5. VEDIC MATHEMATICS - details

This original work on Vedic Mathematics begins with various introductions prefaces etc., illustrative specimen samples and a list of the Sutras and their corollaries. The book covers a considerable range of topics and is intended as an introduction to Vedic Mathematics. The book has 40 chapters, 367 pages and covers arithmetic, solution of equations, factorisation, divisibility, square roots, recurring decimals etc. 1994, revised edition (first published 1965). Author: Bharati Krsna Tirthaji. Approx A5 size. Price 9 pounds (paperback). 

"Vedic Mathematics" - Preface

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 "mental"!

(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 and elsewhere).

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