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INTRODUCTION
The philosophy of education is to acquire right knowledge of different aspects and dimensions of the world around us and to disseminate the same for
the benefit of the individual, society, nation and world.
The present day’s education, though has undergone a radical change, with the development of modern science and technology, especially the
computers, much needs to be done on the holistic approach to education, which can evolve personality. One might get tremendous amounts of information by logging on to the Internet, and the glut of information will
lead to information garbage. In spite of all these we are lagging in imparting right education to right people thereby, creating disharmony. Therefore, one needs to have proper guidance and an evolved system which
will lead to evolution of personality. This system should have a positive impact on the thinking process, beliefs, attitudes with its holistic approach to concepts, methods and techniques; thus creating harmony and
ushering in peace.
Today, in majority of the cases, mathematics is considered to be a dry subject and still notion exists that it is only for the intelligent. Many dread
mathematics, because the teachers themselves teach in an unimpressive way and the joy of making learning of mathematics a pleasurable experience is almost lacking. This is found right from the school to the college
level.
In this regard, Vedic/Ancient Indian Mathematics (VM/AIM) with its wonderful concepts, methods and techniques, which have stood the test of time
comes in handy for imparting a holistic, creative education in mathematics.
The Vedic Seers and Ancient Indian Scientists and Mathematicians perceived mathematics in Nature, in the form of primordial concepts, the different
shapes and patterns symbolising the cosmic truths in the Micro and Macro aspects. This aspect has been very well illustrated in the following Upanisadic Mantra – ’Anoraniyan Mahato Mahiyan’ (Katha Upanishad, II. 20).
Here the ’Supreme being’ is equated in terms of the ’upper bound’ and ’lower bounds’ of a set. That is, HE is lesser than the ’Infimum’ and greater
than the ’Supremum’. The result will be (i) Uncountable (Mathematical) and (ii) Limitless (Metaphysical).
Mathematics was developed right from the fundamental concepts of Numerals, Place value, Zero and Infinity to the advanced concepts finding their
applications in the Artificial Intelligence, computer based numerical methods, philosophy, science, arts, etc. In fact, one will be baffled by the fertile imagination of the ancient seers and scholars, who perceived
mathematics in nature in all its glory and expressed the great mathematical truths and concepts through hymns, theorems and postulates which are lyrically beautiful, yet mathematically precise which speak volumes
about them. It goes without saying that these methods, concepts and techniques of Mathematics in the form of Slokas and Sutras (Aphorisms) were conceived by the Ancient Indian Seers and scholars, centuries before
their modern counterparts.
OBJECTIVE
(a) The present objective is to create an awareness about the vast potential of the Vedic/Ancient Indian Mathematics (V.M./A.I.M.) for
the curriculum development and R & D works.
(b) To demonstrate with practical examples chosen from different levels of mathematics education; the benefits of learning V.M./A.I.N.
like faster computing time, alternative and creative approach to problem solving, etc. This is substantiated by the results of number of workshops and case studies conducted.
(c) To motivate scientists scholars, teachers, philosophers and others to take up an in-depth study of VM/AIM in proper perspective and
adopt it to the modern times, so that a great amount of this wonderful traditional knowledge is revived and developed which would be a true homage to the Seers and Scientists of yore.
(d) Last but not the least, to pave way for the adoption of time tested VM/AIM in the school/college curriculum and establish dedicated
RBD centres.
APPROACH
An Integrated approach for the study and learning VM/AIM is made:
(i) Mathematical concepts embodied in the Vedas, Upanishads.
(ii) ’Vedic Mathematics’ as expounded by H. H. Bharatikrishna Teerthaji consisting of 16 Aphorisms and 16 Corollaries, covering
different topics in Mathematics, right from the fundamental Arithmetic operations to the higher level of Mathematics.
(iii) Ancient Indian Mathematics – contribution by various mathematicians of yore up to the celebrated genius Ramanuja. This integrated
approach is unique of its kind. Various methods techniques are culled out, analysed and are expressed in modern mathematical notations, making it easily adaptable.
SELECTION CRITERIA
The methods and techniques are selected based on the following criteria. If the VM/AIM methods are better than the conventional ones in terms of step
size, length, computation time, elegance and novelty, either individually or collectively, then these are selected in place of the conventional ones.
TECHNIQUE
The teaching of VM/AIM is done following the usual techniques like lecturing, using Audio-visual equipments like OHP, slide projector, computer aided
learning, etc.
Before we go into the subject of VM/AIM in terms of examples, it is imperative to know at least some fundamental facts about the Vedas, their nature,
structure, etc., which would facilitate in understanding and appreciating the mathematical concepts embodied in the Vedas.
