Hindu Vedic Philosophy (Sanatana Dharma, Morals, Rituals and Articles)

Answers to Exercise 1 Tutorial 5
Exercise 2a Tutorial 5
Multiply any 2-figure numbers together by mere mental arithmetic!
If you want 21 stamps at 26 pence each you can
easily find the total price in your head.
There were no carries in the method given above.,/p>
However, there only involve one small extra step.
21 x 26 = 546


The method is the same as above
except that we get a 2-figure number, 14, in the
middle step, so the 1 is carried over to the left
(4 becomes 5).
So 21 stamps cost £5.46.
Practise a few:
1) 21 x 47
2) 23 x 43
3) 32 x 53
4) 42 x 32
5) 71 x 72
Answers to Exercise 2a Tutorial 5
Exercise 2b Tutorial 5
33 x 44 = 1452
There may be more than one carry in a sum:

Vertically on the left we get 12.
Crosswise gives us 24, so we carry 2 to the left
and mentally get 144.
Then vertically on the right we get 12 and the 1
here is carried over to the 144 to make 1452.

6) 32 x 56
7) 32 x 54
8) 31 x 72
9) 44 x 53
10) 54 x 64
Answers to Exercise 2b Tutorial 5
Any two numbers, no matter how big, can be
multiplied in one line by this method.
Tutorial 6
Multiplying a number by 11.
To multiply any 2-figure number by 11 we just put
the total of the two figures between the 2 figures.
26 x 11 = 286
Notice that the outer figures in 286 are the 26
being multiplied.
And the middle figure is just 2 and 6 added up.
So 72 x 11 = 792
Exercise 1 Tutorial 6
Multiply by 11:
1) 43 =
2) 81 =
3) 15 =
4) 44 =
5) 11 =
Answers to Exercise 1 Tutorial 6
77 x 11 = 847
This involves a carry figure because 7 + 7 = 14
we get 77 x 11 = 7₁47 = 847.
Exercise 2 Tutorial 6
Multiply by 11:
1) 11 x  88 =
2)  11 x 84 =
3)  11 x 48 =
4)  11 x 73 =
5)  11 x 56 =
Answers to Exercise 2 Tutorial 6
234 x 11 = 2574
We put the 2 and the 4 at the ends.
We add the first pair 2 + 3 = 5.
and we add the last pair: 3 + 4 = 7.
Exercise 3 Tutorial 6
Multiply by 11:
1) 151 =
2) 527 =
3) 333 =
4) 714 =
5) 909 =
Answers to Exercise 3 Tutorial 6

Tutorial 7
Method for dividing by 9.
23 / 9 = 2 remainder 5
The first figure of 23 is 2, and this is the answer.
The remainder is just 2 and 3 added up!
43 / 9 = 4 remainder 7
The first figure 4 is the answer
and 4 + 3 = 7 is the remainder - could it be easier?
Exercise 1a Tutorial 7
Divide by 9:
1)  61 / 9 =        remainder        
2)  33 / 9 =        remainder        
3)  44 / 9 =       remainder         
4)  53 / 9  =       remainder        
5)  80 / 9  =       remainder        
Answers to Exercise 1a Tutorial 7
134 / 9 = 14 remainder 8
The answer consists of 1,4 and 8.
1 is just the first figure of 134.
4 is the total of the first two figures 1+ 3 = 4,
and 8 is the total of all three figures 1+ 3 + 4 = 8.
Exercise 1b Tutorial 7
Divide by 9:
6)  232 =          remainder        
7)  151 =          remainder        
8)  303 =         remainder         
9)  212 =          remainder         
10)  2121 =      remainder         
Answers to Exercise 1b Tutorial 7
842 / 9 = 8₁2 remainder 14 = 92 remainder 14
Actually a remainder of 9 or more is not usually
permitted because we are trying to find how
many 9's there are in 842.
Since the remainder, 14 has one more 9 with 5
left over the final answer will be 93 remainder 5
Exercise 2 Tutorial 7
Divide these by 9:
1)  771 / 9  =          remainder      
2)  942 / 9  =          remainder      
3)  565 / 9 =           remainder      
4)  555  / 9 =          remainder      
5)  2382 / 9  =        remainder      
6)  7070 / 9  =        remainder      
Answers to Exercise 2 Tutorial 7
Answers
Answers to exercise 1 Tutorial 1
1) 223
2) 717
3) 495
4) 7655
5) 0124
6) 8989
7) 43
8) 943
9) 9679
10) 9962
Return to Exercise 1 Tutorial 1
Answers to exercise 1 tutorial 2
1) 64
2) 63
3) 72
4) 49
5) 81
6)2₁6= 36
Return to Exercise 1 Tutorial 2
Answers to Exercise 2 Tutorial 2
1) 8526
2) 8536
3) 7546
4) 8928
5) 8648
6) 6336
7) 9506 (we put 06 because, like all the others,
we need two figures in each part)
Return to Exercise 2 Tutorial 2

Answers to exercise 3 Tutorial 2
1) 10914
2) 10918
3) 10816
4) 11772
5) 12423
6) 10506 (we put 06, not 6)
Return to Exercise 3 Tutorial 2
Answers to Exercise 1 Tutorial 3
1) 29/30
2) 7/12
3) 20/21
4) 19/30
5) 1/20
6) 13/15
Return to Exercise 1 Tutorial 3
Answers to Exercise 1 Tutorial 4
1) 2025
2) 4225
3) 9025
4) 1225
5) 225
Return to Exercise 1 Tutorial 4
Answers to Exercise 2 Tutorial 4
1) 2021
2) 624
3) 4216
4) 221
5) 3009
6) 5621
Return to Exercise 2 Tutorial 4
Answers to Exercise 1 Tutorial 5
1) 294
2) 682
3) 651
4) 462
5) 672
Return to Exercise 1 Tutorial 5

