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त्रिभुजस्य फलशरीरं समदल•ôôटी भुजार्घ संवर्गः ।
Area = 1/2 AB x CP (Aryabhatta)चतुरघि•¢ शतमष्टगुणं द्वाषष्टिस्तथा सहस्नाणां ।
अयुतद्वय विष्•म्भस्य आसन्नो वृत्तपरिणाहः ॥
AB2 = AC2 + BC2 (Baudhayana)In addition to Mt. Rushmore, one of Gutzon
Borglum's great works as a sculptor is the face of Lincoln which sits
in Washington, D.C. He cut it from a large square block of stone in
his studio. One day, when the face of Lincon was just becoming recognisable
out of the stone, a young girl was visiting the studio with her parents.
She looked at the half-done face of Lincoln, her eyes registering
wonder and astonishment. She stared at the piece for a moment and
then ran to the sculptor.
"Is that Abraham Lincoln?" she asked.
"Yes."
"Well," said the little girl, "how in the world did
you know he was inside there?'
The Vedas are the storehouse of
all the wisdom of the universe. They represent an inexhaustible mine
of the profoundest wisdom in matters both spiritual and temporal.
However, for most of us, the message of our ancient Indian Vedic lore
and the knowledge therein is a cryptic mystery masked in a mystical
language. Yet, the great sages of the past were able to see the portrait
of wisdom concealed within our Vedic scriptures.
They were not sculpted by the chisel and hammer of the ordinary systematic
enquiry methods of modern science. But, they are the eternal truths
revealed by God to the ancient rishis, who through their spiritual
endeavours, were competent enough to receive them from a source which
was perfect and all-knowing. Through their spiritual purity the great
sages of the past had acquired an immense knowledge in all reailms
of life.
That is why, time and again, the great men of antiquity - the sculptors
of wisdom - have carved out from these sacred books, and made recognisable,
the face of knowledge. As a result, the common man's mind has registered
wonder and astonishment at their profound discoveries. The ones that
surprise and interest most are those along scientific lines.
It was Aryabhatta who sculpted the notion that the earth revolves
around the sun, 1000 years before Copernicus.
It was the ancient Indian astronomer, Bhaskaracharya, who discovered
the law of gravity 1200 years before Sir Isaac Newton.
It was the Indian mathematician, Baudhayana, who carved out the so
called Pythagoras' Theorem long before Pythagoras.
It was India that moulded geometry 1200 years before it was introduced
to Europe in the 16th century.
These may seem abstract statements, but they become more plausible
when we consider the great extent to which mathematics was developed
in the Vedas.
The maths was not developed for entertainment, but for a purpose.
The foundation-stone of science is mathematics. To advance scientifically
one has to advance mathematically. The fact that such advanced mathematical
concepts and their amazingly simple application existed in the Vedas
indicates the extent to which science was developed.
'Vedic Mathematics', as it is called, remained hidden in the Vedas
for many centuries. Then the late Shankaracharya, Bharati Krishna
Tirthaji Maharaj, extracted the 'Sixteen Simple Mathematical Formulae'
from the Vedas. He reconstructed them from the 'Atharvaveda' after
assiduous research and penance for about eight years in the forests
surrounding Sringiri.
Initially, he wrote 16 volumes - one for each formula. The manuscripts
were placed in the house of one of his disciples, from where they
were misplaced and lost forever. It was a colossal loss. But the Shankaracharya
was not perturbed, saying that everything was stored in his memory
and that he could re-write the 16 volumes.
In 1957, during his foreign tour, he wrote one volume - an introductory
account of the 16 formulae. A month later his health deteriorated
and a short while later he passed away, leaving only one introductory
volume.
Vedic Mathematics presents a strikingly new theory and method, now
almost unknown. The method is obviously radically different from the
one adopted by the modern mind.
The following are introductory examples which will help demonstrate
this point.
Vulgar Fractions
The conversion of vulgar fractions
whose numerator is 1 and whose denominator ends with a 9 i.e. 1/19,
2/29, 1/39 etc. can be easily done using the 'one more than previous'
formula. To demonstrate let us take 1/19. Converting 1/19 using conventional
methods requires dividing 19 by 100, guessing how many times 19 will
fit into 100, finding the remainder, etc. But the Vedic method is
quite simple. (Follow along on a separate sheet of paper).
-
Step 1) Start by
placing 1 as the last digit (i.e. the right-hand-most digit) of
the answer.
-
Step 2) Proceed to
multiply leftward, continually multiplying by the 'factor'. The
factor = 1 + the penultimate (second-to-last) number of the denominator
(19). In this case the penultimate number is 1. So the factor =
1 + 1 = 2.
-
Step 3) So our first
multiplication should be 1x2 which gives 2. The result should look
as follows: 21
-
Step 4) Next multiply
the 2 by the factor (2) to get 4. The result should look as follows:
421
-
Step 5) The 4 is
multiplied by the factor to get 8. Then when 8 is multiplied by
the factor the result is 16. Here the 6 is placed in the answer
while the 1 is carried. The result should be as follows: 168421.
