[TorryJens]

Shulba Sutra

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.E.), 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.

Math cannot take the mystery out of life without doing away with life itself, for it is life's mystery, its unpredictability — the fact that it is dynamic, not static — that makes it alive and worth living.

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.E. 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. Many of these calculations employ the geometrical formula known as the Pythagorean theorem.

This theorem (c. 540 B.C.E.), equating the square of the hypotenuse of a right angle triangle with the sum of the squares of the other two sides, was utilized in the earliest Shulba Sutra (the Baudhayana) prior to the eighth century B.C.E. Thus, widespread use of this famous mathematical theorem in India several centuries before its being popularized by Pythagoras has been documented. The exact wording of the theorem as presented in the Sulba Sutras is: "The diagonal chord of the rectangle makes both the squares that the horizontal and vertical sides make separately." [2] The proof of this fundamentally important theorem is well known from Euclid's time until the present for its excessively tedious and cumbersome nature; yet the Vedas present five different extremely simple proofs for this theorem. One historian, Needham, has stated, "Future research on the history of science and technology in Asia will in fact reveal that the achievements of these peoples contribute far more in all pre-Renaissance periods to the development of world science than has yet been realized." [3]

The Shulba Sutras have preserved only that part of Vedic mathematics which was used for constructing the altars and for computing the calendar to regulate the performance of religious rituals. After the Shulba Sutra period, the main developments in Vedic mathematics arose from needs in the field of astronomy. The Jyotisha, science of the luminaries, utilizes all branches of mathematics.

The need to determine the right time for their religious rituals gave the first impetus for astronomical observations. With this desire in mind, the priests would spend night after night watching the advance of the moon through the circle of the nakshatras (lunar mansions), and day after day the alternate progress of the sun towards the north and the south. However, the priests were interested in mathematical rules only as far as they were of practical use. These truths were therefore expressed in the simplest and most practical manner. Elaborate proofs were not presented, nor were they desired.

Evolution of Arabic (Roman) Numerals from India

A close investigation of the Vedic system of mathematics shows that it was much more advanced than the mathematical systems of the civilizations of the Nile or the Euphrates. The Vedic mathematicians had developed the decimal system of tens, hundreds, thousands, etc. where the remainder from one column of numbers is carried over to the next. The advantage of this system of nine number signs and a zero is that it allows for calculations to be easily made. Further, it has been said that the introduction of zero, or sunya as the Indians called it, in an operational sense as a definite part of a number system, marks one of the most important developments in the entire history of mathematics. The earliest preserved examples of the number system which is still in use today are found on several stone columns erected in India by King Ashoka in about 250 B.C.E. [4 ] Similar inscriptions are found in caves near Poona (100 B.C.E.) and Nasik (200 C.E.). [5] These earliest Indian numerals appear in a script called brahmi.

After 700 C.E. another notation, called by the name "Indian numerals," which is said to have evolved from the brahmi numerals, assumed common usage, spreading to Arabia and from there around the world. When Arabic numerals (the name they had then become known by) came into common use throughout the Arabian empire, which extended from India to Spain, Europeans called them "Arabic notations," because they received them from the Arabians. However, the Arabians themselves called them "Indian figures" (Al-Arqan-Al-Hindu) and mathematics itself was called "the Indian art" (hindisat).

Evolution of "Arabic numerals" from Brahmi
(250 B.C.E.) to the 16th century.


Mastery of this new mathematics allowed the Muslim mathematicians of Baghdad to fully utilize the geometrical treatises of Euclid and Archimedes. Trigonometry flourished there along with astronomy and geography. Later in history, Carl Friedrich Gauss, the "prince of mathematics," was said to have lamented that Archimedes in the third century B.C.E. had failed to foresee the Indian system of numeration; how much more advanced science would have been.
Prior to these revolutionary discoveries, other world civilizations-the Egyptians, the Babylonians, the Romans, and the Chinese-all used independent symbols for each row of counting beads on the abacus, each requiring its own set of multiplication or addition tables. So cumbersome were these systems that mathematics was virtually at a standstill. The new number system from the Indus Valley led a revolution in mathematics by setting it free. By 500 C.E. mathematicians of India had solved problems that baffled the world's greatest scholars of all time. Aryabhatta, an astronomer mathematician who flourished at the beginning of the 6th century, introduced sines and versed sines-a great improvement over the clumsy half-cords of Ptolemy. A.L. Basham, foremost authority on ancient India, writes in The Wonder That Was India,

Medieval Indian mathematicians, such as Brahmagupta (seventh century), Mahavira (ninth century), and Bhaskara (twelfth century), made several discoveries which in Europe were not known until the Renaissance or later. They understood the import of positive and negative quantities, evolved sound systems of extracting square and cube roots, and could solve quadratic and certain types of indeterminate equations." [6] Mahavira's most noteworthy contribution is his treatment of fractions for the first time and his rule for dividing one fraction by another, which did not appear in Europe until the 16th century.

Equations and Symbols

B.B. Dutta writes: "The use of symbols-letters of the alphabet to denote unknowns, and equations are the foundations of the science of algebra. The Hindus were the first to make systematic use of the letters of the alphabet to denote unknowns. They were also the first to classify and make a detailed study of equations. Thus they may be said to have given birth to the modern science of algebra." [7] The great Indian mathematician Bhaskaracharya (1150 C.E.) produced extensive treatises on both plane and spherical trigonometry and algebra, and his works contain remarkable solutions of problems which were not discovered in Europe until the seventeenth and eighteenth centuries. He preceded Newton by over 500 years in the discovery of the principles of differential calculus. A.L. Basham writes further, "The mathematical implications of zero (sunya) and infinity, never more than vaguely realized by classical authorities, were fully understood in medieval India. Earlier mathematicians had taught that X/0 = X, but Bhaskara proved the contrary. He also established mathematically what had been recognized in Indian theology at least a millennium earlier: that infinity, however divided, remains infinite, represented by the equation oo /X = oo." In the 14th century, Madhava, isolated in South India, developed a power series for the arc tangent function, apparently without the use of calculus, allowing the calculation of pi to any number of decimal places (since arctan 1 = pi/4). Whether he accomplished this by inventing a system as good as calculus or without the aid of calculus; either way it is astonishing.

