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The Vedas, the most
ancient of the
Indian scriptures, are four in number: Rg, Yajur, Sama and Atharva,
but they also have the four Upavedas and the six Vedangas, all of which
form an indivisible corpus of knowledge.
The Sthapatya, Upaveda
of Atharva, comprise all kinds of architectural and structural human
endeavor and all visual arts. Naturally, the science of calculation and
computation falls under this category.
The sixteen Sutras
that deal with Mathematics, form part of the Parisistra (Appendix)
of the Atharveda.
A Sutra is given as a
very short formula for carrying out tedious and cumbersome mathematical
operations, and, to a very large extent, for executing them mentally.
The Sutras were
studied and reconstructed by Jagadguru Swami Sri Bharati Krishna
Tirthaji Maharaja (1884-1960); better known among his disciples by the
name “Gurudeva” or “Jagadguruji”. He spent several years of his life in
the profoundest study of the most advanced Vedanta Philosophy and
spiritual practice. During these years, he taught Sanskrit and Philosophy
in schools and Universities and practiced the highest and most vigorous
Yoga-Sadhana in the nearest forests of his town. Until the end of his life
he traveled all over the world (in the States mainly) delivering lectures
on Sanatana Dharma.
He claimed that:
“the very word “Veda”
has this derivational meaning; i.e. the fountain-head and illimitable
store house of all knowledge. This derivation, in effect, means, connotes
and implies that the Vedas should contain within themselves all the
knowledge needed by mankind relating not only to the so-called spiritual
matters but also to those usually described as purely “secular”,
“temporal”, or “worldly”; and also to the means required by humanity as
such for the achievement of all round, complete and perfect success in all
conceivable directions and that there can be no adjectival or restrictive
epithet calculated to limit that knowledge down in any sphere, any
direction or any respect whatsoever.”
The reconstruction of
the sixteen Sutras from materials scattered here and there in the
Atharvaveda, their translation into English, and their presentation
together with examples and explanations is the result of an eight year
study conducted by
Jagadguru Swami Sri
Bharati Krsna Tirthaji Maharaja, published after his death by his
disciples. These sutras deal with the following:
a)
Very basic mathematical principles of operations such as
multiplication and division (see below).
b)
Factorizations
c)
Recurring Decimals
d)
Algebraic topics such as solutions of simple equations, solution
of system of equations, solution of quadratic equations, solutions of
cubic equations.
e)
Some topics from Geometry like the Pythagorean theorem,
and some of the theorems of Apollonius.
f)
More advanced mathematics such as analytical expressions of
straight lines, analytical Conics
g)
Integration by Partial Fractions.
h)
Differential Calculus
One of these Sutras,
with his elaboration, is as follows:
The Nikhilam Sutra
literally translated means: “All from 9 and the last from 10”
Jagadguru Swami Sri
Bharati Krsna Tirthaji Maharaja claims that this Sutra cryptically
explains how to perform multiplications of numbers above 5 without
previous knowledge of the higher multiplications of the multiplication
tables. Consider the example:
Suppose we want to
multiply 9 by 7. Then:
Select as base for the
calculation that power of 10 which is the nearest to the numbers to be
multiplied (in our example 10 itself).
Put the numbers to be
multiplied, above and below on the left- hand side of a table as:
7
9
Subtract each from the
base (in our example 10) and write down the remainders on the right-hand
side of the table as:
7 – 3
9 – 1
Between each of the
numbers to be multiplied and the remainders put a connecting minus to show
that the remainders are less than the base.
The result of the
multiplication is a two digit number which will be written under the line.
A vertical (/) dividing line may separate the left digit from the right
digit of the product.
The left-hand side
digit can be obtained by cross subtract one deficiency in the second
column (in our example 3) from the original number in the left column (in
our example 9).Both cross subtractions (i.e. 7-1 and 9-3) will give the
same result (it can easily be proved):
7 – 3
9 – 1
6 /
The right-hand digit
of the product is the result of the vertical multiplication of the
remainders in the right column (in our example 3 times 1).
7 – 3
9 – 1
6 /3
Thus, the result is
63.
