'S FOREWORD
The work entitled VEDIC MATHEMATICS or 'Sixteerr
Simple Mathematical Formule from the Vedas' was written by
His Holiness Jagadguru QalikarBcLya 8ri BhLrati Krgqa
Tirthaji Mahiiriija of Govardhana Matha, Puri (18841960).
It forms a class by itself not pragmatically conceived and worked
out as in the caee of other scientific works, but the result of the
intuitional visualisation of fundamental mathematical truths
i and principles during the course of eight years of highly concen
trated mental endeavour on the part of the author and therefore
appropriately given the title of "mental" mathematics appearing
more as miracle than the uaual approach of hardbaked science,
as the author has himself stated in the Preface.
SwBmi Sankartichya was a gifted scholar on many fronts
of learning including science and humanities but his whole
milieu wassomething of s much higher texture vis, that he was
a Rai fulfilling the ideals and attainments of those Seers of
ancient India who discovered the cosmic laws embodied in
tlie Vedas. SwBmi Bhiirati Krgna Tirtha had the same reveren
tial approach towards the Vedas. The qucstion naturally
arises as to whether the Stitras which form the basis of this
treatise exist anywhere in the Vedic literature as known to us.
But this criticism loses all its force if we inform ourselves of the
definition of Veda given by Sri Sa~ikariciir~a himself as quoted
below :
"The very word 'Veda' has this derivational meaning i.e. the fountain
1 head and illimitable storehome of all knowledge. This derivation, in effect,
means, connotes and impliea that the Vedas e M oontoin (~talicsmine) within
themwlvea all the knowledge needed by mankind relating pot only to the 60
called 'spiritual' (or otherworldly) matters but also to those usually dascnbed
aa purely 'neonlar', 'temporal', or 'worldly' and also to the means required by
humaniay ae such for the aohievemsnt of allround, complete and perfect succeas
in all conoeivable directions and that there oan be no adjechval or restrictive
epithet calcnlated (or tending) to limit that knowledge down in any ephere,
any direction or any respect whatsoever.
4.
"In other words, it connotes and impliea that our pncient Indian Vedic mentioned herein do not appear in the hitherto known P a d 
lore shouL1 be (italics mine) allround, complete and perfect and able to throw istas.
the fullest necessary light on all matters which any aspiring seeker after know
ledge can possibly seek to he enlightened on".
A list of these main 16 Siitras and of their subsutras or
corollaries is prefixed in the beginning of the text and
I It is the n hole essence of his assessment of Vedic tradition the style of language also points to their discovery by Sri
1 that it is not to be approached from a factual standpoint but
from the ideal standpoint viz, as the Vedas as traditionally
SwZmiji himself. At any rate, it is needless t o dwell longer on
this point of origin since the vast merit of these rules should be
accepted in India as the repository of all knowledge should be a matter of discovery for each intelligent reader. Whatever is
and not what they are in human possession. That approach
written here by the author stands on its own merits and is
entirely turns the tables on all critics, for the authorship of
presented as such to the mathematical world.
Vedic mathematics then need not be laboriously searched in the
texts as preserved from antiquity. The Vedas are well known Swamiji was a marvellous person with surpassing qualities
as four in nunlher Rk, Yaju, SHma and Atharva but they have and was a prolific writer and eloquent speaker. I had the
also the four Upavedas and the six VedHligas all of which form good fortune of listening t o his discourses for weeks together on
an indivisible corpus of divine knowledge as it once was and as several occasions when he used to visit Lucknow and attracted
it may be revealed. The four Upavedas are as follows : large audiences. He could at a stretch speak for several ~ O I U S
in Sanskrit and English with the same facility and the intonation
Veda Upaveda
of his musical voice left a lasting impression on the minds of
Rgveda hyurveda hie hearers. He was an ardent admirer of Bhartrhari the great
Simaveda GLndharvaveda scientific thinker of the Golden Age of Indian history in a
Yajurveda Dhahurveda different field viz, that of philosophy of grammar.
Atharvaveda SthLpathyaveda
Swamiji had planned t o write 16 volumes on all aspects
In this list the Upaveda of SthHpatya or engineering com and branches of mathematical processes and problems and there
prises all kinds of architectural and structural human endeavour
is no doubt that his mental powers were certainly of that calibre,
and all visual arts. Swamiji naturally regarded mathematics but what has been left to us is this introductory volume which
or the science of calculations and computations to fall under in itself is of the highest merit for reason of presenting a new
this category. technique which the author styles as "mental" mathematics
In the light of the above definition and approach must be different from the orthodox methods of mathematicians all over
understood the author's statement that the sixteen Sfitms on the world. Arithmetical problems usually solved by 18, 28 or
which the present volume is based form part of a Pariiiista of 42 steps in case of such vulgar fractions as 1/19, 1/29, 1/49 are
the Atharvaveda. We are aware that each Veda has its subsi here solved in one simple line and that is possible t o be done
diary apocryphal texts some of which remain in manu8cripts and even by young boys. The truth of these methods was demons
others have been printed but that formulation has not closed. trated by this saintly teacher before many University audiences
For example, some Paribistas of the Atharvaveda were edited in India and in the U.S.A.including learned Professors and every
by G. M. Rolling and J. Von Negelein, Liepzing, 190910. But one present was struck with their originality and simplicity.
this work of Sri SaiikarZo?oBryajideserves to be regarded as a We are told in his Reface by SwHmi $alikariiclya that
new Paribista by itself and it is not surprising that the Sctras he contemplated t o cover all the different branches of mathe
5.
matics such as arithmetic, algebra, geometry (plane and solid) 1 to Varanasi to preside over the Tantric Sammelan of the Varana
trigonometry (plane and spherical) conicsgeometrical and
analytical, astronomy, calculusdifferential and integral etc., 1, 8
seya Sanskrit University (8th t o 11th March 1965) and although
he is now 85 years of age, his innate generosity made him accept
with these basic Siitras. That comprehensive application of our request t o give his foreword.
I the Siitras could not be left by him in writing but if some onc has
the patience and the genius to pursue thc method and impli a I am particularly happy that I am able to publish this
work under the Nepal Endowment Hindu Vishvavidyalaya
catlons of these formulae he may probably be ablc to bring
Publication Series, for I cntcrtained an ardcnt dcsire to do so
these various branches within the orbit of this original style. since our late President Dr. Rajcndra Prasadji spoke to me about
A full flerlged course of his lecturedemonstrations was its existence when I once met him in New Delhi in the lifetime
organised by the Nagpur University in 1952 and some lectures of 6rf Swiimiji.
were delivered by Swamiji at the B.H.U. in 1949. I t is, thele
V. S. ACRAWALA,
fore, in the fitness of things and a happy event for the B.H.U.
M.A., Ph.D., D.Litt.
to be given the opportunity of publi~hingthis book by the
courtesy of Srimati Manjula nevi Trivedi, disciple of Sri SwBmiji
Ranaras Hindu University General Edttor,
who agreed to make over this manuscript to us through the Varanasi5 Hindu Vishwavidyalaya
a March 17, 1965. Nepal Rajya Sanskrit
efforts of Dr. Pt. Omkarhath Thakur. Thc work has been
seen through the Press mainly by Dr. Prem Lata Sharma, Dean, Granthamala Series.
