Archive for the ‘Mathematics’ Category

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Śulbasūtras are collection of sanskrit works from the vedic period which supplements kalpa as appendices. kalpa is one of the six veda-angas which deals with procedures to perform vedic rituals. Śulbasūtras provide as the source of ancient Indian mathematics in the area of geometry developed during the vedic period.

The mathematics in the vedic period should not be confused with the 20th century work titled “Vedic Mathematics” by former Shankaracharya of Puri, the late Jagadguru Swami Shri Bharati Krishna Tirthaji Maharaj. Tirthaji’s “Vedic Mathematics” is neither vedic nor mathematics of any significant importance except for some alternate methods in arithmetics and elementary algebra. The title of the work is actually a misleading one, and people without understanding the facts propagate it as something from ancient Indian epistemology. The claim that the sanskrit aphorisms mentioned in his text were from the appendix (parishishta) of Atharva-Veda is controversial, and so far, no versions of Atharva-Veda contained such aphorisms. Since the book was published posthumously, we are not sure whether the author or the editor is to be blamed for such a misleading book title. For a more detailed discussion about this topic, please refer to the article titled “Myths and reality : On ‘Vedic mathematics’ ” by S.G. Dani, a renowned mathematician at Tata Institute of Fundamental Research, India.

The importance of mathematics were indeed well cherished in ancient Indian mathematical works, and the Jyothisha-Vedanga (attributed with Rig-Veda) glorifies mathematics as follows:

yatha shikha mayurānam nāgānām maṇayo yatḥa |
taḍvad vedāṅga shāstrāṅām gañitham mūrdhin stḥitḥam ||

“Like the crest of the peacock, like the gem in the hood of the king cobra, so is mathematics the top-head of all branches of science/knowledge”.

The geometry in Śulbasūtras particularly laid out details for the design and construction of fire altars for vedic rituals. The vast corpus of works developed in Śulbasūtras are mainly attributed to Baudhāyana, Mānava, Āpastamba and Kātyāyana. The oldest being developed by Baudhāyana during 800 BCE, and the youngest by Kātyāyana during 200 BCE.

One of the most significant work which gained popularity among contemporary mathematician is the statement about hypotenuse theorem (which is currently called as Pythagoras Theorem) contained in Baudhāyana Śulbasūtras which belongs to Taittiriya branch of the Krishna Yajur-Veda.  Though, Baudhāyana did not wrote proof to his theorem, he laid out the sūtra as follows:

dīrgha chaturasrasya akṣaṇayā rajjuḥ pārśvamānī tiryagmānī cha
yat pṛthagbhūte kurutah tat ubhayāṅ karoti. (Chapter 1, sutra 12)

A rope stretched along the diagonal of a rectangle makes a squared length which is made by the squared lengths of the horizontal and vertical sides of the rectangle together.

Other important concepts contained in Śulbasūtras are as follows:
1) Pythagorean triples.
2) Formula to find square roots.
3) Finding a circle whose area is same as a square.
4) Diagonals of rectangle bisecting each other.
5) Diagonals of rhombus bisecting at right angles.
6) Areas associated with squares, rectangles and rhombus.
7) Methodology to handle fractions.

Further reading:
http://www.math.cornell.edu/~dwh/papers/sulba/sulba.html
http://www-history.mcs.st-and.ac.uk/Projects/Pearce/Chapters/Ch4_2.html

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cnMultiplication

There is really an interesting fact associated with the Theory of Computation (ToC). Like we say failures are stepping stones to success, the different aspects of ToC flourished out of failures.

McCulloch and Pitts were initially trying to formalize a mathematical model to simulate the functioning of brain through abstract neuron networks, but it failed to capture the entire notion of brain function, but this failure lead to the development of Finite State Machines.

Noam Chomsky was trying to formalize the human languages, but he realized that to frame the human languages into a mathematical set of rules is extremely complex, but this failure lead to the development of Context Free Grammar (CFG), which eventually lead to Backus Norm Form (a notation technique for CFG). It is surprisingly similar to the grammar of Sanskrit formalized by Panini (4th century BCE). So, CFG shares the grammar of Sanskrit, and this is one way by which Sanskrit is related to computation.

David Hilbert challenged the mathematicians with a famous question – “does there exists a general procedure/method to solve all the mathematical problems?”.  Hilbert believed there exists a procedure to do so (popularly known as entscheidungsproblem). This intrigued many mathematicians, and eventually Church and Turing proved that there is no such method. Hilbert did failed in his belief, but this has lead to the development of an elegant idea called Turing Machines. Turing wanted to first formalize what does Hilbert meant by “general procedure/method”, and he came up with a formal definition for “Algorithms” based on the idea of Turing Machine, and the notion of what is computable (tractable) and what is not computable (intractable) in reasonable amount of time.

How elegant it is to see that such failures had lead to a beautiful subject called Theory of Computation :)which eventually rules the entire modern day computer innovations.

Mathematics

Posted: November 24, 2009 in Mathematics, philosophy
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This would simply translates to –

Like the crest of the peacock, like the gem in the hood of the snake,
So is mathematics the head of all area of knowledge

The source of this aphorism is said to have taken from Vedanga of Rig Veda.