Spiritually advanced cultures were not ignorant of the principles of mathematics, but they saw no necessity to explore those principles beyond that which was helpful in the advancement of God realization.

By the fifteenth century C.E. use of the new mathematical concepts from India had spread all over Europe to Britain, France, Germany, and Italy, among others. A.L. Basham states also that

The debt of the Western world to India in this respect [the field of mathematics] cannot be overestimated. Most of the great discoveries and inventions of which Europe is so proud would have been impossible without a developed system of mathematics, and this in turn would have been impossible if Europe had been shackled by the unwieldy system of Roman numerals. The unknown man who devised the new system was, from the world's point of view, after the Buddha, the most important son of India. His achievement, though easily taken for granted, was the work of an analytical mind of the first order, and he deserves much more honor than he has so far received.

Unfortunately, Eurocentrism has effectively concealed from the common man the fact that we owe much in the way of mathematics to ancient India. Reflection on this may cause modern man to consider more seriously the spiritual preoccupation of ancient India. The rishis (seers) were not men lacking in practical knowledge of the world, dwelling only in the realm of imagination. They were well developed in secular knowledge, yet only insofar as they felt it was necessary within a world view in which consciousness was held as primary.

In ancient India, mathematics served as a bridge between understanding material reality and the spiritual conception. Vedic mathematics differs profoundly from Greek mathematics in that knowledge for its own sake (for its aesthetic satisfaction) did not appeal to the Indian mind. The mathematics of the Vedas lacks the cold, clear, geometric precision of the West; rather, it is cloaked in the poetic language which so distinguishes the East. Vedic mathematicians strongly felt that every discipline must have a purpose, and believed that the ultimate goal of life was to achieve self-realization and love of God and thereby be released from the cycle of birth and death. Those practices which furthered this end either directly or indirectly were practiced most rigorously. Outside of the religio-astronomical sphere, only the problems of day to day life (such as purchasing and bartering) interested the Indian mathematicians.

Poetry in Math

One of the foremost exponents of Vedic math, the late Bharati Krishna Tirtha Maharaja, author of Vedic Mathematics, has offered a glimpse into the sophistication of Vedic math. Drawing from the Atharva-veda, Tirtha Maharaja points to many sutras (codes) or aphorisms which appear to apply to every branch of mathematics: arithmetic, algebra, geometry (plane and solid), trigonometry (plane and spherical), conics (geometrical and analytical), astronomy, calculus (differential and integral), etc.

Utilizing the techniques derived from these sutras, calculations can be done with incredible ease and simplicity in one's head in a fraction of the time required by modern means. Calculations normally requiring as many as a hundred steps can be done by the Vedic method in one single simple step. For instance the conversion of the fraction 1/29 to its equivalent recurring decimal notation normally involves 28 steps. Utilizing the Vedic method it can be calculated in one simple step. (see the next section for examples of how to utilize Vedic sutras)

In order to illustrate how secular and spiritual life were intertwined in Vedic India, Tirtha Maharaja has demonstrated that mathematical formulas and laws were often taught within the context of spiritual expression (mantra). Thus while learning spiritual lessons, one could also learn mathematical rules.

Tirtha Maharaja has pointed out that Vedic mathematicians prefer to use the devanagari letters of Sanskrit to represent the various numbers in their numerical notations rather than the numbers themselves, especially where large numbers are concerned. This made it much easier for the students of this math in their recording of the arguments and the appropriate conclusions.

Tirtha Maharaja states, "In order to help the pupil to memorize the material studied and 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). And this is why we find not only theological, philosophical, medical, astronomical, and other such treatises, but even huge dictionaries in Sanskrit verse! So from this standpoint, they used verse, sutras and codes for lightening the burden and facilitating the work (by versifying scientific and even mathematical material in a readily assimilable form)!" [8] The code used is as follows:

The Sanskrit consonants

ka, ta, pa, and ya all denote 1;
kha, tha, pha, and ra all represent 2;
ga, da, ba, and la all stand for 3;
Gha, dha, bha, and va all represent 4;
gna, na, ma, and sa all represent 5;
ca, ta, and sa all stand for 6;
cha, tha, and sa all denote 7;
ja, da, and ha all represent 8;
jha and dha stand for 9; and
ka means zero.

Vowels make no difference and it is left to the author to select a particular consonant or vowel at each step. This great latitude allows one to bring about additional meanings of his own choice. For example kapa, tapa, papa, and yapa all mean 11. By a particular choice of consonants and vowels one can compose a poetic hymn with double or triple meanings. Here is an actual sutra of spiritual content, as well as secular mathematical significance.

gopi bhagya madhuvrata
srngiso dadhi sandhiga
khala jivita khatava
gala hala rasandara

While this verse is a type of petition to Krishna, when learning it one can also learn the value of pi/10 (i.e. the ratio of the circumference of a circle to its diameter divided by 10) to 32 decimal places. It has a self-contained master-key for extending the evaluation to any number of decimal places.

The translation is as follows:

O Lord anointed with the yogurt of the milkmaids' worship (Krishna), O savior of the fallen, O master of Shiva, please protect me.

At the same time, by application of the consonant code given above, this verse directly yields the decimal equivalent of pi divided by 10: pi/10 = 0.31415926535897932384626433832792. Thus, while offering mantric praise to Godhead in devotion, by this method one can also add to memory significant secular truths.

This is the real gist of the Vedic world view regarding the culture of knowledge: while culturing transcendental knowledge, one can also come to understand the intricacies of the phenomenal world. By the process of knowing the absolute truth, all relative truths also become known. In modern society today it is often contended that never the twain shall meet: science and religion are at odds. This erroneous conclusion is based on little understanding of either discipline. Science is the smaller circle within the larger circle of religion.