It is obvious that the
right-hand side portion of the result must have only one digit, since in
this example our base is 10, and so we are entitled only to one digit
(units).
When the vertical
multiplication of the deficit digits (for obtaining the right-hand side
portion of the answer) gives a product consisting of more than one digit,
then the surplus portion of the left must be “carried” over to the left of
the dividing line. For multiplying 7 times 6 then
7 – 3
6 - 4
3 /12
The number 12 in the
right-hand portion of the product contains both units and tens, though we
want only units. The left-hand side digit of 12, which is 1, will be
carried over to the left of the dividing line and change 3 into 4. The
result will become 3 + 1 /2 so we arrive at the
result 42.
The method
not only
works in all cases but has an
infinite number of applications.
Now, if the numbers 98
and 97 must be multiplied, the base that has to be chosen is 100 and the
Sutra (all from 9 and the last from 10) is used in order to perform on the
spot the subtractions 100-98 and 100-97 and thus determine the numbers in
the right column. In this example for the right-hand side digit a two
digit number must be obtained (since there are 2 zeros in our base not
only one as before) Thus:
98 – 02
97 - 03
95 / 06
Thus, the result is
9506. In order to perform this operation
according to the western way, we
must perform four multiplications (7*8, 7*9, 9*8, 9*9) and then add two
three digit numbers!
Or, if the numbers
99999 and 99994 must be multiplied, then the base is 10000 and the Sutra
finds:
99999 – 00001
99994 - 00006
99993 / 00006
And the result is
9999300006! (The discovery and use of zero from Indians is obviously very
helpful. For example by using zeros, if necessary, we always respect the
condition that the right-hand side of the product must contain the same
number of digits as the number of zeros of the base.) With no calculators,
in the western way, this operation necessitates 25 multiplications from
the multiplication tables and addition of 5 six-digit numbers!!
If the numbers that
must be multiplied are a little bit above a base of 10, then, instead of
cross-subtracting we shall have to cross add, and, instead of using the
minus sign between the numbers on the left side and the right
side, we shall have to use the plus sign
to denote the additional surplus.
So if 12 has to be
multiplied by 11 then:
12 + 2
11 + 1
13 / 2
So the result is 132.
Combining both, if one
number is above and the other is below the suitable base then, the plus
and the minus will, on multiplication, behave as they always do and
produce a minus product. Then the right-hand portion obtained by vertical
multiplication will therefore have to be subtracted. A vinculum may be
used for making this clear. Thus,
12 + 2
8 - 2
10 /
 = 96
Even the subtraction
of the vinculum-portion may be easily done with the aid of the Nikhilam
Sutra.
It can easily be
demonstrated that the previous method always works by using the modern
abstract algebraic notation: If x is the chosen base, the first number to
be multiplied is x-a and the second one is x-b. Then, the rule always
holds because the product is written under the form:
(x-a)*(x-b) =
x*(x-a-b) + ab
Or in the last case
(x+a)*(x+b) = x*(x+a+b)
+ab
Something very
interesting even for a modern mathematician is offered by a Sub-Sutra of
the Nikhilam Sutra. This Upasutra, deals with multiplication of numbers
that are not near a convenient base. In other words, when neither the
multiplicand nor the multiplier is sufficiently near a convenient power of
10 which can suitably serve as a base. Then, in order to perform the
multiplication, we choose as a “working base” a convenient multiple or
sub-multiple of the suitable base, perform the operation with the aid of
the working base and then multiply or divide the result proportionately
i.e. in the same proportion as the original base may bear to the working
base actually used. If for example the numbers 41 and 41 must be
multiplied. The suitable base is 100 and the working base is 50 = 100/2
Then:
41 – 9
41 – 9
32 /81
16/81
The result is 1681
The right-hand side
portion of the product remains unaffected (it must not be divided by 2)
and the explanation is easily offered by the algebraic formalization used
above.
Or, we could have
performed the previous multiplication by choosing as a working base the
number 50 which is 5*10, so in this case we consider as the original base
the number 10. Then:
41
– 9
41 - 9
32
/  1
*5
160/1
168 /1
The result is the
same.