Vedas are the earliest systematic literature in the entire world existing since thousands of years, which have stood the test of time and continue to
challenge the mankind with renewed freshness and vitality. Vedas, which are the fountain head of knowledge and wisdom, are built upon sound structure which are definite, clear, unambiguous, generative and complete,
thus satisfying the properties of an algorithm! This amply proves the fact that the Vedas are the highest revelations to the Rsis (Seers) of Yore and are not a mere compendium and collection of literature from
different parts of the world, and no second version of Vedas exists anywhere. Hence, the Vedas have to be studied and analysed in proper perspective, in order to bring out the truths hidden in them. Now let us take
some stock illustrations from Vedas and other ancient Indian sources in Mathematics at different levels i.e. right from the fundamental level of Arithmetic to Algebra, Calculus, Quadratics, Polynomials, Astronomy,
Geometry, etc. which will give a glimpse of the vast scope of the same.
Now coming to the Vedas, the most outstanding, fundamental contribution for which the entire world is beholden to ancient India is the invention of
decimal numerals (with place values), zero and infinity, for example: 576, 685, 1998, etc. Can there be a more elegant, better method where the digits placed in the units, tenths, hundredths or thousandth place,
indicate their respective values and magnitude?
Right from the Vedic times Dasa (ten) has formed the basis of enumeration in India though we find bases of 9 & 2 in later works. We find
a list of numeral denominations in powers of 10 raised to 12 (i.e. 1 followed by 12 zeros) is called Parardha. [Yaj. Veda Samhita (Vajasaneyi. XVII.2)] and found extended to 1019 (Loka) with some altered terminologies in the Miatrayani (II.8.14) and Kathaka Samhitas (XVII.10). But, in the Tandya Brahmana (VII.14.2) of Samaveda there is a mention of 32 numerical places! (Dvatrimsat Sankhyasthanah) i.e. 1,10,....1000 – 1025, 1031. The famous Valmiki Ramayana there is enumeration up to 1060 (Mahougha) [Val. Ram. Yuddhakanda. 28,33-38]. Similarly in the Jaina and Buddhist works like Tatvarthadigama Sutra, Suryapragnapati, Anuyogadwarasutra (100 – 600 BC) we find numbers of astronomical dimensions.
For example, we find micro and macro numbers known as Avasannasanna
(X-x secs) X = YYY…(134 times) where Y=1010 10…134
whereas ’Sirsaprahelika (a period) is equal to 8,400,00036 years according to Jyotisakarandaka. Similarly in a famous and
valuable Buddhist work called Lalitavistara (120 BC) one finds various series of counting and find numbers extended to a gigantic 10421! and each unit has been given a specific name in all the above
references. But, for the contemporary Greeks, the highest number was just 10,000 i.e. 104 which they called Myriad !
It makes one think seriously why the ancient Indians used such gigantic numbers. These quantities are especially used with Kala (time) and other
astronomy related topics. Similarly one finds very large numbers used on the microscale used in material science, and in time references. Similarly many references can be cited on topics like progressions,
arithmetic operations.
In arithmetic operations one finds use of complimentary numbers very effectively and finds 10 different varieties of multiplication based on the Vedic
mathematical Sutras propounded by H. H. Bharatikrishna Teertha (1880-1964). He being a brilliant scholar and Saint discovered 16 aphorisms and 16 corollaries after 8 years of intense penance, covering variegated
topics (both elementary and advanced) in mathematics. Consider the following examples:
(i) Complimentary numbers Here the objective is to convert all the numerals greater than 5 to less than 5 in order to facilitate speedy
calculation and reduce carry over. For ex. 9,8,7,6 are expressed as 1,2,3 & 4 respectively with base as 10.
(ii) Using complimentary notation known as ’Vinculum’ following examples will make clear, about the usage and benefits of this:
(a) 28 -> 32 [eq. to 30-2]
(b) 19 -> 21 [eq. to 20-1]
(c) 278 -> 322 [eq. 300-22]
(d) 289621 -> 310421
Note: whenever complimentary notations are used, the digit preceding, gets augmented by 1 and while normalising it gets reduced by 1. In
the example (d) we see how only the digits greater than 5 are converted and the others remain as they are.
(iii) Generation of Multiplication Table
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+ -
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In this table of 198 we observe that just by adding 2 & subtracting 2 successively in the hundredths and unit’s place,
this table could be generated in a jiffy. These methods and techniques, if incorporated right at the school level, will make a child learn mathematics in a much better way.