We should never lose sight of our spiritual goals. We should never succumb to the shortsightedness of attempting to exploit the inherent power in the principles of mathematics or any of the natural sciences for ungodly purposes. Our reasoning faculty is but a gracious gift of Godhead intended for divine purposes, and not those of our own design.

Vedic Mathematical Sutras

Consider the following three sutras:

1. "All from 9 and the last from 10," and its corollary: "Whatever the extent of its deficiency, lessen it still further to that very extent; and also set up the square (of that deficiency)."
2. "By one more than the previous one," and its corollary: "Proportionately."
3. "Vertically and crosswise," and its corollary: "The first by the first and the last by the last."

The first rather cryptic formula is best understood by way of a simple example: let us multiply 6 by 8.

1. First, assign as the base for our calculations the power of 10 nearest to the numbers which are to be multiplied. For this example our base is 10.
2. Write the two numbers to be multiplied on a paper one above the other, and to the right of each write the remainder when each number is subtracted from the base 10. The remainders are then connected to the original numbers with minus signs, signifying that they are less than the base 10.

6-4
8-2

3. The answer to the multiplication is given in two parts. The first digit on the left is in multiples of 10 (i.e. the 4 of the answer 48). Although the answer can be arrived at by four different ways, only one is presented here. Subtract the sum of the two deficiencies (4 + 2 = 6) from the base (10) and obtain 10 - 6 = 4 for the left digit (which in multiples of the base 10 is 40).

6-4
8-2
4

4. Now multiply the two remainder numbers 4 and 2 to obtain the product 8. This is the right hand portion of the answer which when added to the left hand portion 4 (multiples of 10) produces 48.

6-4
8-2
----
4/8

Another method employs cross subtraction. In the current example the 2 is subtracted from 6 (or 4 from 8) to obtain the first digit of the answer and the digits 2 and 4 are multiplied together to give the second digit of the answer. This process has been noted by historians as responsible for the general acceptance of the X mark as the sign of multiplication. The algebraical explanation for the first process is

(x-a)(x-b)=x(x-a-b) + ab

where x is the base 10, a is the remainder 4 and b is the remainder 2 so that

6 = (x-a) = (10-4)
8 = (x-b) = (10-2)

The equivalent process of multiplying 6 by 8 is then

x(x-a-b) + ab or
10(10-4-2) + 2x4 = 40 + 8 = 48

These simple examples can be extended without limitation. Consider the following cases where 100 has been chosen as the base:

97 - 3 93 - 7 25 - 75
78 - 22 92 - 8 98 - 2
______ ______ ______

75/66 85/56 23/150 = 24/50

In the last example we carry the 100 of the 150 to the left and 23 (signifying 23 hundred) becomes 24 (hundred). Herein the sutra's words "all from 9 and the last from 10" are shown. The rule is that all the digits of the given original numbers are subtracted from 9, except for the last (the righthand-most one) which should be deducted from 10.

Consider the case when the multiplicand and the multiplier are just above a power of 10. In this case we must cross-add instead of cross subtract. The algebraic formula for the process is: (x+a)(x+b) = x(x+a+b) + ab. Further, if one number is above and the other below a power of 10, we have a combination of subtraction and addition: viz:


108 + 8 and 13 + 3
97 - 3 8 - 2
_______ ______

105/-24 = 104/(100-24) = 104/76 11/-6 = 10/(10-6) = 10/4


The Sub-Sutra: "Proportionately" Provides for those cases where we wish to use as our base multiples of the normal base of powers of ten. That is, whenever neither the multiplicand nor the multiplier is sufficiently near a convenient power of 10, which could serve as our base we simply use a multiple of a power of ten as our working base, perform our calculations with this working base and then multiply or divide the result proportionately.

To multiply 48 by 32, for example, we use as our base 50 = 100/2, so we have

Base 50 48 - 2
32 - 18
______

2/ 30/36 or (30/2) / 36 = 15/36

Note that only the left decimals corresponding to the powers of ten digits (here 100) are to be effected by the proportional division of 2. These examples show how much easier it is to subtract a few numbers, (especially for more complex calculations) rather than memorize long mathematical tables and perform cumbersome calculations the long way.

Squaring Numbers

The algebraic equivalent of the sutra for squaring a number is: (a+-b)2 = a2 +- 2ab + b2 . To square 103 we could write it as (100 + 3 )2 = 10,000 + 600 + 9 = 10,609. This calculation can easily be done mentally. Similarly, to divide 38,982 by 73 we can write the numerator as 38x3 + 9x2 +8x + 2, where x is equal to 10, and the denominator is 7x + 3. It doesn't take much to figure out that the numerator can also be written as 35x3 +36x2 + 37x + 12. Therefore,

38,982/73 = (35x3 + 36x2 +37x + 12)/(7x + 3) = 5x2 + 3x +4 = 534

This is just the algebraic equivalent of the actual method used. The algebraic principle involved in the third sutra, "vertically and crosswise," can be expressed, in one of it's applications, as the multiplication of the two numbers represented by (ax + b) and (cx + d), with the answer acx2 + x(ad + bc) + bd. Differential calculus also is utilized in the Vedic sutras for breaking down a quadratic equation on sight into two simple equations of the first degree. Many additional sutras are given which provide simple mental one or two line methods for division, squaring of numbers, determining square and cube roots, compound additions and subtractions, integrations, differentiations, and integration by partial fractions, factorisation of quadratic equations, solution of simultaneous equations, and many more. For demonstrational purposes, we have only presented simple examples.

--------------

http://www.gosai.com/chaitanya/saran...math/math.html

[doctorzlo]

"Sigh", tonnes of text 'copied and pasted" but not a single quote from the Vedas or any of that Sanskritic works. Many doubts! Many doubts! Many doubts! Vedic Propadanda! Vedic Propadanda!

What a waste of space!

[tgs]

I have seen a few books and articles by enthusiasts. One that I found very readable and reliable with some mention of Indian contributions is V.S. Varadarajan's "Algebra in Ancient and Modern Times". A review can be found in :
http://www.maa.org/reviews/aamt.html
and it is available in India at relatively low price.
Swarup

[Lt_Apple]

Dear SRS,
Very Interesting Article, please connect more sites and informations please.

