ON VEDIC MATHEMATICS: NIKHILAM SUTRA ON MULTIPLICATION
The Vedas, the most ancient of the Indian scriptures, are four in number: Rg, Yajur, Sama and Atharva, but they also have the four Upavedas and the six Vedangas, all of which form an indivisible corpus of knowledge.
The Sthapatya, Upaveda of Atharva, comprise all kinds of architectural and structural human endeavor and all visual arts. Naturally, the science of calculation and computation falls under this category.
The sixteen Sutras that deal with Mathematics, form part of the Parisistra (Appendix) of the Atharveda.
A Sutra is given as a very short formula for carrying out tedious and cumbersome mathematical operations, and, to a very large extent, for executing them mentally.
The Sutras were studied and reconstructed by Jagadguru Swami Sri Bharati Krishna Tirthaji Maharaja (1884-1960); better known among his disciples by the name “Gurudeva” or “Jagadguruji”. He spent several years of his life in the profoundest study of the most advanced Vedanta Philosophy and spiritual practice. During these years, he taught Sanskrit and Philosophy in schools and Universities and practiced the highest and most vigorous Yoga-Sadhana in the nearest forests of his town. Until the end of his life he traveled all over the world (in the States mainly) delivering lectures on Sanatana Dharma.
He claimed that:
“the very word “Veda” has this derivational meaning; i.e. the fountain-head and illimitable store house of all knowledge. This derivation, in effect, means, connotes and implies that the Vedas should contain within themselves all the knowledge needed by mankind relating not only to the so-called spiritual matters but also to those usually described as purely “secular”, “temporal”, or “worldly”; and also to the means required by humanity as such for the achievement of all round, complete and perfect success in all conceivable directions and that there can be no adjectival or restrictive epithet calculated to limit that knowledge down in any sphere, any direction or any respect whatsoever.”
The reconstruction of the sixteen Sutras from materials scattered here and there in the Atharvaveda, their translation into English, and their presentation together with examples and explanations is the result of an eight year study conducted by Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaja, published after
his death by his disciples. These sutras deal with the following:
a) Very basic mathematical principles of operations such as multiplication and division (see below).
c) Recurring Decimals
d) Algebraic topics such as solutions of simple equations, solution of system of equations, solution of quadratic equations, solutions of cubic equations.
e) Some topics from Geometry like the Pythagorean theorem, and some of the theorems of Apollonius.
f) More advanced mathematics such as analytical expressions of straight lines, analytical Conics
g) Integration by Partial Fractions.
h) Differential Calculus
One of these Sutras, with his elaboration, is as follows:
“Nikhilam Sutra” for Multiplication
The Nikhilam Sutra literally translated means: “All from 9 and the last from 10”
Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaja claims that this Sutra cryptically explains how to perform multiplications of numbers above 5 without previous knowledge of the higher multiplications of the multiplication tables. Consider the example:
Suppose we want to multiply 9 by 7. Then:
Select as base for the calculation that power of 10 which is the nearest to the numbers to be multiplied (in our example 10 itself).
Put the numbers to be multiplied, above and below on the left- hand side of a table as:
Subtract each from the base (in our example 10) and write down the remainders on the right-hand side of the table as:
7 – 3
9 – 1
Between each of the numbers to be multiplied and the remainders put a connecting minus to show that the remainders are less than the base.
The result of the multiplication is a two digit number which will be written under the line. A vertical (/) dividing line may separate the left digit from the right digit of the product.
The left-hand side digit can be obtained by cross subtract one deficiency in the second column (in our example 3) from the original number in the left column (in our example 9).Both cross subtractions (i.e. 7-1 and 9-3) will give the same result (it can easily be proved):
7 – 3
9 – 1
The right-hand digit of the product is the result of the vertical multiplication of the remainders in the right column (in our example 3 times 1).
7 – 3
9 – 1
Thus, the result is 63.
It is obvious that the right-hand side portion of the result must have only one digit, since in this example our base is 10, and so we are entitled only to one digit (units).
When the vertical multiplication of the deficit digits (for obtaining the right-hand side portion of the answer) gives a product consisting of more than one digit, then the surplus portion of the left must be “carried” over to the left of the dividing line. For multiplying 7 times 6 then
7 – 3
6 – 4
The number 12 in the right-hand portion of the product contains both units and tens, though we want only units. The left-hand side digit of 12, which is 1, will be carried over to the left of the dividing line and change 3 into 4. The result will become 3 + 1 /2 so we arrive at the result 42.
The method not only works in all cases but has an infinite number of applications.
Now, if the numbers 98 and 97 must be multiplied, the base that has to be chosen is 100 and the Sutra (all from 9 and the last from 10) is used in order to perform on the spot the subtractions 100-98 and 100-97 and thus determine the numbers in the right column. In this example for the right-hand side digit a two digit number must be obtained (since there are 2 zeros in our base not only one as before) Thus:
98 – 02
97 – 03
95 / 06
Thus, the result is 9506. In order to perform this operation according to the western way, we must perform four multiplications (7*8, 7*9, 9*8, 9*9) and then add two three digit numbers!
Or, if the numbers 99999 and 99994 must be multiplied, then the base is 10000 and the Sutra finds:
99999 – 00001
99994 – 00006
99993 / 00006
And the result is 9999300006! (The discovery and use of zero from Indians is obviously very helpful. For example by using zeros, if necessary, we always respect the condition that the right-hand side of the product must contain the same number of digits as the number of zeros of the base.) With no calculators, in the western way, this operation necessitates 25 multiplications from the multiplication tables and addition of 5 six-digit numbers!!