Faculty of Music & Pine Arts in the University. To all of
these our grateful thanks are due. Dr. Brij Mohan, Head of the
Department of Mathematics, B.H.U., took the trouble, a t my
Q
request, ot going through the manuscript and verifying the
calculations for which I offer him my best tlvanks. I also
express gratitude to Sri Lakshmidas, Manager, B.H.U. Press,
for taking great pains in printing this difficult text.
We wish to express our deepest gratitude to Sri Swimi
PratyagBtmHnanda Saraswati for the valuable foreword that
he has written for this work. Today he stands preeminent in
the world of Tantrlc scholars and is a profound mathematician
and scientific thinker himself. His inspiring words are like f
fragrant flowers offered a t the feet of the ancient Vedic Rbis
whose spiritual lineage was revealed m the late 8ankariiciirya
Sri Bhirati Krsna Tirtha. SwEmi PratyagEtmBnandaji has
not only paid a tribute to Sri SankarEcHryaji but his ambrocial 6
words have showered blessings on all those who are lovers
of intuitional experiences in the domain of metaphysics and
physics. Swamiji, by a fortunate chance, travelled from Calcutta
6.
FOREWORD
Vedic Mathematics by the late S a n k a r ~ c i i r ~(BhPrati
a
Krsna Tirtha) of Govardhana Pitha is a monumental work.
In his deeplayer explorations of cryptic Vedic mysteries relat
ing specially to their calculus of shorthand formulae and their
neat and ready application to practical problems, the late
$arikarLcLrya shews the rare combination of the probing insight
and revealing intuition of a Yogi with the analytic acumen and
synthetic talent of a mathematician. With the late $a6kar~
clrya we belong to a race, now fast becoming extinct, of die
hard believers who think that the Vedas represent an inexhaus
tible mine of profoundest,wisdom in matters both spiritual
and temporal ; and that. this store of wisdom was not, as regards
its assets of fundamental validity and value a t lest, gathered
by the laborious inductive and deductive methods of ordinary
systematic enquiry, but was a direct gift of revelation to seers
and sages who in their higher reaches of Yogic realization were
competent to receive it from a Source, perfect and immaculate.
But we admit, and the late $ankadc&ryahas also practically
admitted, that one cannot expect to convert or revert criticism,
much less carry conviction, by merely asserting one's staunchest
beliefs. To meet these ends, one must be prepared to go the
whole length of testing and verification by accepted, accredited
methods. The late $ a ~ i k a r ~ c ~has,
r ~ a by his comparative
and critical study of Vedic mathematics, made this essential
requirement in Vedic studies abundantly clear. So let us agree
to gauge Vedic mysteries not as we gauge the faroff nabulae
with the poet's eye or with that of the seer, but with the alert,
expert, scrutinizing eye of the physical astronomer, if we may
put it as that.
That there is a consolidated metaphysical background in the
Vedas of the objective sciences including mathematics as regards
their basic conceptions is a point that may be granted by a
thinker who has looked broadly and deeply into both the realms.
In our paper recently published'The Metaphysics of
Physics'we attempted to look into the mysteries of creative
enlergeuce as contained in the wellknown cosmogenic Hymn
7.
matical measures and relations. This, however, ore may do
(Rg.X.190) with a view to unveiling the metaphysical background consciously or semiconsciously, systematically or haphazardly.
where both ancient wisdom and modern physics may meet on Even some species of lower animals are by instinct gifted mathe
a common basis of logical understanding, and compare notes, maticians ; for example, the migratory bird which flies t,housands
discovering, where possible, points of significant or suggestive of miles off f r m its nesthome, and after a period, unerringly
parallelism between the two sets of concepts, ancient and modern. returns. This 'implies a subconscious mathematical talent that
That metaphysical background includes mathematics also; works wonder. We may cite the case of a horse who was a
because physics as ever pursued is the application of mathema
mathematical prodigy and could 'tell' the result of a cube root
I tics to given or specified spacetimeevent situations. There we
examined Tapas as a fundamental creative formula whereby
(requiring 32 operations. according t o M. Materlink in his
'Unknown Quest') in a twinkle of the eye. This sounds like magic,
I the Absolute emerges into the realms of measures, variations,
limits, frameworks and relations. And this descent follow^
a logical order which seems to lend itself, within a framework
but it is undeniable that the feat of mathematics does sometimes
assume a magical look. Man, undoubtedly, has been given his
share of this magical gift. And he can improve upon it by
of conditions and specifications, to mathematical analysis.
practice and discipline, by Yoga and allied methods. This is
Rjtri in the Hymn represents the Principle of Limits, for exa
undeniable also. Lztely, he has devised the 'automatic brain'
mple, Rtaiica Satyaiica stand for Becoming (Calanakalana) and
for complicated calculations by science, that looks like magic.
Being (vartanakabna) a t a stage where limits or conditions
But apart from this 'magic', there is and has been, the
or conventions do not yet arise or apply. The former gives the
'logic' of mathematics also. Man works from instinct, t,alent,
unconditioned, unrestricted how or thw of cosmic process ;
or even genius. But ordinarily he works as a logical entity
the latter, what or that of existence. Tapas, which corresponds to requiring specified data or premises to start from, and more
ArdharnitrZ in Tantric symbolism, negotiates, in its rolc specially
or less elaborate steps of reasoning t o arrive at a conclusion.
of critical variation, between what is, abinitio, unconditioned
This is his normal process of induction and deduction. Here
and unrestricted, and what appears otherwise, as for instance,
formulaj (Smras) and relations (e.g. equations) must obtain as
in our own universe of logicomathematical appreciation.
in mathematics. The magic and logic of mathematics in some
This is, necessarily, abstruse metaphysics, but it IS,
nevertheless, the starting backgronnd of both physics and cases get mixed up ; but it is sane to keep them apart. You can
get a result by magic, but when you are called upon to prove,
mathematics. But for all practical purposes we must come down
from mystic nabulae to the terrajrma of our actual apprehension you must have recourse to logic.
and appreciation. That is to say, we must descend to our own Even in this latter case, your logic (your formule and
pragmatic levels of timespaceevent situations. Here we applications) may be either simple a,nd elegant or complicated
face actual problems, and one must meet and deal with these and cumbrous. The former is the ideal to aim at,. We have
aquarely without evasion or mystification. The late SalikarB classical instances of master mathematicians whose methods of
cLya has done this masterly feat with an adroitness that analysis and solution have been regarded as marvels of cogency,
compels admiration. compactness and clcgancc. Some have bcen 'beautiful' as a
It follows from the fundamental premises that the universe poem (e.g. Lagrange's 'Analytical Mechanics.')
we live in must have a basic mathematical structure, and The late Sa1ikar5c5rya has claimed, and rightly we may
consequently, t o know a fact or obtain a result herein, to any think, that the Vedic Sfitras and their applications possess these
required degree of precision, one must obey the rule of mathe
8.
virtues to a degree of eminence that cannot be challenged.
The outstanding merit of his work lies in his actual proving of
this contention. A HUMBLE HOMAGE
!