Pithogorus spent 8 years in INDIA, and he is the father of Greek Philosophy, and MATHS from what he learned from INDIA.

See my Article in INDIAN Heritage post from Wickipedia.

Wikipedia, is like Britanica of Web, for MR.Idiappam.
Uppuma

[radikal]

Dear SRS,
Very Interesting Article, please connect more sites and informations please.

Pithogorus spent 8 years in INDIA, and he is the father of Greek Philosophy, and MATHS from what he learned from INDIA.

See my Article in INDIAN Heritage post from Wickipedia.

Wikipedia, is like Britanica of Web, for MR.Idiappam.
Uppuma

That is interesting. I did not know Pythagorus spent time in India. Maybe if he had spent a little time at Anna U, Mr. Idiappam will sing a different tune. Anyway, it is not surprising the Indians possessed this level of mathematical knowledge. The Chinese were also far ahead of the West (mathematics, physics, astronomy, etc).

[LottiFurmann]

Neither Vedic Nor Mathematics

We, the undersigned, are deeply concerned by the continuing attempts to thrust the so-called `Vedic Mathematics' on the school curriculum by the NCERT.


As has been pointed out earlier on several occasions, the so-called `Vedic Mathematics' is neither 'Vedic' nor can it be dignified by the name of mathematics. `Vedic Mathematics', as is well-known, originated with a book of the same name by a former Sankracharya of Puri (the late Jagadguru Swami Shri Bharati Krishna Tirthaji Maharaj) published posthumously in 1965. The book assembled a set of tricks in elementary arithmetic and algebra to be applied in performing computations with numbers and polynomials. As is pointed out even in the foreword to the book by the General Editor, Dr. A.S. Agarwala, the aphorisms in Sanskrit to be found in the book have nothing to do with the Vedas. Nor are these aphorisms to be found in the genuine Vedic literature.
The term "Vedic mathematics'' is therefore entirely misleading and factually incorrect. Further, it is clear from the notation used in the arithmetical tricks in the book that the methods used in this text have nothing to do with the arithmetical techniques of antiquity. Many of the Sanskrit aphorisms in the book are totally cryptic (ancient Indian mathematical writing was anything but cryptic) and often so generalize to be devoid of any specific mathematical meaning. There are several authoritative texts on the mathematics of Vedic times that could used in part to teach an authoritative and correct account of ancient Indian mathematics but this book clearly cannot be used for any such purpose. The teaching of mathematics involves both the teaching of the basic concepts of the subject as well as methods of mathematical computation. The so-called ``Vedic mathematics'' is entirely inadequate to this task considering that it is largely made up of tricks to do some elementary arithmetic computations. Many of these can be far more easily performed on a simple computer or even an advanced calculator.


The book "Vedic mathematics'' essentially deals with arithmetic of the middle and high-school level. Its claims that "there is no part of mathematics, pure or applied, which is beyond their jurisdiction'' is simply ridiculous. In an era when the content of mathematics teaching has to be carefully designed to keep pace with the general explosion of knowledge and the needs of other modern professions that use mathematical techniques, the imposition of ``Vedic mathematics'' will be nothing short of calamitous.
India today has active and excellent schools of research and teaching in mathematics that are at the forefront of modern research in their discipline with some of them recognised as being among the best in the world in their fields of research. It is noteworthy that they have cherished the legacy of distinguished Indian mathematicians like Srinivasa Ramanujam, V. K. Patodi, S. Minakshisundaram, Harish Chandra, K. G. Ramanathan, Hansraj Gupta, Syamdas Mukhopadhyay, Ganesh Prasad, and many others including several living Indian mathematicians. But not one of these schools has lent an iota of legitimacy to `Vedic mathematics'. Nowhere in the world does any school system teach "Vedic mathematics'' or any form of ancient mathematics for that matter as an adjunct to modern mathematical teaching. The bulk of such teaching belongs properly to the teaching of history and in particular the teaching of the history of the sciences.
We consider the imposition of `Vedic mathematics' by a Government agency, as the perpetration of a fraud on our children, condemning particularly those dependent on public education to a sub-standard mathematical education. Even if we assumed that those who sought to impose `Vedic mathematics' did so in good faith, it would have been appropriate that the NCERT seek the assistance of renowned Indian mathematicians to evaluate so-called "Vedic mathematics" before making it part of the National Curricular framework for School Education. Appallingly they have not done so. In this context we demand that the NCERT submit the proposal for the introduction of `Vedic mathematics in the school curriculum to recognized bodies of mathematical experts in India, in particular the National Board of Higher Mathematics (under the Dept. of Atomic Energy), and the Mathematics sections of the Indian Academy of Sciences and the Indian National Science Academy, for a thorough and critical examination. In the meanwhile no attempt should be made to thrust the subject into the school curriculum either through the centrally administered school system or by trying to impose it on the school systems of various States.
We are concerned that the essential thrust behind the campaign to introduce the so-called `Vedic mathematics' has more to do with promoting a particular brand of religious majoritarianism and associated obscurantist ideas rather than any serious and meaningful development of mathematics teaching in India. We note that similar concerns have been expressed about other aspects too of the National Curricular Framework for School Education. We re-iterate our firm conviction that all teaching and pedagogy, not just the teaching of mathematics, must be founded on rational, scientific and secular principles.




