Introduction
The Sanskrit word Veda is derived from the root Vid, meaning to know without limit. The word Veda covers all Veda-sakhas known to humanity. The Veda is a repository of all knowledge, fathomless, ever revealing as it is delved deeper.
Swami Bharati Krishna Tirtha (1884-1960), former Jagadguru Sankaracharya of Puri culled a set of 16 Sutras (aphorisms) and 13 Sub – Sutras (corollaries) from the Atharva Veda. He developed methods and techniques for amplifying the principles contained in the aphorisms and their corollaries, and called it Vedic Mathematics.
According to him, there has been considerable literature on Mathematics in the Veda-sakhas. Unfortunately most of it has been lost to humanity as of now. This is evident from the fact that while, by the time of Patanjali, about 25 centuries ago, 1131 Veda-sakhas were known to the Vedic scholars, only about ten Veda-sakhas are presently in the knowledge of the Vedic scholars in the country.
The Sutras apply to and cover almost every branch of Mathematics. They apply even to complex problems involving a large number of mathematical operations. Application of the Sutras saves a lot of time and effort in solving the problems, compared to the formal methods presently in vogue. Though the solutions appear like magic, the application of the Sutras is perfectly logical and rational. The computation made on the computers follows, in a way, the principles underlying the Sutras. The Sutras provide not only methods of calculation, but also ways of thinking for their application.
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Application of the Sutras improves the computational skills of the learners in a wide area of problems, ensuring both speed and accuracy, strictly based on rational and logical reasoning. The knowledge of such methods enables the teachers to be more resourceful to mould the students and improve their talent and creativity. Application of the Sutras to specific problems involves rational thinking, which, in the process, helps improve intuition that is the bottom – line of the mastery of the mathematical geniuses of the past and the present such as Aryabhatta, Bhaskaracharya, Srinivasa Ramanujan, etc.
I. Why Vedic Mathematics?
Many Indian Secondary School students consider Mathematics a very difficult subject. Some students encounter difficulty with basic arithmetical operations. Some students feel it difficult to manipulate symbols and balance equations. In other words, abstract and logical reasoning is their hurdle.
Many such difficulties in learning Mathematics enter into a long list if prepared by an experienced teacher of Mathematics. Volumes have been written on the diagnosis of ‘learning difficulties’ related to Mathematics and remedial techniques. Learning Mathematics is an unpleasant experience to some students mainly because it involves mental exercise.
Of late, a few teachers and scholars have revived interest in Vedic Mathematics which was developed, as a system derived from Vedic principles, by Swami Bharati Krishna Tirthaji in the early decades of the 20th century.
Dr. Narinder Puri of the Roorke University prepared teaching materials based on Vedic Mathematics during 1986 – 89. A few of his opinions are stated hereunder:
i) Mathematics, derived from the Veda, provides one line, mental and super- fast methods along with quick cross checking systems.
ii) Vedic Mathematics converts a tedious subject into a playful and blissful one which students learn with smiles.
iii) Vedic Mathematics offers a new and entirely different approach to the study of Mathematics based on pattern recognition. It allows for constant expression of a student’s creativity, and is found to be easier to learn.
iv) In this system, for any problem, there is always one general technique applicable to all cases and also a number of special pattern problems. The element of choice and flexibility at each stage keeps the mind lively and alert to develop clarity of thought and intuition, and thereby a holistic development of the human brain automatically takes place.
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v) Vedic Mathematics with its special features has the inbuilt potential to solve the psychological problem of Mathematics – anxiety.
J.T.Glover (London, 1995) says that the experience of teaching Vedic Mathematics’ methods to children has shown that a high degree of mathematical ability can be attained from an early stage while the subject is enjoyed for its own merits.
A.P. Nicholas (1984) puts the Vedic Mathematics system as ‘one of the most delightful chapters of the 20th century mathematical history’.
Prof. R.C. Gupta (1994) says ‘the system has great educational value because the Sutras contain techniques for performing some elementary mathematical operations in simple ways, and results are obtained quickly’.
Prof. J.N. Kapur says ‘Vedic Mathematics can be used to remove math- phobia, and can be taught to (school) children as enrichment material along with other high speed methods’.
Uses of vedic mathematics
* It helps a person to solve a mathematical problem 10-15 times faster
* It helps in developing intelligent
* It reduces burden
* It is a magical tool which reduce scratch work and finger counting
* It increases concentration
* It helps in reducing silly mistakes
II. Vedic Mathematical Formulae
What we call VEDIC MATHEMATICS is a mathematical elaboration of ‘ Sixteen Simple Mathematical formulae from theVedas ‘ as brought out by Sri Bharati Krishna Tirthaji . In the text authored by the Swamiji, nowhere has the list of the Mathematical formulae (Sutras) been given. But the Editor of the text has compiled the list of the formulae from stray references in the text. The list so compiled contains Sixteen Sutras and Thirteen Sub – Sutras as stated hereunder.
