Thursday, July 01, 2010

Vedic Mathematics - Part I

Introduction

“The so called Vedic Mathematics is neither Vedic nor Mathematics”, says my good friend and a well-known mathematician Prof. S. G. Dani. That is the pedantic view. Nonetheless, it is believed that the strategies provided by Vedic mathematics are creative and useful, and could potentially be applied in a number of ways to advanced calculus and linear algebra, most notably within the education system.

What is Vedic Mathematics?

"Vedic Mathematics" is the name given to the ancient system of mathematics, or, to be precise, a unique technique of calculations based on simple rules and principles, with which any mathematical problem -- be it arithmetic, algebra, geometry or trigonometry -- can be solved, hold your breath, orally! Thousands of years ago, Vedic mathematicians authored various theses and dissertations on mathematics. It is now commonly believed and widely accepted that these treatises laid down the foundations of algebra, algorithm, square roots, cube roots, various methods of calculation, and the concept of zero.

The system is based on 16 Vedic sutras or aphorisms, which are actually word-formula describing natural ways of solving a whole range of mathematical problems. Some examples of sutras are "By one more than the one before", "All from 9 & the last from 10", and "Vertically & Crosswise". These 16 one-line formulas originally written in Sanskrit, which can be easily memorized, enables one to solve long mathematical problems quickly.

History of Vedic Mathematics

In the valley of the Indus River of India, the world's oldest civilization had developed its own system of mathematics. The Vedic Shulba Sutras (fifth to eighth century B.C.), meaning "codes of the rope," show that the earliest geometrical and mathematical investigations among the Indians arose from certain requirements of their religious rituals. When the poetic vision of the Vedic seers was externalized in symbols, rituals requiring altars and precise measurement became manifest, providing a means to the attainment of the unmanifest world of consciousness. "Shulba Sutras" is the name given to those portions or supplements of the Kalpasutras, which deal with the measurement and construction of the different altars or arenas for religious rites. The word shulba refers to the ropes used to make these measurements.

Although Vedic mathematicians are known primarily for their computational genius in arithmetic and algebra, the basis and inspiration for the whole of Indian mathematics is geometry. Evidence of geometrical drawing instruments from as early as 2500 B.C. has been found in the Indus Valley [1]. The beginnings of algebra can be traced to the constructional geometry of the Vedic priests, which are preserved in the Shulba Sutras. Exact measurements, orientations, and different geometrical shapes for the altars and arenas used for the religious functions (yajnas), which occupy an important part of the Vedic religious culture, are described in the Shulba Sutras.

Who developed this?

Vedic mathematics is developed by Jagadguru Swami Sri Bharati Krishna Tirthaji Maharaj in the middle 20th century. It has some similarities to the Trachtenberg system in that it speeds up some arithmetic calculations. It has applications to more advanced mathematics, such as calculus and linear algebra.

Swami Bharati Krishna Tirtha Maharaj, who is generally considered the doyen of this discipline, in his seminal book Vedic Mathematics, wrote about this special use of verses in the Vedic age: "In order to help the pupil memorize the material assimilated, they made it a general rule of practice to write even the most technical and abstruse textbooks in sutras or in verse (which is so much easier -- even for the children -- to memorize). So from this standpoint, they used verse for lightening the burden and facilitating the work (by versifying scientific and even mathematical material in a readily assimilable form)!"

Dr L M Singhvi, the former High Commissioner of India in the UK, who is an avid endorser of the system says: "A single sutra would generally encompass a varied and wide range of particular applications and may be likened to a programmed chip of our computer age". Another Vedic mathematics enthusiast, Clive Middleton of vedicmaths.org feels, "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."

Practitioners of this striking method of mathematical problem-solving opine that Vedic mathematics is far more systematic, coherent, and unified than the conventional system. It is a mental tool for calculation that encourages the development and use of intuition and innovation, while giving the student a lot of flexibility, fun and satisfaction. Therefore, it's direct and easy to implement in schools -- a reason behind its enormous popularity among educationists and academicians.

Bharati Krishna Tirthaji delved into the ancient Vedic texts and established the techniques of this system in his pioneering work -- Vedic Mathematics (1965), which is considered the starting point for all work on Vedic math. It is said that after Bharati Krishna's original 16 volumes of work expounding the Vedic system were lost, in his final years he wrote this single volume, which was published five years after his death [2]. On the other hand, it is also believed by some that he himself developed these techniques from his own ingenuity, thus giving rise to the controversy mentioned in the beginning of this article.

Popularity of Vedic Mathematics

"India was the motherland of our race and Sanskrit the mother of Europe's languages. India was the mother of our philosophy, of much of our mathematics, of the ideals embodied in Christianity...of self-government and democracy. In many ways, Mother India is the mother of us all." Thus wrote the well known American Historian Will Durant in 1885-1981, which indicates the immense popularity of this new system. Quite a few years ago, St James' School, London, and other schools began to teach the Vedic system, with notable success. Today this remarkable system is taught in many schools and institutes in India and abroad, and even to MBA and economics students.

When in 1988, Maharishi Mahesh Yogi brought to light the marvels of Vedic math; Maharishi Schools around the world incorporated it in their syllabi. At the school in Skelmersdale, Lancashire, UK, a full course called "The Cosmic Computer" was written and tested on 11 to 14 year old pupils, and later published in 1998. Maharshi Mahesh Yogi says, "Vedic Mathematics is that one field of knowledge which fulfils the purpose of education by developing the total creative genius of the individual, giving him/her the ability to be always spontaneously right and automatically precise so that his/her action, supported by Natural Law, is always effortlessly fulfilling. The sutras of Vedic Mathematics are the software for the cosmic computer that runs this universe."

Advantages of Vedic Mathematics

There are obviously many advantages of using a flexible, refined and efficient mental system like Vedic math. Pupils can come out of the confinement of the 'only one correct' way, and make their own methods under the Vedic system. Thus, it can induce creativity in intelligent pupils, while helping slow-learners grasp the basic concepts of mathematics. A wider use of Vedic math can undoubtedly generate interest in a subject that is generally dreaded by children.

It helps a person to solve mathematical problems 10-15 times faster.
It helps in Intelligent Guessing
It reduces burden (need to learn tables up to 9 only)
It is a magical tool to reduce scratch work and finger counting
It increases concentration.
Being Unique it brings instant recognition to its practitioner

Vedic Number Representation

Vedic knowledge is in the form of slokas or poems in Sanskrit verse. A number was encoded using consonant groups of the Sanskrit alphabet, and vowels were provided as additional latitude to the author in poetic composition. The coding key is given as Kaadi nav, taadi nav, paadi panchak, yaadashtak ta ksha shunyam.

In other words,

ka, ta, pa, ya = 1;	kha, tha, pha, ra = 2
ga, da, ba, la = 3 ;	gha, dha, bha, va = 4
gna, na, ma, scha = 5;	cha, ta, sha = 6
chha, tha, sa = 7;	ja, da, ha = 8
jha, dha = 9;	ksha = 0

Thus pa pa is 11, ma ra is 52. Words kapa, tapa , papa, and yapa all mean the same that is 11. It was up to the author to choose one that fits the meaning of the verse well. An interesting example of this is a hymn below in the praise of Lord Krishna that gives the value of Pi to the 32 decimal places as: 31415926535897932384626433832792.

gopi bhagya madhuvrata

srngiso dadhi sandhiga

khala jivita khatava

gala hala rasandara

The translation is as follows: "O Lord anointed with the yoghurt of the milkmaids' worship (Krishna), O savior of the fallen, O master of Siva, please protect me."