Whether or not the Vedas be believed as repositories of The late $ankaricirya's epochmaking work on Vcdic
perfect wisdom, it is unquestionable that the Vedic race lived Mathematics brings to the notice of the intelligentsia most
I nd as merely pastoral folk possessing a halforquarterdeveloped strikingly a new theory and method, now almost unknown,
culture and civilization. The Vedic seers were, again, not mere
'navelgazers' or 'nosetipgazers'. They proved themselves
adepta in all levels and branches of knowledge, theoretical and
practical. For example, they had their varied objective science,
both pure and applied.
II of arriving at the truth of things which in this particular case
concerns the truth of numbers and magnitude, but might as well
cover, as it undoubtedly did in a past age in India, all sciences
and arts, with results which do not fail to evoke a sense of awe
and amazement today. The method obviously is radically
I Let us take a concrete illustration. Suppose in a time of
drought we require rains by artificial means. The modern
I differnt from the one adopted by the modern mind.
Music and not Mathematics is my field (although the
scientist has his own theory and art (tecbique) for producing philosophy of numbers, cosmic and metaphysical corres
the result. The old seer scientist had his both also, but pondences w ~ t hmusical numbers, the relation of numbers
different from these now avai1ir.g. He had his science and with consonant, dissonant and assonant tonal intervals etc.,
technique, called Yajiia, in which Mantra, Yantka and other
factors must COoperatewith mathematical determinateness and
precision. For this purpose, he had developed the six auxiliaries
I closely interrelate music and mathematics), but study of the
traditional literature on music and fine arts with which I have
been concerned for the last few years has convinced me of one
of the Vedas in each of which mathematical skill and adroitness, fundamental fact regarding the ancient Indian theory and
occult or otherwise, play the decisive role. The Siitras lay method of knowledge and experience vis a vis the modern.
down the shortest and surest lines. The correct intonation of While all great and true knowledge is born of intuition and
the Mantra, the correct configuration of the Yantra (in the not of any rational process or imagination. there is a radical
making of the Vedi eta., e.g. the quadrature of a circle), the difference between the ancient Indian method and the modem
correct time or astral conjugation factor, the correct rhythms Western method concerning intuition.
etc., all had to be perfected so as to produce the desired result The divergence embraces everything other than the fact
effectively and adequately. Each of these required the calculus
of intuition itselfthe object and field of intuitive vision, the
of mathematics. The modern technician has his logarithmic method of working out experience and rendering it to the
tables and mechanics' manuals ; the old Yijfiika had his Saras. intellect. The modern method is to get the intuition by sugges
How were the Szitras obtained ?by magic or logic or both ?is tion from an appearance in life or nature or from a mental idea
a vital matter we do not dkcuss here. The late S a l i k a r i c ~ r ~ a and even if the source of the intuition ie the soul, the method
has claimed for them cogency, compactness and simplicity. at once relates it to a support external to the soul. The ancient
This is an even more vital point, and we think, he has reasonably Indian method of knowledge had for its business to disclose
made it good. something of the Self, the Infinite or the Divine to the regard
Varanasi, SWAMI P R ATY AGATMANANDA of the soulthe Self through its expressions, the infinite through
2231965 SARA SWAT^ its finite symbols and the Divine through his powers. The
9.
process was one of Integral knowledge and in its subordinate
ranges was instrumental in revealing the truths of cosmic
phenomena and these truths mere utilised for worldly ends. CONVENTIONAL TO UNCONVENTIONALLY ORIGINAL
1 These two methods are based on different theories of
This book Vedic Mathematics deals mainly with various
knowledge and experience, fundamentally divergent in outlook vedic mathematical formulae and their applications for carrying
I and approach. The world as yet knows very little of the out tedious and cumbersome arithmetical operations, and to a
ancient lndian method, much less of its secret techniques.
Sri S a ~ i k a r ~ c k ~remarkably
a's unique work of Vedic mathe
matics has brought t o popular notice demonstrably for the
very large extent, executing them mentally. In this field of mental
arithmetical operations the works of the famous mathemati
cians Trachtenberg and Lester Meyers (High Speed Maths) are
elementary compared to that of Jagadguruji.
first time that the said method was usefully employed ill ancient
India in solving problems of secular knowledge just as for solving Some people may find it ditficult, at first reading, to understand
the arithmetical operations although they have been explained
those of the spiritual domain. very lucidly by Jagadguruji. It is not because the explanations
I am happy that in the printing and publication of this are lacking in any manner but because the methods are totally
monumental work and the preceding spadework I had the unconventional. Some people are so deeply rooted in the con
privilege to render some little service. ventional methods that they, probably, subconsciously reject to
see the logic in unconventional methods.
PREM LATA SHARMA An attempt has been made in this note to explain the un
Varamsi5. Dean, Faculty of Music & Fine Arts, conventional aspects of the methods. Once the reader gets
23365. Banaras Hindu University. used to the unconventional in the beginning itself, he would
find no difficulty in the later chapters. Therefore the explanatory
notes are given for the first few chapters only.
C/iapter I
Chapter I deals with a topic that has been dealt with compre
hensively in the chapter 26 viz. 'Recurring'Decimal'. Gurudeva has
discussed the recurring decimals of 1/19, 1/29, etc. in chapter
I to arouse curiosity and create interest. In conversion of
vulgar fractions into their decimal equivalents Gurudeva has
used very unconventional methods of multiplication and division.
In calculation of decimal equivalent of 1/19, first method of the
'Ekidhika SLitra' requires multiplication of 1 by 2 by a special and
unconventional process. Inconventional method product of 1, the
multiplicand, by 2 the multiplier, is 2 and that is the end of multi
plication process. It is not so in the unconventional 'Ekidhika'
method. In this method, in the above example, 1 is the first multi
plicand and its product with multiplier '2' is 2 which in this special
process becomes the second multiplicand. This when multiplied
by the multiplier (which remains the same) 2 gives the product
as 4 which becomes the third multiplicand. And the process of
10.
multiplication thus goes on till the digits start recurring.
Similarly in the second method of the 'Ek~dhikaSctra' for
calculating the decimal equivalent of 1/19,*it is required to CONTENTS
divide I by 2 by an unconventional and special process. In the
conventional method when 1, the dividend, is to be divided by Page
the divisor 'T,the quotient is 0.5 and the process of division I INTRODUCTORY No.
ends. In the special method of 'Ekfidhika Siitra' for calculating I. My Beloved Gurudeva(Srimati hfanjula Trivedi) ... i
decimal equivalents, the process starts by putting zero as the 2. Author's Preface . .. ... ... xiii
first digit of the quotient, 1 as the first remainder. A decimal A.A Descriptive Prefatory Note ... ... xiii
point is put after the first quotient digit which is zero. NOW, B.Explanatory Exposition ... ... XX
the first remainder digit '1' is prefixed to the first quotient digit
'0' to form '10' as the second dividend. Division of 10 by the
C.Illustrative Specimen Ssmples ... ... xxii
divisor 2 (which does not change) gives 5 as the second quotient I1 TEXT
digit which is put after the decimal point. The second remainder Sixteen S t t r a s and their Corollaries ... 1
digit '0' is prefixed to the second quotient digit 5 to form 5 as
the third dividend digit. Division of 5 by 2 gives 2 as the third
Prolegomena ... ... ... 16
quotient digit and 1 as the third remainder digit which when CHAPTER
prefixed to the third quotient digit '2' gives 12 as the fourth I. Actual Applications of the Vedic SGtras ...