-
Step 6) Then, when
6 is multiplied by the factor the result is 12, but the 1 which
was carried in the previous step has to be added, giving 13. The
3 is placed in the answer and again the 1 is carried. At this point
your answer should look as follows: 1368421.
-
Step 7) This process
of multiplying and carrying should be continued until the multiplication
results in a figure equivalent to the difference between the denominator
(19) and the numerator (1) which is 18 (i.e. 19-1 = 18). So in this
case we stop at 947368421. Since 9 x 2 (the factor) = 18 and at
this point, for this example, our answer should look as follows:
947368421
-
Step 8) Now take
the 9's compliment of this figure to arrive at the first half of
the answer as follows:
999999999
- 947368421
052631578
-
Step 9) Now put the
two halves together to get the complete answer. You should arrive
at the following answer:
.052631578947368421
This can be verified on your calculator.
(If it is capable of being this accurate!)
To arrive at the answer from left to right we use the division process:
-
Step 1) Dividing
1 (the first digit of the dividend) by 2, (the factor) 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 and thus obtain 10 as our next
dividend.
10
-
Step 2) 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.
105
-
Step 3) 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:
10512
-
Step 4) This is continued
until we reach the figure equivalent to the difference between numerator
and denomiator. And then take the 9's compliment as in the first
method. Combining the two figures we get the answer.
Multiplication
Multiplying 12 x 13 (two-digit numbers)
-
Step 1) Multiply
the left-hand-most digit (1) of the multiplicand vertically by the
left-hand-most digit (1) of the multiplier. Their product (1) is
set down as the left-hand-most part of the answer
-
Step 2) Then cross
multiply 1 and 3, and 1 and 2, add the two, get 5 as the sum and
set it down as the middle past of the answer.
-
Step 3) Now multiply
2 and 3 vertically, get 6 as their product and put it down as the
last (the right-hand-most) part of the answer:
12
13
1 : 3+2 : 6 = 156
Other examples: 23x21
23
21
4 : 2+6 : 3 = 483
94 x 81
94
81
72 : 9+32 : 4 = 72:41:4 = 7614
Multiplying 9 by 7
-
Step 1) We should
take, as the base for our calculations, that power of 10 which is
the nearest to the numbers to be multiplied. In this case 10 itself
is that power.
-
Step 2) Put the two
numbers 9 and 7 above and below on the left-hand side
-
Step 3) Subtract
each of them from the base (10) and write down the remainders (1
and 3) on the right-hand side with a connecting minus sign between
them, to show that the numbers to be multiplied are both less than
10:
9-1
7-3
6/3
-
Step 4) The product
will have two parts, one on the left side and one on the right.
-
Step 5) Now, the
left-hand-side digit of the answer can be arrived at in one of four
ways, of which only two will be shown here:
(a) Cross-subtract the deficiency (3) in the second row from the
original number (9) in the first row to get 6.
(b) Cross-subtract the deficiency (1) in the first row from the
original number (7) in the second row to get 6.
-
Step 6) To arrive
at the right-hand-side digit multiply the numbers in the right column
vertically (i.e. 1x3) to give 3. So our answer is 7x9 = 63.
Note: This is a common feature of the Vedic
system and is of great advantage as it enables us to test and verify
the correctness of our answer at each step.
56 x 99
56-44
99- 1
55/44
For multiplicands and multipliers above a certain power all rules remain
the same, except that, instead of cross-subtracting you cross-add. Therefore
for 1005 x 1009 (remember, the closest power of 10 here is 1000):
1005+5
1009+9
1014/045
If one number is above and the other is below a power of 10 then the
cross-adding or substracting will be done according to the sign, and
vertical multiplication will produce a negative product which will be
subtracted from the left-hand side. For example: 107 x 93
107 + 7
93 - 7
100/-49 = 99/51
Squaring
Squaring of numbers ending in 5.
For example, 152. Here the last digit is 5 and the preceeding digit
is 1. So, one more than that is 2. Now, the formula in this context
tells us to multiply the previous digit (1) by one more than itself
(i.e. by 2), so the left-hand side digit is 1x2; and the right hand
side is the vertical - multiplication product (i.e. 5x5 = 25) as usual.
Thus: 152 = 1x2/25 = 2/25
252 = 2x3/25 = 6/25
352 = 3x4/25 = 12/25 etc.
In this way the sixteen formulae of Vedic Mathematics which have been
found apply to everything from simple arithmetical computations (as
we have briefly seen here) to subjects such as analytical conics and
everything in between such as differential calculus, integration, geometry,
cubic equations, and others. Although the introductory volume gives
an overview of all these branches, the remaining 15 volumes have been
lost.
The truths that the Vedic seers intuitively discovered are today being
intellectually established through the discoveries of modern science.
Our ancestors did not rest content with their intuitive realisations.
They wanted to rub it on the touch-stone of reason and establish the
validity of their knowledge. Thus was science born in ancient India.
Written
by: Sadhu Amrutviharidas |
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