VEDIC MATHEMATICS
OR
Sixteen simple Mathematical
Formulae from the Vedas
(For One-Une 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
The Author I
Jagadguru S a i ~ k a r i c i r ~ a MOTILAL BANARSIDASS
Sri Bharati K ~ s n aTirtha MahBrSjtja
(1884-1960)
Delhi :: Varanasi :: Patna
I
, 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 Vice-Chancellor of this University
I accept the donation and offer my heart-felt 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 : 41-U.A., Bungalow Road, Delhi-110007 Date 27-3-65 Vim-Chancellor
Braaches : 1. Chowk, Varanasc-1 (u.P.) Banaras Hindu University
2. Ashok Rajpath. Patna-4 (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
A-45 Phase I adu us trial Area Naraina. 'New Delhi-110 028
Pubished by hlarendra ~ r a k d hJain, for Motilal Banarsidass,
Bungalow Road, Jawahar Nagar. Delhi-110007.
, 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 (1884-1960).
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
I
appropriately given the title of "mental" mathematics appearing
more as miracle than the uaual approach of hard-baked 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 store-home 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 other-worldly) matters but also to those usually dascnbed
aa purely 'neonlar', 'temporal', or 'worldly' and also to the means required by
I
humaniay ae such for the aohievemsnt of all-round, 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.
, "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) all-round, 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 sub-sutras or
corollaries is prefixed in the beginning of the text and
It is the n hole essence of his assessment of Vedic tradition the style of language also points to their discovery by Sri
that it is not to be approached from a factual standpoint but SwZmiji himself. At any rate, it is needless t o dwell longer on
from the ideal standpoint viz, as the Vedas as traditionally 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, 1909-10. 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-
OR
Sixteen simple Mathematical
Formulae from the Vedas
(For One-Une 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
The Author I
Jagadguru S a i ~ k a r i c i r ~ a MOTILAL BANARSIDASS
Sri Bharati K ~ s n aTirtha MahBrSjtja
(1884-1960)
Delhi :: Varanasi :: Patna
I
, 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 Vice-Chancellor of this University
I accept the donation and offer my heart-felt 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 : 41-U.A., Bungalow Road, Delhi-110007 Date 27-3-65 Vim-Chancellor
Braaches : 1. Chowk, Varanasc-1 (u.P.) Banaras Hindu University
2. Ashok Rajpath. Patna-4 (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
A-45 Phase I adu us trial Area Naraina. 'New Delhi-110 028
Pubished by hlarendra ~ r a k d hJain, for Motilal Banarsidass,
Bungalow Road, Jawahar Nagar. Delhi-110007.
, 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 (1884-1960).
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
I
appropriately given the title of "mental" mathematics appearing
more as miracle than the uaual approach of hard-baked 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 store-home 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 other-worldly) matters but also to those usually dascnbed
aa purely 'neonlar', 'temporal', or 'worldly' and also to the means required by
I
humaniay ae such for the aohievemsnt of all-round, 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.
, "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) all-round, 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 sub-sutras or
corollaries is prefixed in the beginning of the text and
It is the n hole essence of his assessment of Vedic tradition the style of language also points to their discovery by Sri
that it is not to be approached from a factual standpoint but SwZmiji himself. At any rate, it is needless t o dwell longer on
from the ideal standpoint viz, as the Vedas as traditionally 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, 1909-10. 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-