dividend and so the process goes on till the digits start recurring. II. Arithmetical Computations ... ...
if
13
Chapter It1 111. Multiplication .. . .. . ... 40
Vinculum is an ingenious device to reduce single digits larger Practical Application (compound multiplication) 49
than 5, thereby facilitatingmultiplication specially for themental Practice C% Proportion ( ,, ) ... 51
oneline method. Vinculum method is based on the fact that
18 is same as (202) and 76 as (10024) or 576 as (60024). Guru
IV. Division by the Nikhilam method ... 55
deva has made this arithmetical fact a powerful device by writing
V. Division by the Parevartpa method . 64
18 as 22; 76 as 1 r 4 arid 576 as 6 2 This device is specially
VI. Argumental Division ... 79
useful in vedic division method. Linking note (Recapitulation $ Conclusion). .. 84
A small note on 'al~quot' may facilitate the study for some. VII. Factorisation (of simple quadratics) .. 86
Aliquot part is the part contained by the whole an integral VIII. Factorisation (of harder quadratics) . 90
number of times, e.g. 12 is contained by the whole number 110, IX. Factorisation of Cubics etc. ...
.. 93
9 times. or in simple words it is the quotient of that fraction. X. Highest Commor~Factor .., ... 98
Chapter I V XI. Simple Equations (First Principles) ... 103
In the division by the Nikhilam.method the dividend is divided
into two portions by a vertical line. This vertical line should
XII.
XIII.
Simple Equations (by Sfinyam etc.)
Merger Type of Easy Simple Equations
...
... 107
126
have as many digits to its right as there can be in the highest possi
ble remainder. In general the number of such digits are the same
Extension mct,hod ... ... 131
as in the figure which is one less than the divisor. Needless t o XIV. Coinplex Mergers ... ... 134
XV. Simultaneous Simple Equations ...
state that the vertical and horizontal lines must be drawn neatly 140
XVI. Miscellaneous (Simple) Equations
when using this method. ... 145
WING. COM. VLSHVA MOHAN TLWARI XVII. ~ u a d r ~Equations
ic ... ... 157
XVIII. Cubic Equations ... ... 168
!
11.

Page
CRAPTERS No. MY BELOVED GURUDEVA
XIX. Biquadratic Equations ... ... 171
XX. Multiple Simultaneous Equations ... 174 SMTI. MANJULA TRIVEDI
XXI. Simultaneous guadratic Equations ... 178 [In the lines that follow the writer gives a short biographical
XXII. Factorisation & Differential Calculus ... 182 sketch of llre illustrious author of Vedic Mathematics and a #hurt
XXIII. Partial Fractions ... ... 186 i
account of the genesis of his work laow published, based on inti
XXIV. Integration by Partial Fractions ... 191 mate personal ~ l u ~ u ~ e d g e  E ~ r ~ o ~ . ]
XXV. The Vedic Numerical Code ... ... 194 Very few persons can there be amongst the cultured people
XXVI. Recurring Decimals ... ... 196 I
of India who have not heard about HIS HOLINESS JAGAD
XXVII. Straight Division ... ... 240
XXVIII. Auxiliary Fractions ... ... 255
GURU SHANKARACHARYA SRI BHARATI KRISHNA
TIRTHAJI MAHARAJ, the magnificent and divine personality
XXIX. Divisibility & Simple Osculators ... 273 that gracefully adorned the famous Govardhan Math, Puri,
XXX. Divisibility & Complex Multiplex Osculators 285
XXXI. Sum & Difference of Squares ... ... 296 his vast and vcrsatile learning, his spiritual and educational
XXXII. Elementary Squaring, Cubing etc. ... 300 attainments, his wonderful research achievements in the field
of Vedic Mathematics and his consecration of all these quali
XXXIII. Straight Squaring ... ... 305
fications to the service of humanity as such.
XXXIV. Vargamfila (square root) ... ... 308
XXXV. Cube Roots of Exact Cubes ... ... 316 His Holiness, better known among his disciples by the
XXXVI. Cube Roots (General) ... ... 827 beloved name 'Jagadguruji' or 'Gurudeva' was born of highly
XXXVII. I?ythagoras' Theorem etc., ... ... 349 learned and pious parents in March, 1884. His father, late
XXXVIII. Apollonius' Theorem ... ... 352 fii P. Narasimha Shastri, was then in service as a Tahsildar a t
XXXIX. Analytical Conics ... ... 354 Tinnivelly (Madras Presidency) who later retired as a Deputy
XL. Miscellaneous Matters ... ... 361 Collector. His uncle, late Shri Chandrashekhar Shastri, was
Press Opinions ... ... 365 the Principal of the Maharaja's College, Vizianagaram and his
greatgrandfather was late Justice C. Ranganath Shastri of the
Madras High Court.
Jagadguruji, named as Venkatraman in his early days,
was an exceptionally brilliant student and invariably won
the first place in a11 the subjects in all the classes throughout his
educational career. During his school days, he was a student
of National College, Trichanapalli; Church Missionary Society
College, Tinnevelli and Hindu College, Tinnevelli. Be passed
his matriculation examination from the Madras University
in January, 1899, topping the list as usual.
He was extraordinarily proficient in Sanskrit and oratory
and on account of this he was awarded the title of 'SARASWATI'
12.
by the Madras Sanskrit Association in July, 1899 when he was of humanity swayed his heart mightily, yet the undoubtedly
still in his 16th year. One cannot fail to mention at this stage deepest attraction that Venkatraman Saraswati felt was that
the profound impremion left on him by his Sanskrit Guru towards the study and practice of the science of sciencesthe
holy ancient Indian epiritual science or AdhyiitmaVidyi. In
Shri Vedam Venkatrai Shastri whom Jagadguruji always
remembered with deepest love, reverence and gratitude, with 1908, therefore, he proceeded to the Sringeri Math in Mysore
tears in hia eyes. to lay himself a t the feet of the renowned late Jagadguru
Shankaracharya Maharaj Shri Satchidiinanda Sivibhinava
After e n i n g the highest place in the B.A. Examinanon,
Nrisimha Bharati Swami:
Shri Venkatraman Saraswati appeared at the M.A. Examination
But he had not stayed there long, before he had to assume
of the American College of Sciences, Rochester, New York, from
Bombay Centre in 1903 ; and in 1904 a t the age of just twenty the post of the first Principal of the newly started National
he passed M.A. Examination in further seven subjects simul College a t Rajmahendri under a preesing and clamant call of
taneously securing the highest honours in all, which is perhaps duty from tthe nationalist leaders. Prof. Venkatraman Saras
the alltime worldrecord of academic brilliance. His subjects wati continued there for three years but in 1911 he could not
included Sanskrit, Philosophy, English, Mathematics, History resist his burning desire for spiritual knowledge, practice and
and hience. attainment any more and, therefore, tsaring himself off suddenly
from the said college he went back t o Shri Satchidhanda
As a etudent Venkatraman was marked for his splendid Siviibhinava Nrisimha Bhkati Swami a t Sringeri.
brilliince, superb retentive memory and everinsatiablt curiosity.
He would deluge his teachers with myriads of piercing questions The next eight years he spent in the profoundest study of
which made them uneasy and forced them frequently to make the most adval~cedVedanta Philosophy and practice of the
Brahmaeadhnna. During these days Prof. Venkatraman
a frank confession of ignorance on their part. In this respect,
used to study Vedanta a t the feet of Shri Nrisimha Bhiirati
he was considered to be a terribly mischievous student.