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SIXTEEN SUTRAS
# | Name | Corollory | Meaning |
1 | Ekadhikena Purvena | Anurupyena | By one more than the previous one |
2 | Nikhilam Navatashcaramam Dashatah | Sisyate Sesasamjnah | All from 9 and the last from 10 |
3 | Urdhva-Tiryagbyham | Adyamadyenantyamantyena | Vertically and crosswise |
4 | Paraavartya Yojayet | Kevalaih Saptakam Gunyat | Transpose and adjust |
5 | Shunyam Saamyasamuccaye | Vestanam | When the sum is the same that sum is zero |
6 | Anurupye Shunyamanyat | Yavadunam Tavadunam | If one is in ratio, the other is zero |
7 | Sankalana-vyavakalanabhyam | Yavadunam Tavadunikritya Varga Yojayet | By addition and by subtraction |
8 | Puranapuranabyham | Antyayordashake’pi | By the completion or non-completion |
9 | Chalana-Kalanabyham | Antyayoreva | Differences and Similarities |
10 | Yaavadunam | Samuccayagunitah | Whatever the extent of its deficiency |
11 | Vyashtisamanstih | Lopanasthapanabhyam | Part and Whole |
12 | Shesanyankena Charamena | Vilokanam | The remainders by the last digit |
13 | Sopaantyadvayamantyam | Gunitasamuccayah Samuccayagunitah | The ultimate and twice the penultimate |
14 | Ekanyunena Purvena | Dhvajanka | By one less than the previous one |
15 | Gunitasamuchyah | Dwandwa Yoga | The product of the sum is equal to the sum of the product |
16 | Gunakasamuchyah | Adyam Antyam Madhyam | The factors of the sum is equal to the sum of the factors |
There are THIRTEEN SUB – SUTRAS
Ekadhikena Purvena
The Sutra (formula) Ekādhikena Pūrvena means: “By one more than the previous one”.
Squares of numbers ending in 5 :
Now we relate the sutra to the ‘squaring of numbers ending in 5’. Consider the example 25 2.
Here the number is 25. We have to find out the square of the number. For the number 25, the last digit is 5 and the ‘previous’ digit is 2. Hence, ‘one more than the previous one’, that is, 2+1=3. The Sutra, in this context, gives the procedure’to multiply the previous digit 2 by one more than itself, that is, by 3 ‘. It becomes the L.H.S (left hand side) of the result, that is, 2 X 3 = 6. The R.H.S (right hand side) of the result is5 2, that is, 25.
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Thus 25 2 = 2 X 3 / 25 = 625.
In the same way,
35 2= 3 X (3+1) /25 = 3 X 4/ 25 = 1225;
65 2= 6 X 7 / 25 = 4225;
105 2= 10 X 11/25 = 11025;
135 2= 13 X 14/25 = 18225;
Urdhva – tiryagbhyam
Urdhva – tiryagbhyam is the general formula applicable to all cases of multiplication and also in the division of a large number by another large number. It means
(a) Multiplication of two 2 digit numbers.
Ex.1: Find the product 14 X 12
i) The right hand most digit of the multiplicand, the first number (14) i.e.,4 is multiplied by the right hand most digit of the multiplier, the second number (12)i.e., 2. The product 4 X 2 = 8 forms the right hand most part of the answer.
ii) Now, diagonally multiply the first digit of the multiplicand (14) i.e., 4 and second digit of the multiplier (12)i.e., 1 (answer 4 X 1=4); then multiply the second digit of the multiplicand i.e.,1 and first digit of the multiplier i.e., 2 (answer 1 X 2 = 2); add these two i.e.,4 + 2 = 6. It gives the next, i.e., second digit of the answer. Hence second digit of the answer is 6.
iii) Now, multiply the second digit of the multiplicand i.e., 1 and second digit of the multiplieri.e., 1 vertically, i.e., 1 X 1 = 1. It gives the left hand most part of the answer.
Thus the answer is 16 8.
Symbolically we can represent the process as follows :
Now in the same process, answer can be written as
23 13 ‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ 2 : 6 + 3 : 9 = 299 (Recall the 3 steps)
Sunyam Samya Samuccaye
The Sutra ‘Sunyam Samyasamuccaye’ says the ‘Samuccaya is the same, that Samuccaya is Zero.’ i.e., it should be equated to zero. The term ‘Samuccaya’ has several meanings under different contexts.
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i) We interpret, ‘Samuccaya’ as a term which occurs as a common factor in all the terms concerned and proceed as follows.
Example 1: The equation 7x + 3x = 4x + 5x has the same factor ‘ x ‘ in all its terms. Hence by the sutra it is zero,i.e., x = 0.
Otherwise we have to work like this:
7x + 3x = 4x + 5x
10x = 9x
10x – 9x = 0
x = 0
This is applicable not only for ‘x’ but also any such unknown quantity as follows.
Conclusion
“Mathematical discoveries, like all others, may come from the need to solve practical problems. But a more important motive is the pure delight some individuals take in invention and discovery.
“A mathematician tries to extend his powers. He knows a trick which allows him to solve one sort of problem – but might there not be other sorts that can be solved in the same way? He tries to find out how his tricks work, to group problems in families, to classify them. When a new problem comes up he will often be able to see immediately what family it belongs to, and what methods are likely to solve it.
