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:

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

Multiply this by "one more", that is, 2 (this is the "key" digit from Ekadhikena)

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.

rgupta@ucsd.edu