Swami, teach Sanskrit and Philosophy in schools there, and
Even from his University days Shri Venkatraman Saras practise the highest and most vigorous Yogasiidhiina in the
wati had started contributing learned articles on religion, nearby forests. Frequently, he was also invited by several
philosophy, sociology, history, politics, literature etc., to institutions to deliver lcctures on philosophy; for example he
late W. T. Stead's "REVIEW OF REVIEWS" and he was delivered a series of sixteen lectures on Shankarachnrya's
specially interested in all the branches of modern science. In Philosophy a t Shankar Institutc of Philosophy, Amalner (Khan
fact, study of the latest researches and discoveries in modern desh) and similar lectures a t several other places like Poona,
science continued t o be Shri Jagadguruji's hobby till his very Bombay etc.
last days.
After several years of the most advanced studies, the deepest
Sri Venkatrarnan started his public life unde the guidance
meditation, and the highest spiritual attainment Prof. Venkatra
of late Hon'ble Shri Gopal Krishna Gokhale, C.I.E. in 1905 in
man Saraswati was initiated into the holy order of SAMNYASA
connection with the National Fdncatlon Movement and the
a t Banaras (Varanasi) by his Holiness Jagadguru Shankara
South African Indian issue. Although, however, on the one charya Sri Trivikram Tirthaji Maharaj of Shgradgpeeth on the
hand, Prof. Venkatribman Sa.raswati had acquired an endless
4th July 1919 and on this occasion he was given the new
fund of learning and his desire to learn ever more was still
name, Swami Bharati Krishna Tirtha.
unquenchable and on the other hand the urge for selfless service
13.
Prom his very early days Jagadguruji was aware of the
This was the starting point of an effulgent manifestation
need for the right interpretation of "Dharma" which he, defined
of Swamiji's real greatness. Within two years of hisatay inthe
holy order, he proved his unique suitability for being installed 8s "the sum total of all the means necessery for speedily making
on the pontifical throne of Sharada Peetha Shankaracharya and and permanently keeping all the people, individually as well
accordingly in 1921, he was so installed with all the formal as collectively superlatively comfortable, prosperous, happy,
ceremonies despite all his reluctance and active resistance. and joyous in all respects (including the physicel, mental,
Immediately, on assuming the pontificate Shri Jagadguruji intellectual, educational, economic, social, political, paycllic,
started touring India from corner to corner and delivering spritual etc. ad injEnitum)". He was painfully aware of the
lectures on SanEtana Dharma and by his scintillating intellectual "escapism" of some from their duties under the garb of spiritua
brilliance, powerful oratory, magnetic personality, sincerity of lity and of the superficial modem educational varnish of the
purpose, indomitable will, purity of thought, and loftiness of others, divorced from spiritual and moral standards. He,
he took the entire intellectual and religious clam of therefore, always laid great emphasis on the necessity of har
the nation by storm. monising the 'spiritual' and the 'material' spheres of daily
Jagadguru Shankaracharya Shri Madhusudan Tirtha of life. He also wanted to remove the falae ideas, on the one
Govardhan Math Puri was at this stage greatly impressed by hand, of those persons who thiilk that Dharma can be practiced
Jagadguruji and when the former was in failing health he by exclusively individual spiritual SSBdhanZ coupled with more
requested. Jagadguruji to succeed him on Govardhan Math honest breadearning, ignoring one's responsibility for rendering
Gadi. Shri Jagadguruji continued to resist his importunate selfleas service to the society and on the other hand of those
requests for a long time but at last when Jagadguru Shri hhdhu who think that the SiidhanZ can be complete by mere service
sudan Tirtha's health took a serious turn in 1925 he virtually of society even without learning or ~ractisingany spirituality
forced Jagadguru Shri Bharati Krishana Tirthaji to accept the oneself. He wanted a happy blending of both. He stood for
Govardhan Matk{s Gadi and accordingly Jagadguruji installed the omnilateral and allround progress aimultenaously of both
Shri Swarupanandji on the Sharadapeeth Gadi and himself the individual sand society towards the speedy realisation
assumed the duties of the ecclesiastical and pontifical head of of India's spiritual and cultural ideal, the lofty Yedantic ideal
Sri Govardhan Math, Puri. of 'Piirnatva' (perfection and harmony allround).
In this capacity of Jagadguru Shankaracharya of Govar
With these ideas agitating his mind for several decades
dhan Math, Puri, he continued to disseminate the holy spiritual
he went on carrying on a laboriou8, elaborate, patient and day
teachings of Ssnatana Dharma in their pristine purity all over
andnight research to evolve 6nally a splendid and perfect scheme
the wortd the rest of his life for 35 years. Months after months for allround reconstruction first of India and through it of the
and years after years he spent in teaching and preaching, talking
world. Consequently Sri Jagadguruji founded in 1953 at Nagpur
and lecturing, discussing and convincing nlillions of people all
an institution named Sri Vishwa Punarnirmana Sangha (World
over the country. He took upon himself the colossal task
Reconstruction Association). The Adn~inistrativeBoard of the
of the renaissance of Indian culture, spreading of Sanatena
Sangha consisted of Jagadguruji's disoiples, devotees and admi
Dharma, revival of the highest human and moral values and
enkindling of the loftiest spiritual enlightenment throughout rers of his idealistic and spiritual ideals for humanihrian service
the world and he dedicated his whole life to this lofty and and included a number of high court judges, ministers, educa
noble mission. tionists, statesmen other personage of the highest calibre
14.
pleasure. To see him was a privilege. To speak to him was
a real blessing and to be granted a specialinterviewAh ! that in adoration of various Devas and Devis. These Slokas have
was the acme of happiness which people coveted most in all been edited and are being translated into Hindi. They are
earnestness. The magnetic force of his wonderful personality proposed to be published in three volumes along with Hindi
was such that one word, one smile, or even one look was quitc translation.
enough to convert even the most sceptic into his most ardent The book on "Sanatana Dharma" by H. H. Swami BhBrati
and obedient disciple. He belonged to all irrespective of caste Krisna T i h a Mahiiriija has been published by Bharatiya
or creed and he was a real Guru to the whole world. Vidya Bhavan, Bombay.
People of all nationalities, religions and climes, Brahmins Above all, his Bhakti towards his VidyHguru was some
and nonBrahmins, Hindus and Mahomedans, Parsis and Chris thing beyond description. He would tak for d a y ~together
tians, Europeans and Americans received equal treatment at about the greatness of his VidyHguru. He would be never
the hands of Mis Holiness. That was the secret of the immense tired of worshipping the Guru. His Guru also was equally
popularity of this great Mahatma. attached to him and called our Swamiji as the own son of the
He was grand in his simplicity. People would give any Goddess of Learning, Shri Sarada. Everyday he would first
thing and everything to get his blessings and he would talk worship hie guru's mndals. His "Gurup6dukii Stotra ' clearly
w6rds of wisdom as freely without fear or favour. He was indicates the qualities he attributed to the sandale of his guru.
most easily accessible to all. Thousands of people visited Shri BhZrati K$pa Tirtha was a great Yogin and a
him and prayed for the relief of their miseries. He had a kind "Siddha" of a very high order. Nothing was impossible for him.
word to say to each, after attentively listening to his or her tale Above all he was a true Samnyasin. He held the world but as
of woe and then give them some 'prasad' which would cure thcir a stage where every one had to play a part. In short, he was
malady whether physical or mental. He would actually undoubtedly a very great Mahgtrnii but without any display of
shed tears when he found people suffering and would pray to mysteries or occultisme.
