A- A DESCRIPTIVE PREFATORY NOTE ON THE ASTOUNDING WONDERS OF ANCIENT
INDIAN VEDIC MATHEMATICS
1. In the course of our discourses on manifold and multifarious subjects
(spiritual, metaphysical, philosophical, psychic, psychological, ethical,
educational, scientific, mathematical, 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 scientifically- nay, mathematically- accurate
lines in them all and so on.
2. The very word "Veda" has this derivational meaning, i.e. the fountain-head
and illimitable store-house of all knowledge. This derivation, in
effect, means, connotes and implies that the Vedas should contain
within themselves all the knowledge needed by mankind relating not
only to the so-called 'spiritual' (or other-worldly) matters but also
to those usually described as purely "secular", "temporal", or "worldly";
and also to the means required by humanity as such for the achievement
of all-round, 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 all-round 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.
4. It is thus in the fitness of things that the Vedas include (i)
Ayurveda (anatomy, physiology, hygiene, sanitary science, medical
science, surgery etc., etc.,) not for the purpose of achieving perfect
health and strength in the after-death future but in order to attain
them here and now in our present physical bodies; (ii) Dhanuveda (archery
and other military sciences) not for fighting with one another after
our transportation to heaven but in order to quell and subdue all
invaders from abroad and all insurgents from within; (iii) Gandharva
Veda (the science and art of music) and (iv) Sthapatya Veda (engineering,
architecture etc., and all branches of mathematics in general). All
these subjects, be it noted, are inherent parts of the Vedas i.e.
are reckoned as "spiritual" studies and catered for as such therein.
5. Similar is the case with regard to the Vedangas (i.e. grammar,
prosody, astronomy, lexicography etc., etc.,) which, according to
the Indian cultural perceptions, are also inherent parts and subjects
of Vedic (i.e. Religious) study.
6. As a direct and unshirkable consequence of this analytical and
grammatical study of the real connotation and full implications of
the word "Veda" and owing to various other historical causes of a
personal character (into details of which we need not now enter),
we have been from our very early childhood, most earnestly and actively
striving to study the Vedas critically from this stand-point and to
realise and prove to ourselves (and to others) the correctness (or
otherwise) of the derivative meaning in question.
7. There were, too, certain personal historical reasons why in our
quest for the discovering of all learning in all its departments,
branches, sub-branches etc., in the Vedas, our gaze was riveted mainly
on ethics, psychology and metaphysics on the one hand and on the "positive"
sciences and especially mathematics on the other.
8. And the contemptuous or, at best patronising attitude adopted by
some so-called Orientalists, Indologists, antiquarians, research-scholars
etc., who condemned, or light-heartedly, nay; irresponsibly, frivolously
and flippantly dismissed, several abstruse-looking and recondite parts
of the Vedas as "sheer-nonsense"- or as "infant-humanity's prattle",
and so on, merely added fuel to the fire (so to speak) and further
confirmed and strengthened our resolute determination to unravel the
too-long hidden mysteries of philosophy and science contained in 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 thereof.
9. And we were agreeably astonished and intensely gratified to find
that exceedingly tough mathematical problems (which the mathematically
most advanced present day Western scientific world had spent huge
lots of time, energy and money on and which even now it solves with
the utmost difficulty and after vast labour and involving large numbers
of difficult, tedious and cumbersome "steps" of working) 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 ATHARVAVEDA in a few simple steps and by methods which can
be conscientiously described as mere "mental arithmetic".
10. 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, by means of lectures, blackboard-demonstrations,
regular classes and so on in schools, colleges, universities etc.,
all over the country and have been astounding our audiences everywhere
with the wonder and marvels not to say, miracles of Indian Vedic Mathematics.
11. We were thus at last enabled to succeed in attracting the more
than passing attention of the authorities of several Indian universities
to this subject. And, in 1952, the Nagpur University not merely had
a few lectures and blackboard-demonstrations given but also arranged
for our holding regular classes in Vedic Mathematics (in the University's
Convocation Hall) for the benefit of all in general and especially
of the University and college professors of mathematics, physics etc.
12. And, consequently, the educationists and the cream of the English
educated section of the people including the 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! and not only the newspapers but even the University's
official reports described the tremendous sensation caused thereby
in superlatively eulogistic terms; and the papers began to refer to
us as " the Octogenarian Jagadguru Shankaracharya who had taken Nagpur
by storm with his Vedic Mathematics", and so on!