Answers to Exercise 2a Tutorial 5
1) 987
2) 989
3) 1696
4) 1344
5) 5112
Return to Exercise 2a Tutorial 5
Answers to Exercise 2b Tutorial 5
6) 1792
7) 1728
8) 2232
9) 2332
10) 3456
Return to Exercise 2b Tutorial 5
Answers to Exercise 1 Tutorial 6
1) 473
2) 891
3) 165
4) 484
5) 121
Return to Exercise 1 Tutorial 6
Answers to Exercise 2 Tutorial 6
1) 968
2) 924
3) 528
4) 803
5) 616
Return to Exercise 2 Tutorial 6
Answers to Exercise 3 Tutorial 6
1) 1661
2) 5797
3) 3663
4) 7854
5) 9999
Return to  Exercise 3 Tutorial 6
Answers to Exercise 1a Tutorial 7
1) 6 r 7
2) 3 r 6
3) 4 r 8
4) 5 r 8
5) 8 r 8
Return to Exercise 1a Tutorial 7
Answers to Exercise 1b Tutorial 7
1) 25 r 7
2) 16 r 7
3) 33 r 6
4) 23 r 5
5) 235 r 6 (we have 2, 2 + 1, 2 + 1 + 2, 2 + 1 + 2 + 1)
Return to Exercise 1b Tutorial 7
Answers to Exercise 2 Tutorial 7
1) 714 r15 = 84 r15 = 85 r6
2) 913 r 15 = 103 r15 = 104 r6
3) 516 r16 = 61 r16 = 62 r7
4) 510 r15 = 60 r15 = 61 r6
5) 714 r21 = 84 r21 = 86 r3
6) 2513 r15 = 263 r15 = 264 r6
7) 7714 r14 = 784 r14 = 785 r5
Return to Exercise 2 Tutorial 7
Tutorial 8
Vedic Maths - Tips & Tricks
Courtesy   www.vedic-maths-ebook.com
By Kevin O'Connor

s it divisible by four?

This little math trick will show you whether
a number is divisible by four or not.
So, this is how it works.
Let's look at 1234
Does 4 divide evenly into 1234?
For 4 to divide into any number we have
to make sure that the last number is even
If it is an odd number, there is no way it will go in evenly.
So, for example, 4 will not go evenly into 1233 or 1235
Now we know that for 4 to divide evenly into any
number the number has to end with an even number.
Back to the question...
4 into 1234, the solution:
Take the last number and add it to 2 times the second last number
If 4 goes evenly into this number then you know
that 4 will go evenly into the whole number.
So
4 + (2 X 3) = 10
4 goes into 10 two times with a remainder of 2 so it does not go in evenly.
Therefore 4 into 1234 does not go in completely.
Let’s try 4 into 3436546
So, from our example, take the last number, 6 and add it to
two times the penultimate number, 4
6 + (2 X 4) = 14
4 goes into 14 three times with two remainder.
So it doesn't go in evenly.
Let's try one more.
4 into 212334436
6 + (2 X 3) = 12
4 goes into 12 three times with 0 remainder.
Therefore 4 goes into 234436 evenly.
So what use is this trick to you?
Well if you have learnt the tutorial at Memorymentor.com
about telling the day in any year, then you can use it in
working out whether the year you are calculating is a leap year or not.

Multiplying by 12 - shortcut

So how does the 12's shortcut work?
Let's take a look.
12 X 7
The first thing is to always multiply the 1 of the twelve by the
number we are multiplying by, in this case 7. So 1 X 7 = 7.
Multiply this 7 by 10 giving 70. (Why? We are working with BASES here.
Bases are the fundamentals to easy calculations for all multiplication tables.
To find out more check out our
Vedic Maths ebook at www.vedic-maths-ebook.com
Now multiply the 7 by the 2 of twelve giving 14. Add this to 70 giving 84.
Therefore 7 X 12 = 84
Let's try another:
17 X 12
Remember, multiply the 17 by the 1 in 12 and multiply by 10
(Just add a zero to the end)
1 X 17 = 17, multiplied by 10 giving 170.
Multiply 17 by 2 giving 34.
Add 34 to 170 giving 204.
So 17 X 12 = 204
lets go one more
24 X 12
Multiply 24 X 1 = 24. Multiply by 10 giving 240.
Multiply 24 by 2 = 48. Add to 240 giving us 288
24 X 12 = 288 (these are Seriously Simple Sums to do aren’t they?!)

Converting Kilos to pounds

In this section you will learn how to convert Kilos to Pounds, and Vice Versa.
Let’s start off with looking at converting Kilos to pounds.
86 kilos into pounds:
Step one, multiply the kilos by TWO.
To do this, just double the kilos.
86 x 2 = 172
Step two, divide the answer by ten.
To do this, just put a decimal point one place in from the right.
172 / 10 = 17.2
Step three, add step two’s answer to step one’s answer.
172 + 17.2 = 189.2
86 Kilos = 189.2 pounds
Let's try:
50 Kilos to pounds:
Step one, multiply the kilos by TWO.
To do this, just double the kilos.
50 x 2 = 100
Step two, divide the answer by ten.
To do this, just put a decimal point one place in from the right.
100/10 = 10
Step three, add step two's answer to step one's answer.
100 + 10 = 110
50 Kilos = 110 pounds

Adding Time

Here is a nice simple way to add hours and minutes together:
Let's add 1 hr and 35 minutes and 3 hr 55 minutes together.
What you do is this:
make the 1 hr 35 minutes into one number, which will give us 135 and do
the same for the other number, 3 hours 55 minutes, giving us 355
Now you want to add these two numbers together:
135
355
___
490
So we now have a sub total of 490.
What you need to do to this and all sub totals is
add the time constant of 40.
No matter what the hours and minutes are,
just add the 40 time constant to the sub total.
490 + 40 = 530
So we can now see our answer is 5 hrs and 30 minutes!

