Vedas- Religious/Rational III :Religious Hymns-=Mathematical Formulas*

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.
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;

andka 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 madhuvratasrngiso

dadhi sandhigakhala jivita khatavagala 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 decimalplaces.
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.
6-48-2
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).
6-48-24
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.
6-48-2----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 or10(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 - 7578 - 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 + 397 - 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 - 232 - 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.

Bibliography
1. E.J.H. Mackay, Further Excavations at Mohenjo-daro, 1938, p. 222.
2. Saraswati Amma, Geometry in Ancient and Medieval India, Motilal Banarsidas, 1979, p. 18.
3. Dr. V. Raghavan, Presidential Address, Technical Sciences and Fine Arts Section, XXIst AIOC, New Delhi, 1961.
4. Herbert Meschkowski, Ways of Thought of Great Mathematicians, Holden-Day Inc., San Francisco, 1964.
5. Howard Eves, An Introduction to the History of Mathematics, Rinehart and Company Inc., New York, 1953, p. 19.
6. A.L. Basham, The Wonder That Was India, Rupa & Co., Calcutta, 1967.
7. B.B. Dutta, History of Hindu Mathematics, Preface.
8. Jagadguru Swami Shri Bharati Krishna Tirthaji Maharaja, Vedic Mathematics, Motilal Banarsidass, Delhi, 1988.

Note*- Source www.gosai.com