Nuggets from Vedic Mathematics
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:
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
1.
adding 8 and -3, or
2.
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.