Temperature Conversions

This is a shortcut to convert Fahrenheit to Celsius and vice versa.
The answer you will get will not be an exact one, but it will
give you an idea of the temperature you are looking at.
Fahrenheit to Celsius:
Take 30 away from the Fahrenheit, and then divide the answer by two.
This is your answer in Celsius.
Example:
74 Fahrenheit - 30 = 44. Then divide by two, 22 Celsius.
So 74 Fahrenheit = 22 Celsius.
Celsius to Fahrenheit just do the reverse:
Double it, and then add 30.
30 Celsius double it, is 60, then add 30 is 90
30 Celsius = 90 Fahrenheit
Remember, the answer is not exact but it gives you a rough idea.

Decimals Equivalents of Fractions

With a little practice, it's not hard to recall
the decimal equivalents of fractions up to 10/11!
First, there are 3 you should know already:
1/2 = .5
1/3 = .333...
1/4 = .25
Starting with the thirds, of which you already know one:
1/3 = .333...
2/3 = .666...
You also know 2 of the 4ths, as well, so there's only one new one to learn:
1/4 = .25
2/4 = 1/2 = .5
3/4 = .75
Fifths are very easy. Take the numerator (the number on top),
double it, and stick a decimal in front of it.
1/5 = .2
2/5 = .4
3/5 = .6
4/5 = .8
There are only two new decimal equivalents to learn with the 6ths:
1/6 = .1666...
2/6 = 1/3 = .333...
3/6 = 1/2 = .5
4/6 = 2/3 = .666...
5/6 = .8333...
What about 7ths? We'll come back to them
at the end. They're very unique.
8ths aren't that hard to learn, as they're just
smaller steps than 4ths. If you have trouble
with any of the 8ths, find the nearest 4th,
and add .125 if needed:
1/8 = .125
2/8 = 1/4 = .25
3/8 = .375
4/8 = 1/2 = .5
5/8 = .625
6/8 = 3/4 = .75
7/8 = .875
9ths are almost too easy:
1/9 = .111...
2/9 = .222...
3/9 = .333...
4/9 = .444...
5/9 = .555...
6/9 = .666...
7/9 = .777...
8/9 = .888...
10ths are very easy, as well.
Just put a decimal in front of the numerator:
1/10 = .1
2/10 = .2
3/10 = .3
4/10 = .4
5/10 = .5
6/10 = .6
7/10 = .7
8/10 = .8
9/10 = .9
Remember how easy 9ths were? 11th are easy in a similar way,
assuming you know your multiples of 9:
1/11 = .090909...
2/11 = .181818...
3/11 = .272727...
4/11 = .363636...
5/11 = .454545...
6/11 = .545454...
7/11 = .636363...
8/11 = .727272...
9/11 = .818181...
10/11 = .909090...
As long as you can remember the pattern for each fraction, it is
quite simple to work out the decimal place as far as you want
or need to go!
Oh, I almost forgot! We haven't done 7ths yet, have we?
One-seventh is an interesting number:
1/7 = .142857142857142857...
For now, just think of one-seventh as: .142857
See if you notice any pattern in the 7ths:
1/7 = .142857...
2/7 = .285714...
3/7 = .428571...
4/7 = .571428...
5/7 = .714285...
6/7 = .857142...
Notice that the 6 digits in the 7ths ALWAYS stay in the same
order and the starting digit is the only thing that changes!
If you know your multiples of 14 up to 6, it isn't difficult to,
work out where to begin the decimal number. Look at this:
For 1/7, think "1 * 14", giving us .14 as the starting point.
For 2/7, think "2 * 14", giving us .28 as the starting point.
For 3/7, think "3 * 14", giving us .42 as the starting point.
For 4/14, 5/14 and 6/14, you'll have to adjust upward by 1:
For 4/7, think "(4 * 14) + 1", giving us .57 as the starting point.
For 5/7, think "(5 * 14) + 1", giving us .71 as the starting point.
For 6/7, think "(6 * 14) + 1", giving us .85 as the starting point.
Practice these, and you'll have the decimal equivalents of
everything from 1/2 to 10/11 at your finger tips!
If you want to demonstrate this skill to other people, and you know
your multiplication tables up to the hundreds for each number 1-9, then give them a
calculator and ask for a 2-digit number (3-digit number, if you're up to it!) to be
divided by a 1-digit number.
If they give you 96 divided by 7, for example, you can think,
"Hmm... the closest multiple of 7 is 91, which is 13 * 7, with 5 left over.
So the answer is 13 and 5/7, or: 13.7142857!"

Converting Kilometres to Miles

This is a useful method for when travelling between imperial
and metric countries and need to know what kilometres to miles are.
The formula to convert kilometres to miles is number of (kilometres / 8 ) X 5
So lets try 80 kilometres into miles
80/8 = 10
multiplied by 5 is 50 miles!
Another example
40 kilometres
40 / 8 = 5
5 X 5= 25 miles
Vedic Mathematics
Master Multiplication tables, division and lots more!
We recommed you check out this ebook, it's packed with tips,
tricks and tutorials that will boost your math ability, guaranteed!
                      www.vedic-maths-ebook.com

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Courtesy   www.vedic-maths-ebook.com
By Kevin O'Connor
_________________

