From: Mermaid . (britannica@hotmail.com)
Date: Sun Jan 06 2002 - 16:48:31 MST
[Mermaid]I just got this link as a response to this whole Vedic Math/Vedic
Code business from someone who swears by it... <Apparently, the Japanese
have a similar system of remembering numbers with certain words in the
Japanese language..he has asked me to check out the record for recitation of
the pi...apparently the title has passed on from a 'vedic mathematician' to
an elderly japanese person who recited the value of pi for approx 18
hours...trivia..if anyone cares... >As far as I can see, it is simply a
manner of remembering numbers and finding shortcuts to mathematical
calculations using the Sanskrit language and its interesting structure. No
end of the world prophecies...no innuendos about the future of the
world...no religious baggage...just pure, simple, wonderful numbers...I
think this proves that certain Doubting Toms should best check before
spouting.
P.S.I havent checked virus despatches this weekend...so if this url or the
information has already been reported, I apologise for the repeat..
http://www1.ics.uci.edu/~rgupta/vedic.html
<begin snip>
The Vedas are ancient holy texts from India than can be legitimately
characterized as the all-encompassing repository of (Hindu) knowledge from
eons past. The term Vedic Mathematics refers to a set of sixteen
mathematical formulae or sutras and their corollaries derived from the
Vedas. The sixteen sutras are:
Ekadhikena Purvena
Nikhilam Navatashcaramam Dashatah
Urdhva-tiryagbhyam
Paraavartya Yojayet
Shunyam Saamyasamuccaye
(Anurupye) Shunyamanyat
Sankalana-vyavakalanabhyam
Puranapuranabhyam
Chalana-Kalanabhyam
Yaavadunam
Vyashtisamanshtih
Shesanyankena Charamena
Sopaantyadvayamantyam
Ekanynena Purvena
Gunitasamuchyah
Gunakasamuchyah
Vedic Number Representation
Vedic knowledge is in the form of slokas or poems in Sanskrit verse. A
number was encoded using consonant groups of the Sanskrit alphabet, and
vowels were provided as additional latitude to the author in poetic
composition. The coding key is given as Kaadi nav, taadi nav, paadi panchak,
yaadashtak ta ksha shunyam Translated as below
letter "ka" and the following eight letters
letter "ta" and the following eight letters
letter "pa" and the following four letters
letter "ya" and the following seven letters, and
letter "ksha" for zero.
In other words,
ka, ta, pa, ya = 1
kha, tha, pha, ra = 2
ga, da, ba, la = 3
gha, dha, bha, va = 4
gna, na, ma, scha = 5
cha, ta, sha = 6
chha, tha, sa = 7
ja, da, ha = 8
jha, dha = 9
ksha = 0
For those of you who don't know or remember the varnmala, here it is:
ka kha ga gha gna
cha chha ja jha inya
Ta Tha Rda Dha Rna
ta tha da dha na
pa pha ba bha ma
ya ra la va scha
sha sa ha chjha tra gna
Thus pa pa is 11, ma ra is 52. Words kapa, tapa , papa, and yapa all mean
the same that is 11. It was upto the author to choose one that fit the
meaning of the verse well. An interesting example of this is a hymn below in
the praise of God Krishna that gives the value of Pi to the 32 decimal
places as .31415926535897932384626433832792.
Gopi bhaagya madhu vraata
Shrngisho dadhisandhiga
Khalajivita khaataava
Galahaataarasandhara
1. Ekadhikena Purvena
or By one more than the previous one.
The proposition "by" means the operations this sutra concerns are either
multiplication or division. [ In case of addition/subtraction proposition
"to" or "from" is used.] Thus this sutra is used for either multiplication
or division. It turns out that it is applicable in both operations.
An interesting application of this sutra is in computing squares of numbers
ending in five. Consider:
35x35 = (3x(3+1)) 25 = 12,25
The latter portion is multiplied by itself (5 by 5) and the previous portion
is multiplied by one more than itself (3 by 4) resulting in the answer 1225.
It can also be applied in multiplications when the last digit is not 5 but
the sum of the last digits is the base (10) and the previous parts are the
same. Consider:
37X33 = (3x4),7x3 = 12,21
29x21 = (2x3),9x1 = 6,09 [Antyayor dashake]
We illustrate this sutra by its application to conversion of fractions into
their equivalent decimal form. Consider fraction 1/19. Using this sutra this
can be converted into a decimal form in a single step. This can be done
either by applying the sutra for a multiplication operation or for a
division operations, thus yielding two methods.
Method 1: using multiplications
1/19, since 19 is not divisible by 2 or 5, the fractional result is a purely
circulating decimal. (If the denominator contains only factors 2 and 5 is a
purely non-circulating decimal, else it is a mixture of the two.)