God to relieve their suffering.
I have not been able to express here even one millionth
He was mighty in hL learning and voracious in his reading.
part of what I feel. His epotless holiness, his deep piety,
A sharp intellect, a retentive memory and a keen zest went to
mark him as the most distinguished scholar of his day. His his endless wisdom, his childlike peacefulness, sportiveness
leisure moments he would never spend in vain. He was always and innocence and his univereal affection beggar all description.
reading something or repeating something. There was no branch His Holiness has left us a noble example of eimplest living and
of knowledge which he did not know and that also 'shastrically'. highest thinking. May all the world benefit by the example
of a life ao nobly and so simply, so spiritually and so lovingly
He was equally learned in Chandahsastra, Ayurveda and
lived.
Jyotish Sastra. He was a poet of uncommon merit and wrote
a number of poems in Sanskrit in the praise of his guru, gods In.trJ* ~ Sthe Present Volume
R ~ W Con
and godesses with a charming flow of Bhakti so conspicuons in I now proceed to give a short account of the genesis
all his writings. of the work published herewith. Revered Guruji used to
I have got a collection of over three thousand slokas for say that he had reconstructed the sixteen mathematical
ming part of the various eulogistic poems composed by Gurudeva formulae (given in this text) from the Atharvaveds after
assiduous rwarch and 'Tapas' for about eight years in the
15.
forests surrounding Sringeri. Obviously these formulae are
had been given to understand that he would have to go to the
not to be found in the present recensions of Atharvaveda ; they
U.S.A. for correction of proofs and personal supervision of
were actually reconstructed, on the basis of intuitive revelation,
printing. But his health deteriorated after his return to India
from materials scattered here and there in the Atharvaveda.
Revered Gurudeva used to say that he had written sixteen and finally the typescript was brought back from the U.S.A.
after his attainment of Mahasamadhi, in 1960.
volumes (one for each Siitra) on these Stitras and that the
manuscripts of the said volumes were deposited at the house ACKNOWLEDGEMENTS
of one of his disciples. Unfortunately, the mid manuscripts I owe a deep debt of gratitude to Justice N. H. Blagwati,
were lost irretrievably from the place of their deposit and this the enlightened ViceChancellor of the Banaras Hindu Univer
colossal loss was &ally confirmed in 1956. Revered Gurudeva sity and other authorities of the B.H.U. who have readily under
was not much perturbed over this irretrievable loss and used to taken the publication of this work which was introduced to them
say that everythmg was there in his memory and that he could by Dr. Pt. Omkarnath Thakur. I am indebted to Dr. Thakur
rewrite the 16 volums !
for this introduction. My hearty and reverent thanks are due
My late husband Sri C. M. Trivedi, Hon. Gen. Secertary to Dr. V. S. Agrawala (Professor, Art & Architecture, B.H.U.)
V. P. Sangh noticed that while Sri Jagadguru Maharaj was the vateran scholar, who took the initiative and throughout
busy demonstratirig before learned people and societies kept up a very keen interest in this publication. I t is my
Vedic Mathematics as discovered and propounded by him, pleasant duty to offer my heartfelt gratitude to Dr. Prem Lata
some persons who had grasped a smattering of the new Sharma, Dean, Faculty of Music and Fine Arts, B.H.U. who
Siitras had already started to dazzle audiences as prodigies volunbarily took over the work of pressdressing of the
claiming occult powers without aknowledging indebtedness typescript and proofreading of this volume after a deadlock
to the Sfitras of Jagadguruji. My husband, therefore, pleaded had come to prevail in the process of printing just at the outset.
earnestly with Gurudeva and persuaded him to arrange for But for her hard labour which she has undertaken out of a
the publication of the Siitras in his own name. sheer sense of reverence for the noble and glorious work of
In 1957, when he had decided finally to undertake a Revered Gurudeva this volume would not have seen the light
tour of the U.S.A. he rewrote from memory the present of the day for a long time. I trust that Revered Gurudeva's
volume, giving an introductory account of the sixteen for Holy Spirit will shower His choicest blessings on her. Mv
mulae reconstructed by him. This volume was written in sincere thanks are also due to Sri S. Nijabodha of the Research
his old age within one month and a half with his failing health Section under the charge of Dr. Sharma, who has ably assisted
and weak eyesight. He had planned to write subsequent volu her in this onerous task.
mes, but his failing health (and cataract developed in both The Humblest of His Disciples
eyes) did not allow the fulfilment of his plans. Now the present Smti. MANJULA TRIVEDI
volume is the only work on Mathematics that has been left over Nagpur, Hony. General Secretary
by Revered Guruji ; all his other writings on Vedic Mathematics 16th March, 1965. SRI Vishwa Punarnirmana
have, alas, been lost for ever. Sangha, Nagpur.
The typescript of the present volume was left over by
Revered Gurudeva in U.S.A. in 1958 for publication. He
16.
AUTIIOR'S PREFACE
A.A DESCRIPTIVE PEEFAWRY MOTE
ON
THE ASTOUNDING WONDERS
OF
ANCIENT INDIAN VEDIC MATHEMATICS
1. In the course of aur di8courses on manifold and
multifarious subjects (spiritual, metaphysical, philosophical,
psychic, psychological, ethical, educational, scientific, mathe
matical, historical, political, economic, social etc., etc., from
time to time and from place to place during the last five decades
and more, we have been repeatedly pointing out that the Vedas
(the most ancient Indian scriptures, nay, the oldest "Religious"
scriptures of the whole world) claim to deal with all branches
of learning (spiritual and temporal) and to give the earnest
seeker after knowledge all the requisite instructions and guidance
in full detail and on scientificallynay, mathematiqally
accurate lines in them all and so on.
2. The very word "Veda" has this derivational meaning
i.e. the fountainhead and illimitable storehouse of all know
ledge. This derivation, in effect, mcans,condotes and implies that
the Vedas should contain within themselves all the knowledge
needed by mankind relating not only to the socalled 'spiritual'
(or otherworldly) matters but also to those usually described
a8 purely "secular", "temporal", or "wordly"; and also to
the means required by humanity as such for the achievement
of allround, complete and perfect success in all conceivable
directions and that there can be no adjectival or restrictive
epithet calculated (or tending) to limit that knowledge down in
any sphere, any direction or any respect whatsoever.
3. In other words, it connotes and implies that our
ancient Indian Vedic lore should be allround complete and
perfect and able to throw the fullest necessary light on all
matters which any aspiring seeker after knowledge can possibly
seek to be enlightened on.
17.