Vedic Mathematics
By Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaj (1884-1960)
Book ref: ISBN 0 8426 0967 9 Published by Motilal Banarasidas
From the Introduction by Smti Manjula Trivedi 16-03-1965. An extract:
Revered Guruji used to say that he had reconstructed the sixteen mathematical formulae from the Atharvaveda after assiduous research and ‘Tapas’ (austerity) for about eight years in the forests surrounding Sringeri. Obviously these formulae are not to be found in the present recensions of Atharvaveda. They were actually reconstructed, on the basis of intuitive revelation, from materials scattered here and there in the Atharvaveda.
From the Preface by the author Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaj
Extracts:
We may however, at this point draw the earnest attention of every one concerned to the following salient items thereof:
The Sutras (aphorisms) apply to and cover each and every part of each and every chapter of each and every branch of mathematics (including Arithmetic, Algebra, Geometry – plane and solid, Trigonometry – plane and spherical, Conics – geometrical and analytical, Astronomy, Calculus – differential and integral etc.) In fact, there is no part of mathematics, pure or applied, that is beyond their jurisdiction.
The Sutras are easy to understand, easy to apply and easy to remember, and the whole work can be truthfully summarised in one word ‘Mental’!
Even as regards complex problems involving a good number of mathematical operations (consecutively or even simultaneously to be performed), the time taken by the Vedic method will be a third, a fourth, a tenth, or even a much smaller fraction of the time required according to modern (i.e. current) Western methods.
And in some very important and striking cases, sums requiring 30, 50, 100 or even more numerous and cumbrous ‘steps’ of working (according to the current Western methods) can be answered in a single and simple step of work by the Vedic method! And little children (of only 10 or 12 years of age) merely look at the sums written on the blackboard and immediately shout out and dictate the answers. And this is because, as a matter of fact, each digit automatically yields its predecessor and its successor! And the children have merely to go on tossing off (or reeling off) the digits one after another (forwards or backwards) by mere mental arithmetic (without needing pen or pencil, paper, slate etc.).
On seeing this kind of work actually being performed by the little children, the doctors, professors and other ‘big-guns’ of mathematics are wonder-struck and exclaim: ‘Is this mathematics or magic’? And we invariably answer and say: ‘It is both. It is magic until you understand it; and it is mathematics thereafter’. And then we proceed to substantiate and prove the correctness of this reply of ours!
As regards the time required by the students for mastering the whole course of Vedic Mathematics as applied to all its branches, we need merely state from our actual experience that 8 months (or 12 months) at an average rate of 2 or 3 hours per day should suffice for completing the whole course of mathematical studies on these Vedic lines instead of 15 or 20 years required according to the existing systems of the Indian and also of foreign universities.
And we were agreeably astonished and intensely gratified to find that exceedingly tough mathematical problems (which the mathematically most advanced present day Western scientific world had spent huge amount of time, energy, and money on and which even now it solves with the utmost difficulty and that also after vast labour involving large numbers of difficult, tedious and cumbersome ‘steps’ of working) can be easily and readily solved with the help of these ultra-easy Vedic Sutras (or mathematical aphorisms) contained in the Parisista (the appendix portion) of the Atharvaveda in a few simple steps and by methods that can be conscientiously described as mere ‘mental arithmetic’.
It is thus in the fitness of things that the Vedas include 1. Ayurveda (anatomy, physiology, hygiene, sanitary science, medical science, surgery etc.), not for the purpose of achieving perfect health and strength in the after-death future but in order to attain them here and now in our present physical bodies. 2.Dhanurveda (archery and other military sciences), not for fighting with one another after our transportation to heaven but in order to quell and subdue all invaders from abroad and all insurgents from within. 3. Gandharva Veda (the science of art and music) and 4. Sthapatya Veda (engineering, architecture etc. and all branches of mathematics in general). All these subjects, be it noted, are inherent parts of the Vedas i.e., are reckoned as ‘spiritual’ studies and catered for as such therein.
Similar is the case with Vedangas (i.e., grammar, prosody, astronomy, lexicography etc.) which according to the Indian cultural conceptions, are also inherent parts and subjects of Vedic (i.e. religious) study.
From the Foreward by Swami Pratyagatmananda Saraswati Varanasi, 22-03-1965
An extract:
Vedic Mathematics by the late Shankaracharya (Bharati Krsna Tirtha) of Govardhan Pitha is a monumental work. In his deep-layer explorations of cryptic Vedic mysteries relating especially to their calculus of shorthand formulae and their neat and ready application to practical problems, the late Shankaracharya shows the rare combination of the probing insight of revealing intuition of a Yogi with the analytic acumen and synthetic talent of a mathematician.
With the late Shankaracharya we belong to a race, now fast becoming extinct, of diehard believers who think that the Vedas represent an inexhaustible mine of profoundest wisdom in matters of both spiritual and temporal; and that this store of wisdom was not, as regards its assets of fundamental validity and value at least, gathered by the laborious inductive and deductive methods of ordinary systemic enquiry, but was direct gift of revelation to seers and sages who in their higher reaches of Yogic realisation were competent to receive it from a source, perfect and immaculate.
Whether or not the Vedas are believed as repositories of perfect wisdom, it is unquestionable that the Vedic race lived not as merely pastoral folk possessing a half or a quarter developed culture and civilisation. The Vedic seers were, again, not mere ‘navel-gazers’ or ‘nose-tip gazers’. They proved themselves adepts in all levels and branches of knowledge, theoretical and practical. For example, they had their varied objective science both pure and applied.
Let us take a concrete illustration. Suppose in a time of drought we require rains by artificial means. The modern scientist has his own theory and art (technique) for producing the result. The old seer scientist had his both also, but different from these now availing. He had his science and technique, called Yajna, in which Mantra, Yantra, and other factors must co-operate with mathematical determinateness and precision. For this purpose, he had developed the six auxiliaries of the Vedas in each of which mathematical skill and adroitness, occult or otherwise, play the decisive role. The Sutras lay down the shortest and surest lines. The correct intonation of the Mantra, the correct configuration of the Yantra (in the making of the Vedi etc., e.g. the quadrate of a circle), the correct time or astral conjunction factor, the correct rhythams etc. All had to be perfected so as to produce the desired results effectively and adequately. Each of these required the calculus of mathematics. The modern technician has his logarithmic tables and mechanic’s manuals. The old Yajnik had his Sutras.