13. It is manifestly impossible, in the course of a short note (in
the nature of a "trailer"), to give a full, detailed, thorough-going,
comprehensive and exhaustive description of the unique features and
startling characteristics of all the mathematical lore in question.
This can and will be done in the subsequent volumes of this series
(dealing seriatim and in extenso with all the various portions of
all the various branches of mathematics).
14. We may, however, at this point, draw the earnest attention of
everyone concerned to the following salient items thereof:-
(i) The Sutras (aphorisms) apply to and cover each and every part
of each and every chapter of each and every branch of mathematics
(including arithmetic, algebra, geometry- plane and solid, trigonometry-
plane and spherical, conics- geometrical and analytical, astronomy,
calculus- differential and integral etc., etc. In fact, there is no
part of mathematics, pure or applied, which is beyond their jurisdiction;
(ii) The Sutras are easy to understand, easy to apply and easy to
remember; and the whole work can be truthfully summarised in one word
(iii) Even as regards complex problems involving a good number of
mathematical operations (consecutively or even simultaneously to be
performed), the time taken by the Vedic method will be a third, a
fourth, a tenth or even a much smaller fraction of time required according
to the modern (i.e. current) Western methods;
(iv) And, in some very important and striking cases, sums requiring
30, 50, 100 or even more numerous and cumbrous "steps" of working
(according to the current Western methods) can be answered in a single
and simple step of work by the Vedic method! And little children (of
only 10 or 12 years of age) merely look at the sums written on the
blackboard (on the platform) and immediately shout out and dictate
the answers from the body of the convocation hall (or other venue
of demonstration). And this is because, as a matter of fact, each
digit automatically yields its predecessor and its successor! and
the children have merely to go on tossing off (or reeling off) the
digits one after another (forwards or backwards) by mere mental arithmetic
(without needing pen or pencil, paper or slate etc)!
(v) On seeing this kind of work actually being performed by the little
children, the doctors, professors and other "big-guns" of mathematics
are wonder struck and exclaim:- "Is this mathematics or magic?" And
we invariably answer and say: "It is both. It is magic until you understand
it; and it is mathematics thereafter"; and then we proceed to substantiate
and prove the correctness of this reply of ours! And
(vi) as regards the time required by the students for mastering the
whole course of Vedic Mathematics as applied to all its branches,
we need merely state from our actual experience that 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 Indian and also of foreign universities.
15. In this connection, it is a gratifying fact that unlike some so-called
Indologists (of the type hereinabove referred to) there have been
some great modern mathematicians and historians of mathematics (like
Prof. G. P. Halstead, Professor Ginsburg, Prof. De Morgan, Prof. Hutton
etc.,) who have, as truth-seekers and truth-lovers, evinced a truly
scientific attitude and frankly expressed their intense and whole-hearted
appreciation of ancient India's grand and glorious contributions to
the progress of mathematical knowledge (in the Western hemisphere
16. The following few excerpts from the published writings of some
universally acknowledged authorities in the domain of the history
of mathematics, will speak eloquently for themselves:-
(i) On page 20 of his book "On the Foundation and Technique of Arithmetic",
we find Prof. G. P. Halstead saying "The importance of the creation
of the zero mark can never be exaggerated. This giving of airy nothing
not merely a local habitation and a name, a picture but helpful power
is the characteristic of the Hindu race whence it sprang. It is like
coining the Nirvana into dynamos. No single mathematical creation
has been more potent for the general on-go of intelligence and power".
(ii) In this connection, in his splendid treatise on "The present
mode of expressing numbers" (the Indian Historical Quarterly Vol.
3, pages 530-540) B. B. Dutta says "The Hindus adopted the decimal
scale very early. The numerical language of no other nation is so
scientific and has attained as high a state of perfection as that
of the ancient Hindus. In symbolism they succeeded with ten signs
to express any number most elegantly and simply. It is this beauty
of the Hindu numerical notation which attracted the attention of all
the civilised peoples of the world and charmed them to adopt it".