S.G.Dani Professor of Mathematics, Tata Institute of Fundamental Research, Mumbai
Madhav M. Deshpande Professor of Sanskrit and Linguistics Department of Asian Languages and Cultures, University of Michigan
Indranil Biswas Professor of Mathematics at TIFR.
Nirmala B. Limaye Professor of Mathematics University of Mumbai
B.V. Limaye Professor of Mathematics Indian Institute of Technology Bombay
Alladi Sitaram, Indian Statistical Institute, B'lore
S. Ramasubramanian, Indian Statistical Inst.,B'ore
V. Pati, Indian Statistical Inst., B'lore
G. Misra, Indian Statistical Inst., B'lore
Jishnu Biswas, Indian Statistical Inst., B'lore
D. P. Sengupta, Indian Inst. of Science(Retd.), B'lore
Alladi Uma, Dept. of English, Univ. of Hyderabad
M. Sridhar, Dept. of English, Univ. of Hyderabad
Amitava Bhattacharya
S.Subramanian, Tata Institute of Fundamental Research Mumbai
Professor Nitin Nitsure,Tata Institute of Fundamental Research, Mumbai

For the article in Frontline on this subject:


http://www.tifr.res.in/~vahia/dani-vmsm.pdf

[Ifroham4]

gopi bhagya madhuvrata
srngiso dadhi sandhiga
khala jivita khatava
gala hala rasandara

While this verse is a type of petition to Krishna, when learning it one can also learn the value of pi/10 (i.e. the ratio of the circumference of a circle to its diameter divided by 10) to 32 decimal places. It has a self-contained master-key for extending the evaluation to any number of decimal places.

The translation is as follows:

O Lord anointed with the yogurt of the milkmaids' worship (Krishna), O savior of the fallen, O master of Shiva, please protect me.

At the same time, by application of the consonant code given above, this verse directly yields the decimal equivalent of pi divided by 10: pi/10 = 0.31415926535897932384626433832792. Thus, while offering mantric praise to Godhead in devotion, by this method one can also add to memory significant secular truths.

why is pi divided by ten? This means that when you want to make a wheel with a radius of 1 meter you'll get only a wheel of 20 cm radius. Even when I chant that verse givven above 10 times, I'll get ten wheels each with 40 cm diameter.

And long before the constant pi was known to humans, they had the knowledge to make wheels. They simply calculated 6 fold radius plus a bit to get the circumference of a wheel.

And Krishna being the lord of Shiva? I think I should run to my red phone and call Siva and clarify this.

[Big A]

why is pi divided by ten? This means that when you want to make a wheel with a radius of 1 meter you'll get only a wheel of 20 cm radius. Even when I chant that verse givven above 10 times, I'll get ten wheels each with 40 cm diameter.

SRS the great vedic mathematician,

can you perhaps tell me what is wrong there in the above?

This is what happened to so called great vedic mathematics: They did not know to make the best out of it!

(Vedic ?) Indians "invented" the 0 and had the knowledge about 1 but it was Leibniz the first to define the dual system - YES & NO, TRUE & FALSE, 0 & 1, To Be and Not to Be. Okay, you may argue that to be and not to be is from Shakespeare. And again you are wrong because William Shakespeare invented to be or not to be!

[brraverishhh]

[tscii]

Originally Posted by SRS gopi bhagya madhuvrata
srngiso dadhi sandhiga
khala jivita khatava
gala hala rasandara

While this verse is a type of petition to Krishna, when learning it one can also learn the value of pi/10 (i.e. the ratio of the circumference of a circle to its diameter divided by 10) to 32 decimal places. It has a self-contained master-key for extending the evaluation to any number of decimal places.

The translation is as follows:

O Lord anointed with the yogurt of the milkmaids' worship (Krishna), O savior of the fallen, O master of Shiva, please protect me.

At the same time, by application of the consonant code given above, this verse directly yields the decimal equivalent of pi divided by 10: pi/10 = 0.31415926535897932384626433832792. Thus, while offering mantric praise to Godhead in devotion, by this method one can also add to memory significant secular truths.

why is pi divided by ten? This means that when you want to make a wheel with a radius of 1 meter you'll get only a wheel of 20 cm radius. Even when I chant that verse givven above 10 times, I'll get ten wheels each with 40 cm diameter.

And long before the constant pi was known to humans, they had the knowledge to make wheels. They simply calculated 6 fold radius plus a bit to get the circumference of a wheel. The point of the pi calculation is not to make bicycle wheels. pi/10 gives the same value as pi to 32 decimal places. Pi is an "irrational" number; some type of numerical technique is needed to approximate its value. These days, of course, the approximation is done using a computer. It is amazing that a simple mantra can do exactly what a computer does, yielding the exact same result:
It has a self-contained master-key for extending the evaluation to any number of decimal places.

[Lillie_Steins]

Neither Vedic Nor Mathematics

We, the undersigned, are deeply concerned by the continuing attempts to thrust the so-called `Vedic Mathematics' on the school curriculum by the NCERT.


As has been pointed out earlier on several occasions, the so-called `Vedic Mathematics' is neither 'Vedic' nor can it be dignified by the name of mathematics. `Vedic Mathematics', as is well-known, originated with a book of the same name by a former Sankracharya of Puri (the late Jagadguru Swami Shri Bharati Krishna Tirthaji Maharaj) published posthumously in 1965. The book assembled a set of tricks in elementary arithmetic and algebra to be applied in performing computations with numbers and polynomials. As is pointed out even in the foreword to the book by the General Editor, Dr. A.S. Agarwala, the aphorisms in Sanskrit to be found in the book have nothing to do with the Vedas. Nor are these aphorisms to be found in the genuine Vedic literature.
The term "Vedic mathematics'' is therefore entirely misleading and factually incorrect. Further, it is clear from the notation used in the arithmetical tricks in the book that the methods used in this text have nothing to do with the arithmetical techniques of antiquity. Many of the Sanskrit aphorisms in the book are totally cryptic (ancient Indian mathematical writing was anything but cryptic) and often so generalize to be devoid of any specific mathematical meaning. There are several authoritative texts on the mathematics of Vedic times that could used in part to teach an authoritative and correct account of ancient Indian mathematics but this book clearly cannot be used for any such purpose. The teaching of mathematics involves both the teaching of the basic concepts of the subject as well as methods of mathematical computation. The so-called ``Vedic mathematics'' is entirely inadequate to this task considering that it is largely made up of tricks to do some elementary arithmetic computations. Many of these can be far more easily performed on a simple computer or even an advanced calculator.