Pages from the history of the Indian sub-continent:
Science and Mathematics in India
History of Mathematics in India
In all early civilizations, the first expression of mathematical understanding appears in the form of counting systems. Numbers in vey early societies were typically represented by groups of lines, though later different numbers came to be assigned specific numeral names and symbols (as in India) or were designated by alphabetic letters (such as in Rome). Although today, we take our decimal system for granted, not all ancient civilizations based their numbers on a ten-base system. In ancient Babylon, a sexagesimal (base 60) system was in use.
The Decimal System in Harappa
In India a decimal system was already in place during the Harappan period, as indicated by an analysis of Harappan weights and measures. Weights corresponding to ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500 have been identified, as have scales with decimal divisions. A particularly notable characteristic of Harappan weights and measures is their remarkable accuracy. A bronze rod marked in units of 0.367 inches points to the degree of precision demanded in those times. Such scales were particularly important in ensuring proper implementation of town planning rules that required roads of fixed widths to run at right angles to each other, for drains to be constructed of precise measurements, and for homes to be constructed according to specified guidelines. The existence of a gradated system of accurately marked weights points to the development of trade and commerce in Harappan society.
Mathematical Activity in the Vedic Period
In the Vedic period, records of mathematical activity are mostly to be found in Vedic texts associated with ritual activities. However, as in many other early agricultural civilizations, the study of arithmetic and geometry was also impelled by secular considerations. Thus, to some extent early mathematical developments in India mirrored the developments in Egypt, Babylon and China . The system of land grants and agricultural tax assessments required accurate measurement of cultivated areas. As land was redistributed or consolidated, problems of mensuration came up that required solutions. In order to ensure that all cultivators had equivalent amounts of irrigated and non-irrigated lands and tracts of equivalent fertility - individual farmers in a village often had their holdings broken up in several parcels to ensure fairness. Since plots could not all be of the same shape - local administrators were required to convert rectangular plots or triangular plots to squares of equivalent sizes and so on. Tax assessments were based on fixed proportions of annual or seasonal crop incomes, but could be adjusted upwards or downwards based on a variety of factors. This meant that an understanding of geometry and arithmetic was virtually essential for revenue administrators. Mathematics was thus brought into the service of both the secular and the ritual domains.
Arithmetic operations (Ganit) such as addition, subtraction, multiplication, fractions, squares, cubes and roots are enumerated in the Narad Vishnu Purana attributed to Ved Vyas (pre-1000 BC). Examples of geometric knowledge (rekha-ganit) are to be found in the Sulva-Sutras of Baudhayana (800 BC) and Apasthmaba (600 BC) which describe techniques for the construction of ritual altars in use during the Vedic era. It is likely that these texts tapped geometric knowledge that may have been acquired much earlier, possibly in the Harappan period. Baudhayana's Sutra displays an understanding of basic geometric shapes and techniques of converting one geometric shape (such as a rectangle) to another of equivalent (or multiple, or fractional) area (such as a square). While some of the formulations are approximations, others are accurate and reveal a certain degree of practical ingenuity as well as some theoretical understanding of basic geometric principles. Modern methods of multiplication and addition probably emerged from the techniques described in the Sulva-Sutras.
Pythagoras - the Greek mathematician and philosopher who lived in the 6th C B.C was familiar with the Upanishads and learnt his basic geometry from the Sulva Sutras. An early statement of what is commonly known as the Pythagoras theorem is to be found in Baudhayana's Sutra: The chord which is stretched across the diagonal of a square produces an area of double the size. A similar observation pertaining to oblongs is also noted. His Sutra also contains geometric solutions of a linear equation in a single unknown. Examples of quadratic equations also appear. Apasthamba's sutra (an expansion of Baudhayana's with several original contributions) provides a value for the square root of 2 that is accurate to the fifth decimal place. Apasthamba also looked at the problems of squaring a circle, dividing a segment into seven equal parts, and a solution to the general linear equation. Jain texts from the 6th C BC such as the Surya Pragyapti describe ellipses.
Modern-day commentators are divided on how some of the results were generated. Some believe that these results came about through hit and trial - as rules of thumb, or as generalizations of observed examples. Others believe that once the scientific method came to be formalized in the Nyaya-Sutras - proofs for such results must have been provided, but these have either been lost or destroyed, or else were transmitted orally through the Gurukul system, and only the final results were tabulated in the texts. In any case, the study of Ganit i.e mathematics was given considerable importance in the Vedic period. The Vedang Jyotish (1000 BC) includes the statement: "Just as the feathers of a peacock and the jewel-stone of a snake are placed at the highest point of the body (at the forehead), similarly, the position of Ganit is the highest amongst all branches of the Vedas and the Shastras."
(Many centuries later, Jain mathematician from Mysore, Mahaviracharya further emphasized the importance of mathematics: "Whatever object exists in this moving and non-moving world, cannot be understood without the base of Ganit (i.e. mathematics)".)
Panini and Formal Scientific Notation
A particularly important development in the history of Indian science that was to have a profound impact on all mathematical treatises that followed was the pioneering work by Panini (6th C BC) in the field of Sanskrit grammar and linguistics. Besides expounding a comprehensive and scientific theory of phonetics, phonology and morphology, Panini provided formal production rules and definitions describing Sanskrit grammar in his treatise called Asthadhyayi. Basic elements such as vowels and consonants, parts of speech such as nouns and verbs were placed in classes. The construction of compound words and sentences was elaborated through ordered rules operating on underlying structures in a manner similar to formal language theory.