( xiv )
heartedly, nay; irresponsibly, frivolously and flippantly dis
4. I t is thus in the fitness of things that the Vedas include
missed, several abstruselooking and recpndite parts of the
(i) Ayurveda (anatomy, physiology, hygiene, sanitary science,
Vedas as "sheernonsensemor au "infanthumanity's prattle",
medical science, surgery etc., etc.,) not far the purpose of achic
ttnd so on, merely added fuel to the fire (so to speak) and further
ving perfect health and strength in the afterdeath future but
confirmed and strengthened our resolute determination to
in order to attain them here and now in our present physical
I bodies; (ii) Dhanurveda (archcry and other military sciences) unravel the toolong hidden mysteries of philosophy and science
contained in ancient India's Vedic lore, with the consequence
not for fighting with one another after our transportation to
heaven but in order to quell and subdue all invaders from that, after eigM years of concentrated eontemplation in forest
abroad and all insurgents from within; (iii) GCndharva Veda solitude, we were at long last able to recover the long lost keya
(the science and art of music) and (iv) Sthcipatya Veda (engineer which alone could unlock the portals thereof.
ing, architecture etc.,antl alI branches of mathematics in general). 9 . And we were agreeably astonished and intensely gra
All these subjects, be it noted, are inherent parts of the Vedas tified to find that exceedingly tough mathematical problems
, i.e. arc reckoned as "spiritual" studies and catered for as (which the mathematically most advanced present day Wes
I such therein.
tern scientific world had spent huge lots of time, energy and
money on and which even now it solves with the utmost difficulty
5. Similar is the case with regard to t)lle Vedciligas (i.e.
and after vast labour involving large numbers of difficult, tedious
grammar, prosody, astronomy, lexicography etc., etc.,) which,
and cumbeisome "steps" of working) can be easily and readily
according to the Indian cultural conceptions, are also inherent
solved with the help of these ultraeasy Vedic Siitras (or mathe
parts and subjects of Vedic (i.e. Religious) study.
matical aphorisms) contained in the Paribiata (the Appendix
6. As a direct and unshirkable consequence of this portion) of the ATHARVAVEDA in a few simple steps and by
analytical and grammatical study of the real connotation mothods which can be conscientiouslydescribed as mere "mental
and full implications of the word "Veda" and owing to various a~ithmetic".
other historical causes of a personal character (into details of 10. Eversince (i.e. since several decades ago), we have
which we need not now enter), we have been from our very
been carrying on an incessant and strenuous campaign for
early childhood, most earnestly and actively striving to study
the Indiawide diffusion of all this scientific knowledge, by
the Vedas critically from this standpoint and to realise and
means of lectures, blackboard demonstrations, regular classes
I prove to ourselves (and to others) the correctness (or otherwise)
of the derivative meaning in question.
7. There were, too, certain personal historical reasons
and so on in schools, colleges, universities etc., all over the
country and have been astounding our audiences everywhere
with the wonders and marvels not to say, miracles of Indian
why in our quest for the discovering of all learning in all its Vedic mathematics.
departments, branches, subbranches etc., in the Vedas, our 11. We were thus a t last enabled to succeed in attracting
gaze was riveted mainly on ethics, psychology and metaphysics
the more than passing attention of the authorities of several
on the one hand and on the "positive" sciences and especially
Indian universities to this eubject. And, in 1962, the Namur
mathematics on the other.
University not meely had a few lectures and blackboard
8. And the contemptuous or, at best patronising attitude demonstrations given but also arranged for our holding regular
adopted by some socalled Orientalists, Indologists, anti classes in Vedir mathematics (in the University's Convomtion
quarians, researchscholars etc., who condemned, or light
18.
( xvi )
Hall) for the benefit of all in general and especially of the Uni (iii) Even as regards complex problems involving a good
versity and college professors of mathematics, physics etc. number of mathematical operations (consecutively
12. And, consequently, the educationists and the cream or even simultaneously to be performed), the time
of the English educated sect,ion of the people including the taken by the Vedic method ~villbe a third, a fourth,
highest officials (e.g. the highcourt judges, the ministers a tenth or even a much smaller fraction of the time
etc.,) and the general public as such were all highly impressed ; required according to modern (i.e. current) Western
nay, thrilled, wonderstruck and flabbergasted! And not methods :
only the newspapers but even the University's official reports (iv) And, in some very important and striking cases,
described the tremendous sensation caused thereby in superlati sums requiring 30, 50, 100 or even more numerous
vely eulogistic terms ; and the papers began to refer to us as and cumbrous "steps" of working (according to the
"the Octogenarian Jagadguru Shankaracharya who hat1 taken current Western methods) car1 be answered in a
Nagpur by storm with his Vedic mathematics", and so on ! single and simple step of work by the Vedic method !
And little children (of only 10 or 12 years of age)
13. I t is manifestly impospible, in the course of a short
merely look a t the sums written on the blackboard
note [in the nature of a "trailer"), to give a full, detailed, tho
roughgoing, comprehensive and exhaustive tlescription of (on the platform) and immediately shout out and
dictate the answers from the body of the convocation
the unique features and stArtling characteristics of all the
mathematical lore in question. Tllis call and will )Je done hall (or other vcnue of the demonstration). And 
this is because, as a matter of fact, each digit automa
in the subsequent volumes of this series (dealing seriatim and
tically yields its predecessor and its successor ! and
in extenso with all the various portions of all the various branches
the children have merely to go on tossing off !or
of mathematics).
reeling off) the digits one after another (forwards or
14. We may, however, a t this point, draw the earnest backwards) by mere ~nental arithmetic (without
attention of every one concerned to the following salient items needing pen or pencil, paper or slate etc) !
thereof : (v) On seeing this kind of work actually being performed
(i) The Siitras (aphorisms) apply to and cover each by the little children, the doctors, professors and
and every part of each and every chapter of each other "bigguns" of mat,liematics are wonder struck
and every branch of mathematics (including ari and exclaim :"Is this mathematics or magic" ? hrld
thmetic, algebra. geometryplane and solid, trigo
we invariably answer and suy : "It is both. I t is
nometryplane and spherical, conicsgeometrical magic until you understand it ; and i t is mathematics
and analytical, astronomy, calculusdifferential thereafter" ; and then we proceed to substantiate
and integral etc., etc. I n fact, there is no part of
and prove t,he correctness of t,his reply of ours ! And
~i~athematics, pure or applied, which is beyond their
jurisdiction ; (vi) As regards the time required by the students for
mastering the whole course of Vedic mathematics
(ii) The S ~ t r a sare easy to understand, easy to apply
as applied to all its branches, we need merely state
and easy to remember ; and the whole work can be
from our actual experience that 8 months (or 12
truthfully summarised in one word "mental" !
months) a t an average rate of 2 or 3 hours per day
19.
In symbolism they succeeded with ten signs to express
should s&ce for completing the whole course of any number most elegantly and simply. I t is this
mathematical studies on these Vedic lines instead of beauty of the Hindu numerical notation which attrac
16 or 20 years required according to the existing ted the attention of all the civilised peoples of the
systems of the Indian and also of foreign uni world and charmed them to adopt it"
versities.