The book "Vedic mathematics'' essentially deals with arithmetic of the middle and high-school level. Its claims that "there is no part of mathematics, pure or applied, which is beyond their jurisdiction'' is simply ridiculous. In an era when the content of mathematics teaching has to be carefully designed to keep pace with the general explosion of knowledge and the needs of other modern professions that use mathematical techniques, the imposition of ``Vedic mathematics'' will be nothing short of calamitous.
India today has active and excellent schools of research and teaching in mathematics that are at the forefront of modern research in their discipline with some of them recognised as being among the best in the world in their fields of research. It is noteworthy that they have cherished the legacy of distinguished Indian mathematicians like Srinivasa Ramanujam, V. K. Patodi, S. Minakshisundaram, Harish Chandra, K. G. Ramanathan, Hansraj Gupta, Syamdas Mukhopadhyay, Ganesh Prasad, and many others including several living Indian mathematicians. But not one of these schools has lent an iota of legitimacy to `Vedic mathematics'. Nowhere in the world does any school system teach "Vedic mathematics'' or any form of ancient mathematics for that matter as an adjunct to modern mathematical teaching. The bulk of such teaching belongs properly to the teaching of history and in particular the teaching of the history of the sciences.
We consider the imposition of `Vedic mathematics' by a Government agency, as the perpetration of a fraud on our children, condemning particularly those dependent on public education to a sub-standard mathematical education. Even if we assumed that those who sought to impose `Vedic mathematics' did so in good faith, it would have been appropriate that the NCERT seek the assistance of renowned Indian mathematicians to evaluate so-called "Vedic mathematics" before making it part of the National Curricular framework for School Education. Appallingly they have not done so. In this context we demand that the NCERT submit the proposal for the introduction of `Vedic mathematics in the school curriculum to recognized bodies of mathematical experts in India, in particular the National Board of Higher Mathematics (under the Dept. of Atomic Energy), and the Mathematics sections of the Indian Academy of Sciences and the Indian National Science Academy, for a thorough and critical examination. In the meanwhile no attempt should be made to thrust the subject into the school curriculum either through the centrally administered school system or by trying to impose it on the school systems of various States.
We are concerned that the essential thrust behind the campaign to introduce the so-called `Vedic mathematics' has more to do with promoting a particular brand of religious majoritarianism and associated obscurantist ideas rather than any serious and meaningful development of mathematics teaching in India. We note that similar concerns have been expressed about other aspects too of the National Curricular Framework for School Education. We re-iterate our firm conviction that all teaching and pedagogy, not just the teaching of mathematics, must be founded on rational, scientific and secular principles.




S.G.Dani Professor of Mathematics, Tata Institute of Fundamental Research, Mumbai
Madhav M. Deshpande Professor of Sanskrit and Linguistics Department of Asian Languages and Cultures, University of Michigan
Indranil Biswas Professor of Mathematics at TIFR.
Nirmala B. Limaye Professor of Mathematics University of Mumbai
B.V. Limaye Professor of Mathematics Indian Institute of Technology Bombay
Alladi Sitaram, Indian Statistical Institute, B'lore
S. Ramasubramanian, Indian Statistical Inst.,B'ore
V. Pati, Indian Statistical Inst., B'lore
G. Misra, Indian Statistical Inst., B'lore
Jishnu Biswas, Indian Statistical Inst., B'lore
D. P. Sengupta, Indian Inst. of Science(Retd.), B'lore
Alladi Uma, Dept. of English, Univ. of Hyderabad
M. Sridhar, Dept. of English, Univ. of Hyderabad
Amitava Bhattacharya
S.Subramanian, Tata Institute of Fundamental Research Mumbai
Professor Nitin Nitsure,Tata Institute of Fundamental Research, Mumbai

For the article in Frontline on this subject:


http://www.tifr.res.in/~vahia/dani-vmsm.pdf

If you are willing to call the Chinese liers as well, then I am willing to debate the matter further with you.

[NeroASERCH]

SRS Wrote: 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.E. has been found in the Indus Valley

Who said that Indus valley is vedic ? if the indus people knew such things then it is highly likely that the vedics borowed and collated those ideas from the natives.

SRS wrote: The first rather cryptic formula is best understood by way of a simple example: let us multiply 6 by 8. ..............

Hey your method does not work for x X y when x + y =10/100/100, it goes through an infinite spiral. And it does not simplify anything it keeps the complexity the same, multiplication of 2 single/double/.. numbers remains the same after the step one. Neverthless I have seen some other more elegant techniques for multiplying large numbers, not sure if they are vedic or native Indian.

Regarding the Zero we have to give it to the desis, in general and not necessarily to the vedics.

[Drugmachine]

The point of the pi calculation is not to make bicycle wheels. pi/10 gives the same value as pi to 32 decimal places. Pi is an "irrational" number; some type of numerical technique is needed to approximate its value. These days, of course, the approximation is done using a computer. It is amazing that a simple mantra can do exactly what a computer does, yielding the exact same result:
It has a self-contained master-key for extending the evaluation to any number of decimal places.

Who said that I wanted to produce bicycle wheels? Did the vedic mathematicians invent bicycle with the pi-mantra?

ka, ta, pa, and ya all denote 1;
kha, tha, pha, and ra all represent 2;
ga, da, ba, and la all stand for 3;
Gha, dha, bha, and va all represent 4;
gna, na, ma, and sa all represent 5;
ca, ta, and sa all stand for 6;
cha, tha, and sa all denote 7;
ja, da, and ha all represent 8;
jha and dha stand for 9; and
ka means zero.

Vedic Star,
unlike you I spent a bit time in decoding the pi-mantra you have provided and located some errors there. So I analyzed your decoding table and coloured the mismatch.
Can you clarify the mismatch to the hubbers? Will you ever refrain from copy and paste, you clown?

AS I have to assume that your vedic brain does not understand the mismatch, I better explain further:
According to you
ka can be 0 and 1
ta can be 1 and 6
tha can be 2 and 7
da can be 3 and 8
sa can be 5, 6, and 7

Why can't you apply all numbers to all consonants and say PaPa x MaMa is Star! ==> 00x00=*! ==> 0 (read Zero)

No one wonders why you were not educated in India. Anyway, copy and paste you have learned very well but nothing else!