Today, Panini's constructions can also be seen as comparable to modern definitions of a mathematical function. G G Joseph, in The crest of the peacock argues that the algebraic nature of Indian mathematics arises as a consequence of the structure of the Sanskrit language. Ingerman in his paper titled Panini-Backus form finds Panini's notation to be equivalent in its power to that of Backus - inventor of the Backus Normal Form used to describe the syntax of modern computer languages. Thus Panini's work provided an example of a scientific notational model that could have propelled later mathematicians to use abstract notations in characterizing algebraic equations and presenting algebraic theorems and results in a scientific format.
Philosophy and Mathematics
Philosophical doctrines also had a profound influence on the development of mathematical concepts and formulations. Like the Upanishadic world view, space and time were considered limitless in Jain cosmology. This led to a deep interest in very large numbers and definitions of infinite numbers. Infinite numbers were created through recursive formulae, as in the Anuyoga Dwara Sutra. Jain mathematicians recognized five different types of infinities: infinite in one direction, in two directions, in area, infinite everywhere and perpetually infinite. Permutations and combinations are listed in the Bhagvati Sutras (3rd C BC) and Sathananga Sutra (2nd C BC).
Jain set theory probably arose in parallel with the Syadvada system of Jain epistemology in which reality was described in terms of pairs of truth conditions and state changes. The Anuyoga Dwara Sutra demonstrates an understanding of the law of indeces and uses it to develop the notion of logarithms. Terms like Ardh Aached , Trik Aached, and Chatur Aached are used to denote log base 2, log base 3 and log base 4 respectively. In Satkhandagama various sets are operated upon by logarithmic functions to base two, by squaring and extracting square roots, and by raising to finite or infinite powers. The operations are repeated to produce new sets. In other works the relation of the number of combinations to the coefficients occurring in the binomial expansion is noted.
Since Jain epistemology allowed for a degree of indeterminacy in describing reality, it probably helped in grappling with indeterminate equations and finding numerical approximations to irrational numbers.
Buddhist literature also demonstrates an awareness of indeterminate and infinite numbers. Buddhist mathematics was classified either as Garna (Simple Mathematics) or Sankhyan (Higher Mathematics). Numbers were deemed to be of three types: Sankheya (countable), Asankheya (uncountable) and Anant (infinite).
Philosophical formulations concerning Shunya - i.e. emptiness or the void may have facilitated in the introduction of the concept of zero. While the zero (bindu) as an empty place holder in the place-value numeral system appears much earlier, algebraic definitions of the zero and it's relationship to mathematical functions appear in the mathematical treatises of Brahmagupta in the 7th C AD. Although scholars are divided about how early the symbol for zero came to be used in numeric notation in India, (Ifrah arguing that the use of zero is already implied in Aryabhatta) tangible evidence for the use of the zero begins to proliferate towards the end of the Gupta period. Between the 7th C and the 11th C, Indian numerals developed into their modern form, and along with the symbols denoting various mathematical functions (such as plus, minus, square root etc) eventually became the foundation stones of modern mathematical notation.
The Indian Numeral System
Although the Chinese were also using a decimal based counting system, the Chinese lacked a formal notational system that had the abstraction and elegance of the Indian notational system, and it was the Indian notational system that reached the Western world through the Arabs and has now been accepted as universal. Several factors contributed to this development whose significance is perhaps best stated by French mathematician, Laplace: "The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. It's simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions."
Brilliant as it was, this invention was no accident. In the Western world, the cumbersome roman numeral system posed as a major obstacle, and in China the pictorial script posed as a hindrance. But in India, almost everything was in place to favor such a development. There was already a long and established history in the use of decimal numbers, and philosophical and cosmological constructs encouraged a creative and expansive approach to number theory. Panini's studies in linguistic theory and formal language and the powerful role of symbolism and representational abstraction in art and architecture may have also provided an impetus, as might have the rationalist doctrines and the exacting epistemology of the Nyaya Sutras, and the innovative abstractions of the Syadavada and Buddhist schools of learning.
Influence of Trade and Commerce, Importance of Astronomy
The growth of trade and commerce, particularly lending and borrowing demanded an understanding of both simple and compound interest which probably stimulated the interest in arithmetic and geometric series. Brahmagupta's description of negative numbers as debts and positive numbers as fortunes points to a link between trade and mathematical study. Knowledge of astronomy - particularly knowledge of the tides and the stars was of great import to trading communities who crossed oceans or deserts at night. This is borne out by numerous references in the Jataka tales and several other folk-tales. The young person who wished to embark on a commercial venture was inevitably required to first gain some grounding in astronomy. This led to a proliferation of teachers of astronomy, who in turn received training at universities such as at Kusumpura (Bihar) or Ujjain (Central India) or at smaller local colleges or Gurukuls. This also led to the exchange of texts on astronomy and mathematics amongst scholars and the transmission of knowledge from one part of India to another. Virtually every Indian state produced great mathematicians who wrote commentaries on the works of other mathematicians (who may have lived and worked in a different part of India many centuries earlier). Sanskrit served as the common medium of scientific communication.
The science of astronomy was also spurred by the need to have accurate calendars and a better understanding of climate and rainfall patterns for timely sowing and choice of crops. At the same time, religion and astrology also played a role in creating an interest in astronomy and a negative fallout of this irrational influence was the rejection of scientific theories that were far ahead of their time. One of the greatest scientists of the Gupta period - Aryabhatta (born in 476 AD, Kusumpura, Bihar) provided a systematic treatment of the position of the planets in space. He correctly posited the axial rotation of the earth, and inferred correctly that the orbits of the planets were ellipses. He also correctly deduced that the moon and the planets shined by reflected sunlight and provided a valid explanation for the solar and lunar eclipses rejecting the superstitions and mythical belief systems surrounding the phenomenon. Although Bhaskar I (born Saurashtra, 6th C, and follower of the Asmaka school of science, Nizamabad, Andhra ) recognized his genius and the tremendous value of his scientific contributions, some later astronomers continued to believe in a static earth and rejected his rational explanations of the eclipses. But in spite of such setbacks, Aryabhatta had a profound influence on the astronomers and mathematicians who followed him, particularly on those from the Asmaka school.