So we start with the last digit
1
Multiply this by "one more", that is, 2 (this is the "key" digit from
Ekadhikena)
21
Multiplying 2 by 2, followed by multiplying 4 by 2
421 => 8421
Now, multiplying 8 by 2, sixteen
68421
1 <= carry
multiplying 6 by 2 is 12 plus 1 carry gives 13
368421
1 <= carry
Continuing
7368421 => 47368421 => 947368421
1
Now we have 9 digits of the answer. There are a total of 18 digits
(=denominator-numerator) in the answer computed by complementing the lower
half:
052631578
947368421
Thus the result is .052631578,947368421
Method 2: using divisions
The earlier process can also be done using division instead of
multiplication. We divide 1 by 2, answer is 0 with remainder 1
.0
Next 10 divided by 2 is five
.05
Next 5 divided by 2 is 2 with remainder 1
.052
next 12 (remainder,2) divided by 2 is 6
.0526
and so on.
As another example, consider 1/7, this same as 7/49 which as last digit of
the denominator as 9. The previous digit is 4, by one more is 5. So we
multiply (or divide) by 5, that is,
...7 => 57 => 857 => 2857 => 42857 => 142857 => .142,857 (stop after 7-1
digits)
3 2 4 1 2
2. Nikhilam Navatashcaramam Dashatah
or All from nine and the last from ten.
This sutra is often used in special cases of multiplication.
Corollary 1: Yavdunam Jaavdunikritya Varga Cha Yojayet
or Whatever the extent of its deficiency, lessen it still further to that
very extent; and also set up the square of that deficiency.
For instance: in computing the square of 9 we go through the following
steps:
The nearest power of 10 to 9 is 10. Therefore, let us take 10 as our base.
Since 9 is 1 less than 10, decrease it still further to 8. This is the
left side of our answer.
On the right hand side put the square of the deficiency, that is 1^2.
Hence the answer is 81.
Similarly, 8^2 = 64, 7^2 = 49
For numbers above 10, instead of looking at the deficit we look at the
surplus. For example:
11^2 = 12 1^2 = 121
12^2 = (12+2) 2^2 = 144
14^2 = (14+4) 4^2 = 18 16 = 196
and so on.
3. Urdhva-tiryagbhyam
or Vertically and cross-wise.
This sutra applies to all cases of multiplication and is very useful in
division of one large number by another large number.
4. Paraavartya Yojayet
or Transpose and apply.
This sutra complements the Nikhilam sutra which is useful in divisions by
large numbers. This sutra is useful in cases where the divisor consists of
small digits. This sutra can be used to derive the Horner's process of
Synthetic Division.
5. Shunyam Saamyasamuccaye
or When the samuccaya is the same, that samuccaya is zero.
This sutra is useful in solution of several special types of equations that
can be solved visually. The word samuccaya has various meanings in different
applicatins. For instance, it may mean a term which occurs as a common
factor in all the terms concerned. A simple example is equation "12x + 3x =
4x + 5x". Since "x" occurs as a common factor in all the terms, therefore,
x=0 is a solution. Another meaning may be that samuccaya is a product of
independent terms. For instance, in (x+7)(x+9) = (x+3)(x+21), the samuccaya
is 7 x 9 = 3 x 21, therefore, x = 0 is a solution. Another meaning is the
sum of the denominators of two fractions having the same numerical
numerator, for example: 1/(2x-1) + 1/(3x-1) = 0 means 5x - 2 = 0.
Yet another meaning is "combination" or total. This is commonly used. For
instance, if the sum of the numerators and the sum of denominators are the
same then that sum is zero. Therefore,
2x + 9 2x + 7
------ = ------
2x + 7 2x + 9
therefore, 4x + 16 = 0 or x = -4
This meaning ("total") can also be applied in solving quadratic equations.
The total meaning can not only imply sum but also subtraction. For instance
when given N1/D1 = N2/D2, if N1+N2 = D1 + D2 (as shown earlier) then this
sum is zero. Mental cross multiplication reveals that the resulting equation
is quadratic (the coefficients of x^2 are different on the two sides). So,
if N1 - D1 = N2 - D2 then that samuccaya is also zero. This yield the other
root of a quadratic equation.
Yet interpretation of "total" is applied in multi-term RHS and LHS. For
instance, consider
1 1 1 1
--- + ----- = ----- + ------
x-7 x-9 x-6 x-10
Here D1 + D2 = D3 + D4 = 2 x - 16. Thus x = 8.
There are several other cases where samuccaya can be applied with great
versatility. For instance "apparently cubic" or "biquadratic" equations can
be easily solved as shown below:
(x-3)^3 + (x-9)^3 = 2 (x-6)^3
Note that x -3 + x - 9 = 2 (x - 6). Therefore (x - 6) = 0 or x = 6.
consider
(x+3)^3 x+1
-------- = --------
(x+5)^3 x + 7
Observe: N1 + D1 = N2 + D2 = 2x + 8.
Therefore, x = -4.
This sutra has been extended further.
6. (Anurupye) Shunyamanyat
or If one is in ratio, the other one is zero.
This sutra is often used to solve simultaneous simple equations which may
involve big numbers. But these equations in special cases can be visually
solved because of a certain ratio between the coefficients. Consider the
following example:
6x + 7y = 8
19x + 14y = 16
Here the ratio of coefficients of y is same as that of the constant terms.