Obviously, this
sub-sutra is an excellent pedagogical example of both the use of the base
and the importance of the proportion of the original base to the working
base. Pupils are allowed to “play” with the power of 10 which is close to
the numbers that should be multiplied, and they feel free to change the
power-base of their numbers. They become familiar with the fact that the
same collection of units is represented by different numerals when
written under different bases. Above all they don’t get stuck on (and in)
the use of the decimal system. On the other hand one can easily notice the
property “the proportion of bases is transferred to the numbers”: 1681 =
16*100+81 = 32*50 + (50 +31) (which in the system with a base of 50 is, in
reality, 3331 and it can be seen in the first multiplication; but since
everything will be written in the decimal system the right-hand portion of
the 81 units is not described as 1*50+31 but it remains unaffected equal
to 81).
“whatever the extent
of its deficiency, lessen it still further to that very extent; and also
set up the square of that deficiency”.
It is not said, but
this evidently deals with the squaring of numbers. In his book, Jagadguru
Swami Sri Bharati Krsna Tirthaji Maharaja gives the following example to
make this clear.
Suppose we want to
find the square of 9. Take the nearest power of 10 as
a base, in this case 10.
As 9 is 1 less than
10, we decrease it still further by 1,
and set 8 down as the
left-side portion of the answer.
And on the right-hand,
we put down the square of the deficiency i.e. 1
Thus the square of 9
is 81.
Similarly,
It is admirable that
all these operations can be performed mentally!!
Nevertheless,
it has to be mentioned that the Nikhilam Sutra does not apply to all
cases. If for example, one of the numbers that have to be multiplied is a
three-digit number and the second one a two digit number (example 968*56)
no suitable base can be found. Then, we necessarily choose as a base the
number 1000 and so the right-hand multiplication, i.e. on the right
column, becomes cumbersome.
Then the most general
Sutra has to be used:
The Sutra literally
means “vertically and cross-wise”.
Jagadguru Swami Sri
Bharati Krsna Tirthaji Maharaja gives the following example to clarify
the Sutra:
Suppose we want to
multiply 12 times 13.
Multiply the left-hand
most digit of the multiplicand vertically by the left-hand-most
digit of the multiplier.
Set down their product
as the left-hand-most part of the answer.
Multiply 1 times 3 and
1 times 2 cross-wise, add the two results and set the middle digit of the
answer.
Multiply 2 times 3
vertically, get 6 as their product and put it down as
the right-hand-most part of the answer.
12
13
1 / 1*3+2*1 / 2*3
=156.
Similarly, if numbers
with more digits must be multiplied, we perform from the left to the right
all possible combinations for the cross-wise multiplications of the middle
digits.
122
131
1*1 / 1*3+2*1
/1*1+2*1+2*3 /2*1+2*3 /2*1 =15982
Or
582
231
5*2 / 5*3+8*2 /
5*1+2*2+8*3 /8*1+2*3 /2*1 =10 / / / /2 =134442
This Sutra explains
multiplication in a way very similar to the western way only the start of
the operations from left to right helps the mental performance.
These were two of the
very elementary Sutras of Vedic Mathematics.
Historians of
mathematics know that the greatest difficulty, when studying ancient
scriptures, is to rediscover the knowledge and purposes of the Ancients.
To project modern
knowledge and ideas on to the scriptures
effectively becomes a very damaging
act to such a
study.
Accordingly,
a modern
mathematician may justifiably be skeptical about the statements found in
the book “Vedic Mathematics” which claim that a large part of our
mathematical knowledge is included in the Vedas. On the other hand, the
sixteen Sutras are very short and cryptic and they are not found all
together in a chapter concerning mathematics, but scattered here and there
as has already been mentioned. It is difficult
then, even for a modern mathematician, to collect them all and then
to explain such a large part of our Mathematics based only on
these Sutras.
In
effect,
this work is
not only one of the rarest that effectively
help people in performing a vast variety of
mathematical
operations mentally, but it recognizes the profound structure of
several modern theorems in the sixteen Sutras.
That is why the work
of the author of “Vedic Mathematics” is formidable.
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