Mathematics played a vital role in Aryabhatta's revolutionary understanding of the solar system. His calculations on pi, the circumferance of the earth (62832 miles) and the length of the solar year (within about 13 minutes of the modern calculation) were remarkably close approximations. In making such calculations, Aryabhatta had to solve several mathematical problems that had not been addressed before, including problems in algebra (beej-ganit) and trigonometry (trikonmiti).
Bhaskar I continued where Aryabhatta left off, and discussed in further detail topics such as the longitudes of the planets; conjunctions of the planets with each other and with bright stars; risings and settings of the planets; and the lunar crescent. Again, these studies required still more advanced mathematics and Bhaskar I expanded on the trigonometric equations provided by Aryabhatta, and like Aryabhatta correctly assessed pi to be an irrational number. Amongst his most important contributions was his formula for calculating the sine function which was 99% accurate. He also did pioneering work on indeterminate equations and considered for the first time quadrilaterals with all the four sides unequal and none of the opposite sides parallel.
Another important astronomer/mathematician was Varahamira (6th C, Ujjain) who compiled previously written texts on astronomy and made important additions to Aryabhatta's trigonometric formulas. His works on permutations and combinations complemented what had been previously achieved by Jain mathematicians and provided a method of calculation of nCr that closely resembles the much more recent Pascal's Triangle. In the 7th century, Brahmagupta did important work in enumerating the basic principles of algebra. In addition to listing the algebraic properties of zero, he also listed the algebraic properties of negative numbers. His work on solutions to quadratic indeterminate equations anticipated the work of Euler and Lagrange.
Emergence of Calculus
In the course of developing a precise mapping of the lunar eclipse, Aryabhatta was obliged to introduce the concept of infinitesimals - i.e. tatkalika gati to designate the infinitesimal, or near instantaneous motion of the moon, and express it in the form of a basic differential equation. Aryabhatta's equations were elaborated on by Manjula (10th C) and Bhaskaracharya (12th C) who derived the differential of the sine function. Later mathematicians used their intuitive understanding of integration in deriving the areas of curved surfaces and the volumes enclosed by them.
Applied Mathematics, Solutions to Practical Problems
Developments also took place in applied mathematics such as in creation of trigonometric tables and measurement units. Yativrsabha's work Tiloyapannatti (6th C) gives various units for measuring distances and time and also describes the system of infinite time measures.
In the 9th C, Mahaviracharya ( Mysore) wrote Ganit Saar Sangraha where he described the currently used method of calculating the Least Common Multiple (LCM) of given numbers. He also derived formulae to calculate the area of an ellipse and a quadrilateral inscribed within a circle (something that had also been looked at by Brahmagupta) The solution of indeterminate equations also drew considerable interest in the 9th century, and several mathematicians contributed approximations and solutions to different types of indeterminate equations.
In the late 9th C, Sridhara (probably Bengal) provided mathematical formulae for a variety of practical problems involving ratios, barter, simple interest, mixtures, purchase and sale, rates of travel, wages, and filling of cisterns. Some of these examples involved fairly complicated solutions and his Patiganita is considered an advanced mathematical work. Sections of the book were also devoted to arithmetic and geometric progressions, including progressions with fractional numbers or terms, and formulas for the sum of certain finite series are provided. Mathematical investigation continued into the 10th C. Vijayanandi (of Benares, whose Karanatilaka was translated by Al-Beruni into Arabic) and Sripati of Maharashtra are amongst the prominent mathematicians of the century.
The leading light of 12th C Indian mathematics was Bhaskaracharya who came from a long-line of mathematicians and was head of the astronomical observatory at Ujjain. He left several important mathematical texts including the Lilavati and Bijaganita and the Siddhanta Shiromani, an astronomical text. He was the first to recognize that certain types of quadratic equations could have two solutions. His Chakrawaat method of solving indeterminate solutions preceded European solutions by several centuries, and in his Siddhanta Shiromani he postulated that the earth had a gravitational force, and broached the fields of infinitesimal calculation and integration. In the second part of this treatise, there are several chapters relating to the study of the sphere and it's properties and applications to geography, planetary mean motion, eccentric epicyclical model of the planets, first visibilities of the planets, the seasons, the lunar crescent etc. He also discussed astronomical instruments and spherical trigonometry. Of particular interest are his trigonometric equations: sin(a + b) = sin a cos b + cos a sin b; sin(a - b) = sin a cos b - cos a sin b;
The Spread of Indian Mathematics
The study of mathematics appears to slow down after the onslaught of the Islamic invasions and the conversion of colleges and universities to madrasahs. But this was also the time when Indian mathematical texts were increasingly being translated into Arabic and Persian. Although Arab scholars relied on a variety of sources including Babylonian, Syrian, Greek and some Chinese texts, Indian mathematical texts played a particularly important role. Scholars such as Ibn Tariq and Al-Fazari (8th C, Baghdad), Al-Kindi (9th C, Basra), Al-Khwarizmi (9th C. Khiva), Al-Qayarawani (9th C, Maghreb, author of Kitab fi al-hisab al-hindi), Al-Uqlidisi (10th C, Damascus, author of The book of Chapters in Indian Arithmetic), Ibn-Sina (Avicenna), Ibn al-Samh (Granada, 11th C, Spain), Al-Nasawi (Khurasan, 11th C, Persia), Al-Beruni (11th C, born Khiva, died Afghanistan), Al-Razi (Teheran), and Ibn-Al-Saffar (11th C, Cordoba) were amongst the many who based their own scientific texts on translations of Indian treatises. Records of the Indian origin of many proofs, concepts and formulations were obscured in the later centuries, but the enormous contributions of Indian mathematics was generously acknowledged by several important Arabic and Persian scholars, especially in Spain. Abbasid scholar Al-Gaheth wrote: " India is the source of knowledge, thought and insight”. Al-Maoudi (956 AD) who travelled in Western India also wrote about the greatness of Indian science. Said Al-Andalusi, an 11th C Spanish scholar and court historian was amongst the most enthusiastic in his praise of Indian civilization, and specially remarked on Indian achievements in the sciences and in mathematics. Of course, eventually, Indian algebra and trigonometry reached Europe through a cycle of translations, travelling from the Arab world to Spain and Sicily, and eventually penetrating all of Europe. At the same time, Arabic and Persian translations of Greek and Egyptian scientific texts became more readily available in India.
