Tattva

What is Vedic Mathematics?
Vedic Mathematics is the name given to the ancient system of Mathematics which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krishna Tirthaji (1884-1960). According to his research all of mathematics is based on 16 Sutras or aphorisms. For example, ‘Vertically and Crosswise` is one of these Sutras. These formulae describe the way the mind naturally works and are therefore a great help in directing the student to the appropriate method of solution.
Swami Bharati Krishna Tirtha’s Vedic mathematics is a system of mathematics consisting of a list of 16 basic sūtras, or aphorisms, that allegedly encompass all mathematics.

Tirthaji claims that he found the sūtras after years of studying the Vedas, a set of sacred ancient Hindu texts. However, labeling the mathematics he presented as ‘Vedic’ provoked great controversy amongst Indian mathematicians who question both the Vedic origin of the mathematics, and whether the sūtras can fulfill the claim of encompassing all mathematics. Nonetheless, the calculation strategies provided by Vedic mathematics are creative and useful, and can be applied in a number of ways to calculation methods in arithmetic and algebra.

Vedic math has some similarities to the Trachtenberg system and many of the arithmetic computational strategies are based on the same concepts.

Tirthaji and the discovery of Vedic mathematics

Jagadguru Swami Sri Bharati Krishna Tirthaji is described as having the “rare combination of the probing insight and revealing intuition of a Yogi with the analytical acumen and synthetic talent of a mathematician” (Pratyagatmananda, 1965). Born in India in 1884, Tirthaji was an exceptional scholar; by age twenty he had studied at a number of colleges and universities throughout the country, been awarded the title of ‘Saraswati’ by the Madras Sanskrit Association for his remarkable proficiency in Sanskrit, and had completed seven masters degrees, including Sanskrit, Philosophy, English, Mathematics, History and Science, with the American College of Sciences (Trivedi, 1965).
Around 1911, Tirthaji resolved to study several sections of the Atharva-veda that had been dismissed by Orientalists, Indologists and antiquarian scholars as nonsensical. He was part of a shrinking group of Indian scholars who believed that the Vedas represented an “inexhaustible mine of profound wisdom” both spiritual and secular. Tirthaji claimed that there were sections of the Atharva-veda labeled “ganita sūtras” or “mathematical formulae” that mysteriously made implicit references to mathematics. Tirthaji explains that he was determined to understand the “ganita sūtras” , and began studying ancient lexicons and lexicography in more detail. With this resolve, Tirthaji went to Sringeri, Karnataka, where he began years of solitary study and meditation.
Eight years later, Tirthaji emerged claiming to have deciphered 16 fundamental mathematical sūtras in the Vedas, which today have become the foundation of Vedic mathematics. According to Tirthaji, the sūtras cover every branch of mathematics, from arithmetic to spherical conics, and that “there is no mathematics beyond their jurisdiction”.
After discovering the sūtras, Tirthaji traveled around India presenting Vedic mathematics, and even lectured in the United States and England in 1958. In addition to lecturing, Tirthaji also wrote sixteen volumes, one for each basic sūtra, explaining their applications. Unfortunately, before they were published, the manuscripts were lost irretrievably. Before falling ill and passing away in 1960, Tirthaji was able to rewrite the first of the sixteen volumes he had composed. This text — simply titled Vedic Mathematics, and published in 1965 — has become the basis for all study in the area.

The sūtras (formulas or aphorisms)

Vedic mathematics is based on sixteen sūtras which serve as somewhat cryptic instructions for dealing with different mathematical problems. Below is a list of the sūtras, translated from Sanskrit into English:

1.    “By one more than the previous one”
2.    “All from 9 and the last from 10″
3.    “Vertically and crosswise (multiplications)”
4.    “Transpose and apply”
5.    “Transpose and adjust (the coefficient)”
6.    “If the Samuccaya is the same (on both sides of the equation, then) that Samuccaya is (equal to) zero”
7.    By the Parāvartya rule
8.    “If one is in ratio, the other one is zero.”
9.    “By addition and by subtraction.”
10.    By the completion or non-completion (of the square, the cube, the fourth power, etc.)
11.    Differential calculus
12.    By the deficiency
13.    Specific and general
14.    The remainders by the last digit
15.    “The ultimate (binomial) and twice the penultimate (binomial) (equals zero),”
16.    “Only the last terms,” by one less than the one before the product of the sum all the multipliers