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202
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01
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198
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02
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396
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03
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594
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04
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792
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05
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990
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06
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1188
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07
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1386
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08
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1584
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09
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1782
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10
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1980
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(iv) Let us consider some typical examples in multiplication. We know how cumbersome,
the conventional method whenever multiplication involves more digits.
(a) Multiply 88 by 96. Base 100
88 12
96 04
84/48 i.e. 8448
The VM method gives straight away the answer in a jiffy!
(b) Base 100,000: Multiply 96578 by 99997
96 578 99 997
99 997 00 003
96575/10266
(c) Base 100: Multiply 107 by 105
107 07
105 05
112/35
The solution of these makes use of 3 Sutras; viz., Nikhilam .... Urdhwa Tiryagbhyam and
Yavadunam Tavadunikrtya Vargam ca Yojayet. This method is adopted to multiply any two numbers which are nearer to the base, the base being selected according to the magnitude
of the numbers. Here in example (a) the deficiencies 12 & 04 are cross subtracted and then multiplied to get the answer 8448. Same procedure is adopted to the other example (c).
Algebraically, this is of the form (x-a) (x-b). If the numbers are above the base, then it becomes (x+a) (x+b) i.e., the differences are further added as illustrated in example (b).
Division
In division also we have various methods to suit the problems. Let us consider the following examples:
Divide 1122 by 89.
In a conventional way it will take lot of time and also quite cumbersome when we are dealing
with higher digits. The VM method as we shall see now is very elegant and faster than the conventional one.
89) 1 1 2 2
1 1
11 2 2
1 2 5 4
Method : Take the complement of 89 i.e. 11. Bring down first 1 into the answer, multiply
each of the deficiency-digits by it, and put the two products under the next two figures. Add the second column, put down the result, multiply the deficiency-digits by it and put the
products in the last two columns. Add up the remainder. So q = 12 and r = 54 !
Divide 121234 by 8998
Even in this case, the answer can be easily written down as c = 13 and r = 4260. The
speciality of the method is when the procedure is continued one gets the accuracy up to any number of decimal places.
8998) 1 2 1 2 3 4
1002 1 0 0 2
3 0 0 6
1 3 4 2 6 0
The procedure followed here is same as the above.
Now let us consider the division of polynomials.
Divide x3+6x2+13x+13 by x2+2x+3
x2 +2x +3) x3 +6x2 +13x +13
-2 -3 -2 -3
-8 -12
x +4 R 2x +1
Here we have two transposed figures -2 and -3 : x3 + x2 = x which goes into the answer. Its
coefficient, 1, then multiplies both the transposed figures, so we put -2, -3 in the next two columns, and the second column to get +4, put it down and multiply both transposed figures
by it, put -8, -12, in the last two columns, and add up the remainder. So we have Q = x+4 and R = 2x+l.
This method can also be adopted in computer based numerical methods.
We see here how one VM method can be very effectively used to solve such problems in an elegant and faster way.
Similarly, there are lot of interesting methods to solve quadratics and biquadratics. Now let us consider a few examples.
x – 2 x – 3 x – l x – 4
----- + ----- = ----- + -----
x – 3 x – 4 x – 2 x – 5
The value of X can be straight away written down as
x = 7/2 & 5/2
The formula used is D1 + D2 = D3 + D4 where D1....D4 are denominators. Also, N1 + N2 = N3
+ N4 where Nl .... N4 are numerators. So we get two values of x as indicated above.
Before we take up other topics in Mathematics we cannot help but quote some examples on
the philosophical aspect of numbers, which the ancient seers very effectively utilised.
Now let us consider this very interesting example. The idea of counting nine numbers and
zero which is an Indian system is closely connected with nine months of development of human embryo. Man develops in nine months through nine successive stages, like the nine
number and completes his birth in the 10th and this is the 10th Avatara of ’MAN’ to put it figuratively. Similarly the body has nine openings (Navadvara) and the 10th being the
Brahmadvara which can be located in the soft region in the head in a new born infant. This small micro-opening called the Brahmarandhra. At the top of the head through which a yogi
leaves the body. Thus the 10th opening is the gate to the highest stage of development, similar to his birth from the womb in the 10th month. In the former he is delivered downwards
and in the lateral upwards! This is the essence of story of deliverance of man from the imprisoning restrictions of matter which has been beautifully delineated in many philosophical Tantric and musical texts.
Similarly the concept of Sets and cardinal numbers are conceived by the Seers and scholars.