[NeroASERCH]

SRS Wrote: 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.E. has been found in the Indus Valley

Who said that Indus valley is vedic ? if the indus people knew such things then it is highly likely that the vedics borowed and collated those ideas from the natives.

Then show me the literature of the so-called "natives" where the same ideas can be found. It seems like you subscribe to AIT, which has been debunked by genetic evidence. Caste system was not imposed on anyone, but native to India itself. Sandeep? has said in another thread that Manu was a South Indian king. So your attempts to separate the "Vedics" from the "natives" leads to nowhere.

[Lt_Apple]

Originally Posted by SRS The point of the pi calculation is not to make bicycle wheels. pi/10 gives the same value as pi to 32 decimal places. Pi is an "irrational" number; some type of numerical technique is needed to approximate its value. These days, of course, the approximation is done using a computer. It is amazing that a simple mantra can do exactly what a computer does, yielding the exact same result:
It has a self-contained master-key for extending the evaluation to any number of decimal places.

Who said that I wanted to produce bicycle wheels? Did the vedic mathematicians invent bicycle with the pi-mantra?

ka, ta, pa, and ya all denote 1;
kha, tha, pha, and ra all represent 2;
ga, da, ba, and la all stand for 3;
Gha, dha, bha, and va all represent 4;
gna, na, ma, and sa all represent 5;
ca, ta, and sa all stand for 6;
cha, tha, and sa all denote 7;
ja, da, and ha all represent 8;
jha and dha stand for 9; and
ka means zero.

Vedic Star,
unlike you I spent a bit time in decoding the pi-mantra you have provided and located some errors there. So I analyzed your decoding table and coloured the mismatch.
Can you clarify the mismatch to the hubbers? Will you ever refrain from copy and paste, you clown?

AS I have to assume that your vedic brain does not understand the mismatch, I better explain further:
According to you
ka can be 0 and 1
ta can be 1 and 6
tha can be 2 and 7
da can be 3 and 8
sa can be 5, 6, and 7

Why can't you apply all numbers to all consonants and say PaPa x MaMa is Star! ==> 00x00=*! ==> 0 (read Zero)

No one wonders why you were not educated in India. Anyway, copy and paste you have learned very well but nothing else!
I am not following your logic. The use of variables to represent numbers is basic to algebra. The difference is that the Vedics were doing this 2000 yrs before Westerners (Greeks did not know of algebra). Of course the choice of which variable to represent which number is arbitrary.

[HedgeYourBets]

Originally Posted by Bärlin Originally Posted by SRS The point of the pi calculation is not to make bicycle wheels. pi/10 gives the same value as pi to 32 decimal places. Pi is an "irrational" number; some type of numerical technique is needed to approximate its value. These days, of course, the approximation is done using a computer. It is amazing that a simple mantra can do exactly what a computer does, yielding the exact same result:
It has a self-contained master-key for extending the evaluation to any number of decimal places.

Who said that I wanted to produce bicycle wheels? Did the vedic mathematicians invent bicycle with the pi-mantra?

ka, ta, pa, and ya all denote 1;
kha, tha, pha, and ra all represent 2;
ga, da, ba, and la all stand for 3;
Gha, dha, bha, and va all represent 4;
gna, na, ma, and sa all represent 5;
ca, ta, and sa all stand for 6;
cha, tha, and sa all denote 7;
ja, da, and ha all represent 8;
jha and dha stand for 9; and
ka means zero.

Vedic Star,
unlike you I spent a bit time in decoding the pi-mantra you have provided and located some errors there. So I analyzed your decoding table and coloured the mismatch.
Can you clarify the mismatch to the hubbers? Will you ever refrain from copy and paste, you clown?

AS I have to assume that your vedic brain does not understand the mismatch, I better explain further:
According to you
ka can be 0 and 1
ta can be 1 and 6
tha can be 2 and 7
da can be 3 and 8
sa can be 5, 6, and 7

Why can't you apply all numbers to all consonants and say PaPa x MaMa is Star! ==> 00x00=*! ==> 0 (read Zero)

No one wonders why you were not educated in India. Anyway, copy and paste you have learned very well but nothing else!
I am not following your logic. The use of variables to represent numbers is basic to algebra. The difference is that the Vedics were doing this 2000 yrs before Westerners (Greeks did not know of algebra). Of course the choice of which variable to represent which number is arbitrary. Yes, I do understand that you don't understand vedic moron.
The variables you gave to represent numbers are misleading. For example sa can mean 5 or 6 or 7 at the same time according to you. It is not unique. This means that any time a sequence sasa occurs it can mean 55, 56, 57, 65, 66, 67, 75, 76 or 77! Do you get the point or are you already that much confused by your vedic idealogy?

gala hala rasandara - this is the last line of your pi-mantra. According to my calculator the 32th digit must be 5 but yours is 2, why? what happens to the "n" in rasandara? Should it not be rasa dara? Does it make any sense then? When you translate the pi-mantra consonant by consonant then I may perhaps accept but do it first, vedic master!

[Big A]

Originally Posted by SRS Originally Posted by Bärlin Originally Posted by SRS The point of the pi calculation is not to make bicycle wheels. pi/10 gives the same value as pi to 32 decimal places. Pi is an "irrational" number; some type of numerical technique is needed to approximate its value. These days, of course, the approximation is done using a computer. It is amazing that a simple mantra can do exactly what a computer does, yielding the exact same result:
It has a self-contained master-key for extending the evaluation to any number of decimal places.

Who said that I wanted to produce bicycle wheels? Did the vedic mathematicians invent bicycle with the pi-mantra?

ka, ta, pa, and ya all denote 1;
kha, tha, pha, and ra all represent 2;
ga, da, ba, and la all stand for 3;
Gha, dha, bha, and va all represent 4;
gna, na, ma, and sa all represent 5;
ca, ta, and sa all stand for 6;
cha, tha, and sa all denote 7;
ja, da, and ha all represent 8;
jha and dha stand for 9; and
ka means zero.