Applications of Vedic mathematics

The most notable application of Vedic mathematics is in education. Vedic mathematical strategies may prove to be a useful resource for teachers and students, who may find elements of it easier and more accessible to teach and learn than conventional mathematics. In particular, these strategies may be an invaluable resource to students that already struggle with mathematics, and could benefit from alternative approaches.
One attempt at incorporating Vedic mathematics into education was made by Mark Gaskell, the head of mathematics at the Maharishi School Lancashire, England. The school has developed a Vedic mathematics curriculum equivalent to the national one with impressive results. According to Gaskell, the alternative curriculum has resulted in livelier classes, greater student enjoyment and understanding, and improved academic performance. In fact, the first set of students to complete the course were each able to not only pass, but achieve over 80%, on the General Certificate of Secondary Education, a proficiency test taken by all secondary school British students, a year earlier than their peers in the regular curriculum. If harnessed appropriately, there seems to be great potential for how Vedic mathematics can be used to teach, learn and understand mathematics. Perhaps the most important aspect of including Vedic mathematics in an education system will be taking the step towards becoming open to conceptually different mathematical approaches — approaches that could one day free and transform mathematics education.

Few Practical Demonstrations
Controversy and criticism

There has been much controversy amongst Indian scholars about Tirthaji’s claims that the mathematics is Vedic and that it encompasses all aspects of mathematics. First, Tirthaji’s description of the mathematics as Vedic is most commonly criticised on the basis that, thus far, none of the sūtras can be found in any extant Vedic literature. However, trying to locate Tirthaji’s references in the Vedic literature would be extremely difficult as it is possible that Tirthaji rediscovered and reconstructed the sūtras from stray references scattered throughout the Atharva-veda, making it difficult to trace them.
In response to criticisms that the sūtras cannot be located within the texts, several people have explained how textual references should not be the basis for evaluating the Vedicity of the mathematics. Some propose that Vedic mathematics is different than other scientific work because it is not logically worked out, but is based on a direct revelation, or an “intuitional visualisation” of fundamental mathematical truths. Tirthaji has been described as having the same “reverential approach” towards the Vedas as the ancient rishis that formed them. Thus, it seems as though some believe that Tirthaji may not have found the sūtras within the Vedas, but that he received them spiritually as the rishis did, which should validate them as Vedic.
The controversy about the Vedicity of the mathematics is further confused by the double meaning of veda. Since veda can be translated to mean ‘knowledge’, it is also possible that Vedic mathematics simply refers to the fact that the sūtras are supposed to present all knowledge of mathematics. Tirthaji’s definition of veda does not clearly clarify whether he uses it to represent ‘all knowledge’ or the Vedic texts; rather, it seems that he uses it to refer to both.
Considering the lack of references to the sūtras, coupled with the fact that the language style does not seem Vedic, some propose that the sūtras were simply composed by Tirthaji himself. In that case, one must consider what motivated Tirthaji to attribute the mathematical sūtras to the ancient texts. Was it because they are from the Vedas, or does claiming so give them more credibility? Other areas of controversy regarding Vedic mathematics focus on the actual mathematics itself. Tirthaji’s assertion that the 16 sutras of Vedic mathematics encompass all branches of mathematics is an extreme one even if true, and so it is not surprising that many mathematicians challenge it. They point to the inconsistency between the topics addressed by the system (such as decimal fractions) and the known mathematics of early India, the substantial extrapolations from a few words of a sūtra to complex arithmetic strategies, and the restriction of applications to convenient, special cases. They further say that such arithmetic as is sped up by application of the sūtras can be performed on a computer or calculator anyway, making their knowledge rather irrelevant in the modern world.
They are also worried that it deflects attention from genuine achievements of ancient and modern Indian mathematics and mathematicians.

References

•    Wikipedia
•    www.hinduism.co.za/index.html
•    “Aryabhatiya” by Aryabhatta

Written and Compiled by
Divya Alok
EC – I 1st year

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