Therefore, the "other" is zero, i.e., x = 0. Hence the solution of the
equations is x = 0 and y = 8/7.
This sutra is easily applicable to more general cases with any number of
variables. For instance
ax + by + cz = a
bx + cy + az = b
cx + ay + bz = c
which yields x = 1, y = 0, z = 0.
A corollary (upsutra) of this sutra says Sankalana-Vyavakalanaabhyam or By
addition and by subtraction. It is applicable in case of simultaneous linear
equations where the x- and y-coefficients are interchanged. For instance:
45x - 23y = 113
23x - 45y = 91
By addition: 68x - 68 y = 204 => 68(x-y) = 204 => x - y = 3
By subtraction: 22x + 22y = 22 => 22(x+y) = 22 => x + y = 1
8. Puranapuranabhyam
or By the completion or non-completion.
14. Ekanynena Purvena
It is converse of the Ekaadhika sutra. It provides for multiplications
wherein the multiplier digits consist entirely of nines.
--------------------------------------------------------------------------------
"Rules of Thumb"
Many of the basic sutras have been applied to devise commonly used rules of
thumb. For instance, the Ekanyuna sutra can be used to derive the following
results:
Kevalaih Saptakam Gunyaat, or in the case of seven the multiplicand should
be 143
Kalau Kshudasasaih, or in the case of 13 the multiplicand should be 077
Kamse Kshaamadaaha-khalairmalaih, or in the case of 17 the multiplicand
should be 05882353 (by the way, the literal meaning of this result is "In
king Kamsa's reign famine, and unhygenic conditions prevailed." -- not
immediately obvious what it had to do with Mathematics. These multiple
meanings of these sutras were one of the reasons why some of the early
translations of Vedas missed discourses on vedaangas.)
These are used to correctly identify first half of a recurring decimal
number, and then applying Ekanyuna to arrive at the complete answer
mechanically. Consider for example the following visual computations:
1/7 = 143x999/999999 = 142857/999999 = 0.142857
1/13 = 077x999/999999 = 076923/999999 = 0.076923
1/17 = 05882353x99999999/9999999999999999 = 0.05882352 94117647
Note that
7x142857 = 999999
13x076923 = 999999
17x05882352 94117647 = 9999999999999999
which says that if the last digit of the denominator is 7 or 3 then the last
digit of the equivalent decimal fraction is 7 or 3 respectively.
Some Interesting Nuggets and Examples:
The Multiplication Sign "X" as a Cross-Addition: Let us multiply (decimal
numbers) 8 by 7: first column lists the numbers and the second column the
deficits (from base = 10):
8 -2
X 7 -3
---------
The multiplication proceeds from the most signficant digit to least
significant digit (which is natural since the positional numbers are also
read from MSD to LSD, thus the result can be produced "on-line"). The first
digit (most significant digit) is obtained by
adding 8 and -3, or
adding 7 and -2, or
that is,
8 -2
\/
/\
7 -3
This process of obtaining MSD of a multiplication by cross-addition is said
to be the origin of the conventional cross sign for multiplication. BTW, you
can generate the following digit by multiplication and (if necessary) by
forwarding the carry to more significant digits. This method (derived from
Nikhilam sutra) works multiplication of multidigit numbers and numbers
greater than as well as less than the base (or half the base). Consider bit
more complex examples below:
97 -3 102 2 888 -112
X 98 -2 X 104 4 X997 -003
----- ------ ---------
95,06 106,08 885,336
For cases when the numbers are closer to the middle of the base, Anurupyena
sutra (according to the ratio) can be used to compute deficit/excess from a
ratio of the base and then ratio the result:
48 -2 (base/2 = 50)
X46 -4
------
44,08 => 22,08
Division using "Seshaanyankaani charamena": to carry out a division first
compute remainders and then multiply the remainders by the last digit and
put down the last digit of the multiplicand. Consider: 1/7. When divising
1(0) by 7 the remainder is 3. Therefore, dividing 3 by 7 will subsequently
lead to remainder 9 (= 3x3). But since 9 is more than 7 the remainder would
be 2, so the remainder sequence is:
3, 2
Now 2 divided by 7 will have remainder of 6 (3x2), that is
3, 2, 6
Continuing
3, 2, 6, 4, 5, 1
We stop when the remainder sequence starts to repeat. Now, multiply these
remainders by the last digit (7) of the denominator and keep only the first
digit (LSD). So we have:
7x3 = 21 => put down 1
.1
3, 2, 6, 4, 5, 1
7x2 = 14 => put down 4
.1 4
3, 2, 6, 4, 5, 1
7x6 = 42 => put down 2
.1 4 2
3, 2, 6, 4, 5, 1
Continuing
.1 4 2 8 5 7
3, 2, 6, 4, 5, 1
So the answer is 1/7 = .142857142857...
Acknowledgments
The illustrations are taken from the book Vedic Mathematics by Jagadguru
Swami Shri Bharati Krishna Tirthaji Maharaja published by Motilal
Banarasidass Publishers, Delhi, India.
--------------------------------------------------------------------------------
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