Vedic Star,
unlike you I spent a bit time in decoding the pi-mantra you have provided and located some errors there. So I analyzed your decoding table and coloured the mismatch.
Can you clarify the mismatch to the hubbers? Will you ever refrain from copy and paste, you clown?

AS I have to assume that your vedic brain does not understand the mismatch, I better explain further:
According to you
ka can be 0 and 1
ta can be 1 and 6
tha can be 2 and 7
da can be 3 and 8
sa can be 5, 6, and 7

Why can't you apply all numbers to all consonants and say PaPa x MaMa is Star! ==> 00x00=*! ==> 0 (read Zero)

No one wonders why you were not educated in India. Anyway, copy and paste you have learned very well but nothing else!
I am not following your logic. The use of variables to represent numbers is basic to algebra. The difference is that the Vedics were doing this 2000 yrs before Westerners (Greeks did not know of algebra). Of course the choice of which variable to represent which number is arbitrary. Yes, I do understand that you don't understand vedic moron.
The variables you gave to represent numbers are misleading. For example sa can mean 5 or 6 or 7 at the same time according to you. It is not unique. This means that any time a sequence sasa occurs it can mean 55, 56, 57, 65, 66, 67, 75, 76 or 77! Do you get the point or are you already that much confused by your vedic idealogy?

gala hala rasandara - this is the last line of your pi-mantra. According to my calculator the 32th digit must be 5 but yours is 2, why? what happens to the "n" in rasandara? Should it not be rasa dara? Does it make any sense then? When you translate the pi-mantra consonant by consonant then I may perhaps accept but do it first, vedic master! What are you trying to say? Double meanings are allowed. The article says very clearly:

For example kapa, tapa, papa, and yapa all mean 11. By a particular choice of consonants and vowels one can compose a poetic hymn with double or triple meanings.

[Fegasderty]

Friends,

It has been pointed out in various threads, all English number's names are from Sanskrit.

The names of number
Two from Tuvithiyai
Three from Thrithiyai; also Tri came from this
Four- IT is Chathurthi- became Quathur- to Quadra and Square to Four
Five- Panchami- became PENTA - TO five and Pancha,p gone tamil anju cames
Six is from SASHTI- became hexta- hexagon and six
Seven from Septhami- and September in calender means seventh month.
Eight- From Oshtami- becaME 0CTA- Eight
Nine - from Navami
Ten- thasami
One- Ptatham became PRIME

This expalains origin of MATHEMATICS.
uPPUMA

[Slonopotam845]

Summa what do you think about that ?

The Sankrit numbers(2-10) Dwi, Tri, Chatur, Pancha, Shat, Sapta, Ashta, Nava, Dasha

The numbers in other language, there is only some interchangability:

The Hindi numbers: Ek, Do, Tin, Char, Panja, che, sat, aat, nuo, das

The French numbers: Un, Deux, Trois, Quatre, cinq, six, sept, huit, neuf, dix

The German numbers: eins, zwei, drei, vier, fünf, sechs, sieben, acht, neun, zehn

The Latin numbers: unus, duo, tres, quattuor, quinque, sex, septem, octo, novem

There is a strong common link between Sanskrit and the European languages....
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I like this topic because i love mathematic..
There is no other logical language than mathematic..
If you know well vedic mathematic , you don't need a calculator to do complex oprerations...

For who it may interest, there is a full course in vedic mathematic at this link :

www.vedamu.org

Look an example :

For a fraction of the form in whose denominator 9 is the last digit, we take the case of 1 / 19 as follows:

For 1 / 19, 'previous' of 19 is 1. And one more than of it is 1 + 1 = 2.

Therefore 2 is the multiplier for the conversion. We write the last digit in the numerator as 1 and follow the steps leftwards.

Step. 1 : 1

Step. 2 : 21(multiply 1 by 2, put to left)

Step. 3 : 421(multiply 2 by 2, put to left)

Step. 4 : 8421(multiply 4 by 2, put to left)

Step. 5 : 168421 (multiply 8 by 2 =16, 1 carried over, 6 put to left)

Step. 6 : 1368421 ( 6 X 2 =12,+1 [carry over]

= 13, 1 carried over, 3 put to left )

Step. 7 : 7368421 ( 3 X 2, = 6 +1 [Carryover]

= 7, put to left)

Step. 8 : 147368421 (as in the same process)

Step. 9 : 947368421 ( Do – continue to step 18)

Step. 10 : 18947368421

Step. 11 : 178947368421

Step. 12 : 1578947368421

Step. 13 : 11578947368421

Step. 14 : 31578947368421

Step. 15 : 631578947368421

Step. 16 : 12631578947368421

Step. 17 : 52631578947368421

Step. 18 : 1052631578947368421

Now from step 18 onwards the same numbers and order towards left continue.

Thus 1 / 19 = 0.052631578947368421

Beautiful !!!!! There is no other word to describe this genious operation !!!

[Big A]

Hi Uppuma
English belongs to Indo European group of languages, the same group that prakrits belong to and sanskrit came from prakrit and tamil. So it would surprising only if sanskrit numbers did not resemble english.

[S.T.D.]

ka can be 0 and 1
ta can be 1 and 6
tha can be 2 and 7
da can be 3 and 8
sa can be 5, 6, and 7

What are you trying to say? Double meanings are allowed. The article says very clearly:

For example kapa, tapa, papa, and yapa all mean 11. By a particular choice of consonants and vowels one can compose a poetic hymn with double or triple meanings. [/quote]

SRS>kapa, tapa, papa, and yapa all mean 11
yes, but Bärlin says something different

rasandara - 2792 ought to be the last 4 digits of the 32 digits you gave and rasandara represents.

But as Bärlin explains that "rasandra" represents 2592, 2692 and 2792. This simply means that rasandra can be any of these 4 numbers. If nobody knows that this mantra is pi-mantra then it simply means nothing. Something of no value. In other words 2592, 2692 and 2792 all deliver rasandra but only 2792 are the last 4 digits of the pi/10 you have givven above.

So this is just misleading and confusing. That is why you are not understanding anything!