The Kerala School
Although it appears that original work in mathematics ceased in much of Northern India after the Islamic conquests, Benaras survived as a center for mathematical study, and an important school of mathematics blossomed in Kerala. Madhava (14th C, Kochi) made important mathematical discoveries that would not be identified by European mathematicians till at least two centuries later. His series expansion of the cos and sine functions anticipated Newton by almost three centuries. Historians of mathematics, Rajagopal, Rangachari and Joseph considered his contributions instrumental in taking mathematics to the next stage, that of modern classical analysis. Nilkantha (15th C, Tirur, Kerala) extended and elaborated upon the results of Madhava while Jyesthadeva (16th C, Kerala) provided detailed proofs of the theorems and derivations of the rules contained in the works of Madhava and Nilkantha. It is also notable that Jyesthadeva's Yuktibhasa which contained commentaries on Nilkantha's Tantrasamgraha included elaborations on planetary theory later adopted by Tycho Brahe, and mathematics that anticipated work by later Europeans. Chitrabhanu (16th C, Kerala) gave integer solutions to twenty-one types of systems of two algebraic equations, using both algebraic and geometric methods in developing his results. Important discoveries by the Kerala mathematicians included the Newton-Gauss interpolation formula, the formula for the sum of an infinite series, and a series notation for pi. Charles Whish (1835, published in the Transactions of the Royal Asiatic Society of Great Britain and Ireland) was one of the first Westerners to recognize that the Kerala school had anticipated by almost 300 years many European developments in the field.
Yet, few modern compendiums on the history of mathematics have paid adequate attention to the often pioneering and revolutionary contributions of Indian mathematicians. But as this essay amply demonstrates, a significant body of mathematical works were produced in the Indian subcontinent. The science of mathematics played a pivotal role not only in the industrial revolution but in the scientific developments that have occurred since. No other branch of science is complete without mathematics. Not only did India provide the financial capital for the industrial revolution (see the essay on colonization) India also provided vital elements of the scientific foundation without which humanity could not have entered this modern age of science and high technology.
Notes:
Mathematics and Music: Pingala (3rd C AD), author of Chandasutra explored the relationship between combinatorics and musical theory anticipating Mersenne (1588-1648) author of a classic on musical theory.
Mathematics and Architecture: Interest in arithmetic and geometric series may have also been stimulated by (and influenced) Indian architectural designs - (as in temple shikaras, gopurams and corbelled temple ceilings). Of course, the relationship between geometry and architectural decoration was developed to it's greatest heights by Central Asian, Persian, Turkish, Arab and Indian architects in a variety of monuments commissioned by the Islamic rulers.
Transmission of the Indian Numeral System: Evidence for the transmission of the Indian Numeral System to the West is provided by Joseph (Crest of the Peacock):
Quotes Severus Sebokht (662) in a Syriac text describing the "subtle discoveries" of Indian astronomers as being "more ingenious than those of the Greeks and the Babylonians" and "their valuable methods of computation which surpass description" and then goes on to mention the use of nine numerals.
· Quotes from Liber abaci (Book of the Abacus) by Fibonacci (1170-1250): The nine Indian numerals are ...with these nine and with the sign 0 which in Arabic is sifr, any desired number can be written. (Fibonaci learnt about Indian numerals from his Arab teachers in North Africa)
Influence of the Kerala School: Joseph (Crest of the Peacock) suggests that Indian mathematical manuscripts may have been brought to Europe by Jesuit priests such as Matteo Ricci who spent two years in Kochi (Cochin) after being ordained in Goa in 1580. Kochi is only 70km from Thrissur (Trichur) which was then the largest repository of astronomical documents. Whish and Hyne - two European mathematicians obtained their copies of works by the Kerala mathematicians from Thrissur, and it is not inconceivable that Jesuit monks may have also taken copies to Pisa (where Galileo, Cavalieri and Wallis spent time), or Padau (where James Gregory studied) or Paris (where Mersenne who was in touch with Fermat and Pascal, acted as an agent for the transmission of mathematical ideas).
References:
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8. P Z Ingerman: 'Panini-Backus form', Communications of the ACM 10 (3)(1967), 137.
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Another view on Indian Mathematics:
Dr. David Gray writes:
"The study of mathematics in the West has long been characterized by a certain ethnocentric bias, a bias which most often manifests not in explicit racism, but in a tendency toward undermining or eliding the real contributions made by non-Western civilizations. The debt owed by the West to other civilizations, and to India in particular, go back to the earliest epoch of the "Western" scientific tradition, the age of the classical Greeks, and continued up until the dawn of the modern era, the renaissance, when Europe was awakening from its dark ages."
Dr Gray goes on to list some of the most important developments in the history of mathematics that took place in India, summarizing the contributions of luminaries such as Aryabhatta, Brahmagupta, Mahavira, Bhaskara and Maadhava. He concludes by asserting that "the role played by India in the development (of the scientific revolution in Europe) is no mere footnote, easily and inconsequentially swept under the rug of Eurocentric bias. To do so is to distort history, and to deny India one of its greatest contributions to world civilization."
-Indic Mathematics
Related Essays:
The following essays can be found at http://india_resource.tripod.com/indianhistory.html
Development of Philosophical Thought and Scientific Method in Ancient india
Philosophical Development from Upanishadic Theism to Scientific Realism
History of the Physical Sciences in India











 
_{(My humble salutations to  Swamy jis , Vedic Mathematicians  and Hinduism com  for the collection)}