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Sunday, February 15, 2009

Long time after

haiiii...

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Thursday, August 28, 2008

The first suthra of vedic maths

The First Sutra: Ekādhikena Pūrvena
The relevant Sutra reads Ekādhikena Pūrvena which rendered
into English simply says “By one more than the previous one”.
Its application and “modus operandi” are as follows.
(1) The last digit of the denominator in this case being 1 and the
previous one being 1 “one more than the previous one”
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evidently means 2. Further the proposition ‘by’ (in the sutra)
indicates that the arithmetical operation prescribed is either
multiplication or division. We illustrate this example from pp. 1
to 3. [51]
Let us first deal with the case of a fraction say 1/19. 1/19
where denominator ends in 9.
By the Vedic one - line mental method.
A. First method
1/
19
=
.0 5 2 6 315 7 89 4 7 368 4 2 i
1 1 111 1 1 11
B. Second Method
1/
19
=
.0 5 2 6 3 1 5 7 8 / 9 4 7 3 6 8 4 2 i
1 1 11 1 1 1 1 1
This is the whole working. And the modus operandi is
explained below.
A. First Method
Modus operandi chart is as follows:
(i) We put down 1 as the right-hand most digit 1
(ii) We multiply that last digit 1 by 2 and put the 2
down as the immediately preceding digit. 2 1
(iii) We multiply that 2 by 2 and put 4 down as the
next previous digit. 4 2 1
(iv) We multiply that 4 by 2 and put it down thus 8 4 2 1
(v) We multiply that 8 by 2 and get 16 as the
product. But this has two digits. We therefore
put the product. But this has two digits we
therefore put the 6 down immediately to the
left of the 8 and keep the 1 on hand to be
carried over to the left at the next step (as we
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always do in all multiplication e.g. of 69 × 2 =
138 and so on). 6 8 4 2 1
1
(vi) We now multiply 6 by 2 get 12 as product, add
thereto the 1 (kept to be carried over from the
right at the last step), get 13 as the
consolidated product, put the 3 down and keep
the 1 on hand for carrying over to the left at
the next step. 3 6 8 4 2 1
1 1
(vii) We then multiply 3 by 2 add the one carried
over from the right one, get 7 as the
consolidated product. But as this is a single
digit number with nothing to carry over to
the left, we put it down as our next
multiplicand. 7 3 6 8 4 2 1
1 1
((viii) and xviii) we follow this procedure
continually until we reach the 18th digit
counting leftwards from the right, when we
find that the whole decimal has begun to
repeat itself. We therefore put up the usual
recurring marks (dots) on the first and the last
digit of the answer (from betokening that the
whole of it is a Recurring Decimal) and stop
the multiplication there.
Our chart now reads as follows:
1/
19
= . 0 5 2 6 3 1 5 7 8 / 9 4 7 3 6 8 4 2 i .
1 1 1 1 1 1 / 1 1 1
B. Second Method
The second method is the method of division (instead of
multiplication) by the self-same “Ekādhikena Pūrvena” namely
2. And as division is the exact opposite of multiplication it
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stands to reason that the operation of division should proceed
not from right to left (as in the case of multiplication as
expounded here in before) but in the exactly opposite direction;
i.e. from left to right. And such is actually found to be the case.
Its application and modus operandi are as follows:
(i) Dividing 1 (The first digit of the dividend) by
2, we see the quotient is zero and the
remainder is 1. We therefore set 0 down as the
first digit of the quotient and prefix the
remainder 1 to that very digit of the quotient
(as a sort of reverse-procedure to the carrying
to the left process used in multiplication) and
thus obtain 10 as our next dividend. 0
1
(ii) Dividing this 10 by 2, we get 5 as the second
digit of the quotient, and as there is no
remainder to be prefixed thereto we take up
that digit 5 itself as our next dividend. . 0 5
1
(iii) So, the next quotient – digit is 2, and the
remainder is 1. We therefore put 2 down as the
third digit of the quotient and prefix the
remainder 1 to that quotient digit 2 and thus
have 12 as our next dividend. . 0 5 2
1 1
(iv) This gives us 6 as quotient digit and zero as
remainder. So we set 6 down as the fourth
digit of the quotient, and as there is no
remainder to be prefixed thereto we take 6
itself as our next digit for division which gives
the next quotient digit as 3. . 0 5 2 6 3 1
1 1 1
(v) That gives us 1 and 1 as quotient and
remainder respectively. We therefore put 1
down as the 6th quotient digit prefix the 1
thereto and have 11 as our next dividend. . 0 5 2 6 3 1 5
1 1 1 1
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(vi to xvii) Carrying this process of straight continuous
division by 2 we get 2 as the 17th quotient digit and 0 as
remainder.
(xviii) Dividing this 2 by 2 are get 1 as 18th
quotient digit and 0 as remainder. But this is
exactly what we began with. This means that
the decimal begins to repeat itself from here.
So we stop the mental division process and
put down the usual recurring symbols (dots)
on the 1st and 18th digit to show that the
whole of it is a circulating decimal.
Now if we are interested to find 1/29 the student should
note down that the last digit of the denominator is 9, but the
penultimate one is 2 and one more than that means 3. Likewise
for 1/49 the last digit of the denominator is 9 but penultimate is
4 and one more than that is 5 so for each number the
observation must be memorized by the student and remembered.
The following are to be noted
1. Student should find out the procedure to be followed.
The technique must be memorized. They feel it is
difficult and cumbersome and wastes their time and
repels them from mathematics.
2. “This problem can be solved by a calculator in a time
less than a second. Who in this modernized world take
so much strain to work and waste time over such simple
calculation?” asked several of the students.
3. According to many students the long division method
was itself more interesting.

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