(iii) In this very context, Prof. Ginsburg says:- "The Hindu notation
was carried to Arabia about 770 A.D. by a Hindu scholar named KANKA
who was invited from Ujjain to the famous court of Baghdad by the
Abbaside Khalif Al-MANSUR. Kanka taught Hindu astronomy and mathematics
to the Arabian scholars; and, with his help, they translated into
Arabic the Brahma-Sphuta-Siddhanta of Brahma Gupta. The recent discovery
by the French savant M. F. NAU proves that the Hindu numerals were
well known and much appreciated in Syria about the middle of the 7th
Century A.D. (GINSBURG'S "New light on our numerals", Bulletin of
the American Mathematical Society, Second Series, Vol. 25, pages 366-369).
(iv) On this point, we find B. B. Dutta further saying: "From Arabia,
the numerals slowly marched towards the West through Egypt and Northern
Arabia; and they finally entered Europe in the 11th Century. The Europeans
called them the Arabic notations, because they received them from
the Arabs. But the Arabs themselves, the Eastern as well as the Western,
have unanimously called them the Hindu figures. (Al-Arqan-Al-Hindu)."
17. The above-cited passages are, however, in connection with, and
in appreciation of India's invention of the "ZERO" mark and her contributions
of the 7th century A.D. and later to world mathematical knowledge.
In the light , however, of the hereinabove given detailed description
of the unique merits and characteristic excellences of the still earlier
Vedic Sutras dealt with in the 16 volumes of this series, the conscientious
(truth-loving and truth-telling) historians of mathematics (of the
lofty eminence of Prof. De Morgan etc.) have not been guilty of even
the least exaggeration in their candid admission that "even the highest
and farthest reaches of modern Western mathematics have not yet brought
the Western world even to the threshold of Ancient Indian Vedic Mathematics".
18. It is our earnest aim and aspiration, in these 16 volumes, to
explain and expound the contents of the Vedic Mathematical Sutras
and bring them within the easy intellectual reach of every seeker
after mathematical knowledge.
"Vedic Mathematics" - Contents
1 ACTUAL APPLICATIONS OF THE VEDIC SUTRAS
2 ARITHMETICAL COMPUTATIONS
3 MULTIPLICATION PRACTICAL APPLICATION IN "COMPOUND MULTIPLICATION"
PRACTICE AND PROPORTION IN "COMPOUND MULTIPLICATION"
4 DIVISION BY THE NIKHILAM METHOD
5 DIVISION BY THE PARAVARTYA METHOD
6 ARGUMENTAL DIVISION LINKING NOTE (Recapitulation and Conclusion)
7 FACTORISATION (of Simple Quadratics)
8 FACTORISATION (of Harder Quadratics)
9 FACTORISATION OF CUBICS ETC.
10 HIGHEST COMMON FACTOR
11 SIMPLE EQUATIONS (First Principles)
12 SIMPLE EQUATIONS (by Sunyam etc.)
13 MERGER TYPE OF SIMPLE EASY EQUATIONS
14 COMPLEX MERGERS
15 SIMULTANEOUS SIMPLE EQUATIONS
16 MISCELLANEOUS (Simple) EQUATIONS
17 QUADRATIC EQUATIONS
18 CUBIC EQUATIONS
19 BI-QUADRATIC EQUATIONS
20 MULTIPLE SIMULTANEOUS EQUATIONS
21 SIMULTANEOUS QUADRATIC EQUATIONS
22 FACTORISATION AND DIFFERENTIAL CALCULUS
23 PARTIAL FRACTIONS
24 INTEGRATION BY PARTIAL FRACTIONS
25 THE VEDIC NUMERICAL CODE
26 RECURRING DECIMALS
27 STRAIGHT DIVISION
28 AUXILIARY FRACTIONS
29 DIVISIBILITY AND SIMPLE OSCULATORS
30 DIVISIBILITY AND COMPLEX MULTIPLEX OSCULATORS
31 SUM AND DIFFERENCE OF SQUARES
32 ELEMENTARY SQUARING, CUBING ETC.
33 STRAIGHT SQUARING
34 VARGAMULA (Square root)
35 CUBE ROOTS OF EXACT CUBES
36 CUBE ROOTS (General)
37 PYTHAGORAS' THEOREM ETC.
38 APOLLONIUS' THEOREM
39 ANALYTICAL CONICS
40 MISCELLANEOUS MATTERS
RECAPITULATION AND CONCLUSION