(iii) In this very context, Prof. Ginsburg says:
15. In thia connection, it is a gatifping fact that unlike "The Hindu notation was carried to Arabia about.
some socalled Indologists (of the type hereinabove referred to) 770 A.D. by a Hindu scholar named K ~ K who A
there have been some great modern mathematicians and his was invited from Ujjain to the famous Court of Bagh
torians of mathematics (like Prof. G. P. Halstead, Professor dad by the Abbaside Khalif Al MANS^^. KalYra
Ginsburg, Prof. De Moregan, Prof. Hutton etc.,) who have, taught Hindu astronomy and mathematics to the
as truthseekers and truthlovers, evinced a truly scientific Arabian scholars ; and, with his help, they translated
attitude and frankly expressed their intenee and wholehearted into Arabic the BrahmaSphutaSiddhHntaof Brahma
appreciation of ancient India's grand and glorious contributions Gupta. The recent discovery by the French savant
to the progress of mathematical knowledge (in the Western M.F.NAUproves that the Hindu numerals were well
hemisphere and elsewhere).
known and much appreciated in Syria about the middle
16. The following few excerpts from the published writings of the 7th Century AD". (GINSBURQ'S "NEW LIGHT
of some universally acknowledged authorities in the domain on our numerals", Bulletin of the American Mathe
of the history of mathematics, will speak eloquently fw matical Society, Second series, Vol. 25, pages 3663691.
themselves : (iv) On this point, we find B. B. Dutta further saying :
(i) On page 20 of his book "On the Foundation and "From Arabia, the numerals slowly marched
Technique of Arithmetic", we find Prof. G.P. Halstead towards the West through Egypt and Northem
saying "The importance of the creation of the Arabia; and they h a l l y entered Europe in the
ZERO mark can never be exaggerated. This giving 11th Century. The Europeans called them the Arabic
of airy nothing not merely a local habitation and a notations, because they received them from the
name, a picture but helpful power is the characteristic Arabs. But the Arabs themselves, the Eastern a6
of the Hindu race whence it sprang. It is like
weH as the Western, have unanimously called them
coining the NirvBgB into dynamos. No single
the Hindu figures. (AlArqanAlHindu".)
mathematical creation has been more potent for the
general ongo of intelligence and power". 17. The abovecited passages are, however, in connection
(ii) In this connection, in his splendid treatise on "The with and in appreciation of India's invention of the "ZERO"
present mode of expressing numbers" (the Indian mark and her contributions of the 7th century A.D. and later
Historical Quarterly Vol. 3, pages 630540) B. B. to world mathematical knowledge.
Dutta says: "The Hindus adopted the decimal In the light, however, of the hereinabove given detailed
scale vary early. The nuinerical language of no description of the unique merits and characteristic excellences
other nation is so scientific and has attained as high of the still earlier Vedic Sfitras dealt with in the 16 volumes of
a state of perfection as that of the ancient Hindus.
20.
this series1, the conscientious (truthloving and truthtelling) 3. i4rid all that the student of these Siitras has t o do
historians ~f Mathematics (of the lofty eminence of Prof. De is to look for the special characteristics in question, recognisc
Morgan etc.) have not been guilty of even the least exaggeration the particular type before him and determi~leand npply the
in their randid admission that "even the highest and farthest special formula prescribed therefor.
reaches of modern Western mathematics have not yet brought 4. And, generally speaking it is only in case no special
the Western world even t o the threshold of Ancient Indian Vedic case is involved, that the general formula has to be resorted to.
And this process is naturally a little longer. But i t nced
18. It is our earnest aim and aspiration, in these 16 hardly be pointed out that, even then, the longest of the methods
volumesl, t o explain and expound the contents of the Vedic according t o the Vedic system comes nowhere (in respect of
mathematical Siitras and bring them qithin the easy intellectual length, cumbrousness and tediousness etc.,) near the correspol~d
reach of every seeker after mathematical knowledge. irlg process according to the system now current everywhere.
5. For instance, the conversion of a vulgar fraction
(say & or Jv or 2v etc.,) t o its equivalent recurring decimal
B.EXPLANATORY EXPOSITION shape involves 18 or 28 or 42 or more steps of cumbrous work
OF ing (according t o the current system) but requires only one
SOME SALIENT, INSTRUCTIVE AND single and simple step of nlental working (nccording to
INTERESTING ILLUFTRATIVE SAMPLE SPECIMENS the Vedic Sfitras) !
BY WAY OF 6. This is not all. There are still other methods and
COMPARISON AND CONTRAST processes (in the latter system) whereby even that very small
(mental) working can be rendered shorter still ! This nnd
Preliminary Note :
herein is the beatific beauty of t,he whole scheme.
With regard t o every subject dealt with in the Vedic
Mathematical Siitras, the rule generally holds good that the 7. To start with, we should naturally have liked t o begin
Siitras have always provided for what may be termed the this explanatory and illustrative expositior~wit11 a few pro
'General Case' (by means of simple processes which can be easily cesses in arithmetical corilputatior~srelating to multiplications
and readilynay, instantaneously applied to any and every and divisions of huge nu~nbersby big multipliers and big divisors
question which can possibly arise under any particular heading. respectively and then go on to ot,her branches of mathematical
calculation.
2. But, a t the same time, we often come across special
cases which, although classifiable under the general heading 8. Rnt, as we have just hereinabove referred to a parti
In question, yet present certain additional and typical charac culno hnt wonderful t3ype of mathematiral work wherein 18,
terestics which render them still easier t o solve. And, therefore, 28, 42 or even more steps of working can be condensed into a
special provision is found t o have been made for such special singlestep answer which can be written dowl~ immediately
(by means of what we have been describing as straight, singlc
cases by means of special Siitras, subSEtras, corollaries etc.,
line, mental arithmet,ic) ; nncl, as this statement 111ust naturally
relating and applicable t o those articular types alone.
have aroused intense eagerness and curiosity in the minds of the
' Only one volume has been bequeathed by His Holiness to posterity
students (and thc teachers too) and especially as the process is
cf p. x aboveGeneral Editor.
21.
( d ( d )
based on elementary and basic fundamental principles and 11. Division:
no previous knowledge of anything in the nature (2) Express & in its full recurring decimal shape (18digits) :
of an indispensable and inmapable prerequisite ohapter, BY the current method : The "Sanskrit (Formula) is;
subjeot and 80 on, we are beginning this exposition here with
19) 1 '~('05263157894736842i 11 rpqj%&qIpinlll
an may explanation and a aimple elucidation of that particular 95 By the Vedic mental oneline meth~d:
illustrative specimen. 
50 (by the EkidhiLP&va S u r a )
9. And then we shall take up the other various parts, 38
(forwards or backwards), we merely
one by one, of the various branches of mathematical computation write down the 18dzgitanswer :
and hope to throw sufficient light thereon to enable the students
to make their own comparison and contrast and arrive at
correct conclusions on a11 the various points dealt with. 57

30
 19

110
95
150
C. ILLUSTRATIVE SPECIMEN SAMPLES
(Comparison and Contra&)

133
170
152
SAMPLE SPECIMENS 
180
OR
ARITHMETICAL COMPUTATIONS

171
00
76
I . Multiplication:
ti) Multiply 87265 @/ 32117
..
The "Sanskrit Sfitra" (Formula) is
II aimhwmy 11
140
133
: B y Vedic mental oneline method : 
B~ 70
87265 87285 57
32117 32117 
 2802690005
130 @
610855 114
87285 
87265 Note : Only the answer is writ 160
174530 ten automatically down 152 38
261795 by Ordhwa Tiryak

80 
20
2802690005 S a r a (forwards or back 76 19
wards).

40
1 ,