For example we have deep significance for numbers like seven as Saptaswara (seven basic notes in Music), Sapta Rsis (seven great sages), Saptavarna (seven colours), nine planets
(Navagrahas), Navanidhi (nine treasures), Dasadik (ten directions), Ekadasa Rudra (eleven Rudras), Dvadasamasa (twelve months), Saptavimsat Nakshatrah (27 stars) etc.
Similarly we can quote many examples from different branches of mathematics, even in
geometry: the Sulva Sutras (which are the origin of geometry) make use of this branch of mathematics to construct varieties of altars for the purposes of sacrifices. The magnificent
temples which stand testimony to the skills and designs of the workers are the living examples of applied mathematics.
Mathematics was also brought into worship by use of Mantrabandhas which are nothing but
mystical geometric diagrams inscribed with digits or cosmic energy capsules (beejaksharas). These diagrams are of various sizes, shapes and colours. The word of Mantra and Tantra is
indeed a network of energies, forces and vectors and are very dynamic. These geometric forms indicate outward manifestation of the spirit so as to exert its influence in the desired
manner. The power possessed by the Yantra (graphical representation) is sought to be stepped up by their inscription and infusion of the appropriate Mantra. In short these can be
termed in the modern context as the printed circuit boards which vibrate into light through the electrical energy passed and performs the desired functions independently and
collectively. The Yantra is the basic hardware and the Mantra is the software.
These clearly indicate the multidimensional approach of the ancient seers and scholars
towards mathematics. Hence, one may conclude mathematics edifices philosophy and philosophy beautifies mathematics. So, one can say that philosophy in mathematics are supplementary in complementary to each other.
Resume in a Life’s Equations for excellence
(+) Add good friends
(-) Subtract bad qualities
(x) Multiply one’s strengths
(:) Divide the work
(d) Differentiate right and wrong
( ) Integrated personality.
At this juncture one cannot help but quote the great Jaina Mathematician Mahaviracarya
(9th cent. AD) who has extolled the inevitability and greatness of mathematics in the following resonant verse:
Lokike Vaidike Va apitatha Samayike apiva |
Vyaparastatra Sarvatra Samkhyanam Upayujyate |
Bahubhirvipralpaih Kim trailokye Sacharachare |
yatkincit vastu tatsarvam Ganitena vina na hi ||
(Ganitasarasangrah. 1.9,16)
In all the worldly life or vedic matters or even in the religious things whatever be the
dealings, everywhere enumeration is essential. Why say much? In the entire three worlds, living or non-living, whatever is transacted, that cannot be without mathematics
CONCLUSIONS
1. Vedic/Ancient Indian Mathematics (VM/AIM) with their novel methods, techniques
and approach help the learners to develop their aptitude and creative rational thinking.
2. The VM/AIM which have stood the test of time for centuries with their novel
creative approach and techniques help the students to develop their intuitive capacity, which in turn has a positive impact on the right half of the brain, paving way for the holistic development.
3. Many workshops and courses conducted for students of different classes, be it
urban or rural, have proved the efficacy of these and the results are very encouraging.
4. It has been found from practical trials that one year portion of school curriculum in
mathematics can be completed in about 60% of the time, thus making learning faster and enjoyable. In fact VM/AIM has been popularly known as ’Maths with Smiles’.
5. This will prove to be a boon for all the competitive exams, with its much faster and accurate results and can be adopted globally.
6. Learning of VM/AIM leads to education in different dimensions and appreciate
philosophy, religion in better perspective which in turn has a positive impact on the personality development, leading to harmony and peace.
7. VM/AIM open up new vistas for R&D, especially on computer based numerical
methods, development of new algorithms apart from opening up of frontier areas of interdisciplinary research. Example: Nyayasastra (one of the six Darshanas has
wonderful scope in Artificial Intelligence). The author has already culled out and developed about 15 computer based numerical methods based on VM/AIM which are unique of their kind.
8. The Ancient Indian Mathematics not only kept up the curriculum development, but
also made mathematics interesting to house wives, common people and people from non-mathematical areas of study. They composed aphorisms, slokas and hymns, which
were lyrically beautiful yet mathematically precise, thus contributing to the holistic development of the subject.
9. The Mantra/Yantrabandhas are the repository of Esoteric geometry thus bringing
Mathematics into worship. These mystic diagrams indicate outward manifestation of the spirit, so as to exert its influence in the desired manner. The ancients had also
developed excellent mathematical coding system apart from the literary codes in order to preserve the vast amount of literature in their pristine form through generations.
10. Last but not the least, study and practice of VM/AIM with their creative and novel
approach can aid in holistic development of personality. This will not only revive the ancient treasure of knowledge and wisdom, but also will be a true homage to the great Seers and Scholars of Yore.
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