If the numbers that must be multiplied are a little bit above a base of 10, then, instead of cross-subtracting we shall have to cross add, and, instead of using the minus sign between the numbers on the left side and the right side, we shall have to use the plus sign to denote the additional surplus.
So if 12 has to be multiplied by 11 then:
12 + 2
11 + 1
13 / 2
So the result is 132.
Combining both, if one number is above and the other is below the suitable base then, the plus and the minus will, on multiplication, behave as they always do and produce a minus product. Then the right-hand portion obtained by vertical multiplication will therefore have to be subtracted. A vinculum may be used for making this clear. Thus,
12 + 2
8 – 2
10 / ‾4= 96
Even the subtraction of the vinculum-portion may be easily done with the aid of the Nikhilam Sutra.
It can easily be demonstrated that the previous method always works by using the modern abstract algebraic notation: If x is the chosen base, the first number to be multiplied is x-a and the second one is x-b. Then, the rule always holds because the product is written under the form:
(x-a)*(x-b) = x*(x-a-b) + ab
Or in the last case
(x+a)*(x+b) = x*(x+a+b) +ab
Something very interesting even for a modern mathematician is offered by a Sub-Sutra of the Nikhilam Sutra. This Upasutra, deals with multiplication of numbers that are not near a convenient base. In other words, when neither the multiplicand nor the multiplier is sufficiently near a convenient power of 10 which can suitably serve as a base. Then, in order to perform the multiplication, we choose as a “working base” a convenient multiple or sub-multiple of the suitable base, perform the operation with the aid of the working base and then multiply or divide the result proportionately i.e. in the same proportion as the original base may bear to the working base actually used. If for example the numbers 41 and 41 must be multiplied. The suitable base is 100 and the working base is 50 = 100/2 Then:
41 – 9
41 – 9
The result is 1681
The right-hand side portion of the product remains unaffected (it must not be divided by 2) and the explanation is easily offered by the algebraic formalization used above.
Or, we could have performed the previous multiplication by choosing as a working base the number 50 which is 5*10, so in this case we consider as the original base the number 10. Then:
41 – 9
41 – 9
32 / 81
168 / 1
The result is the same.
Obviously, this sub-sutra is an excellent pedagogical example of both the use of the base and the importance of the proportion of the original base to the working base. Pupils are allowed to “play” with the power of 10 which is close to the numbers that should be multiplied, and they feel free to change the power-base of their numbers. They become familiar with the fact that the same collection of units is represented by different numerals when written under different bases. Above all they don’t get stuck on (and in) the use of the decimal system. On the other hand one can easily notice the property “the proportion of bases is transferred to the numbers”: 1681 = 16*100+81 = 32*50 + (50 +31) (which in the system with a base of 50 is, in reality, 3331 and it can be seen in the first multiplication; but since everything will be written in the decimal system the right-hand portion of the 81 units is not described as 1*50+31 but it remains unaffected equal to 81).
The first corollary of the Nikhilam Sutra can be translated as:
“whatever the extent of its deficiency, lessen it still further to that very extent; and also set up the square of that deficiency”.
It is not said, but this evidently deals with the squaring of numbers. In his book, Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaja gives the following example to make this clear.
Suppose we want to find the square of 9.
Take the nearest power of 10 as a base, in this case 10.
As 9 is 1 less than 10, we decrease it still further by 1,
and set 8 down as the left-side portion of the answer.
And on the right-hand, we put down the square of the deficiency i.e. 1
Thus the square of 9 is 81.
It is admirable that all these operations can be performed mentally!!
Nevertheless, it has to be mentioned that the Nikhilam Sutra does not apply to all cases. If for example, one of the numbers that have to be multiplied is a three-digit number and the second one a two digit number (example 968*56) no suitable base can be found. Then, we necessarily choose as a base the number 1000 and so the right-hand multiplication, i.e. on the right column, becomes cumbersome.
Then the most general Sutra has to be used:
URDHVA-TIRYAK SUTRA. MULTIPLICATION.
The Sutra literally means “vertically and cross-wise”.
Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaja gives the following example to clarify the Sutra:
Suppose we want to multiply 12 times 13.
Multiply the left-hand most digit of the multiplicand vertically by the left-hand-most digit of the multiplier.
Set down their product as the left-hand-most part of the answer.
Multiply 1 times 3 and 1 times 2 cross-wise, add the two results and set the middle digit of the answer.
Multiply 2 times 3 vertically, get 6 as their product and put it down as the right-hand-most part of the answer.
1 / 1*3+2*1 / 2*3 =156.
Similarly, if numbers with more digits must be multiplied, we perform from the left to the right all possible combinations for the cross-wise multiplications of the middle digits.
1*1 / 1*3+2*1 /1*1+2*1+2*3 /2*1+2*3 /2*1 =15982
5*2 / 5*3+8*2 / 5*1+2*2+8*3 /8*1+2*3 /2*1 =10 / 31/ 33/ 14/2 =134442
This Sutra explains multiplication in a way very similar to the western way only the start of the operations from left to right helps the mental performance.
These were two of the very elementary Sutras of Vedic Mathematics.
Historians of mathematics know that the greatest difficulty, when studying ancient scriptures, is to rediscover the knowledge and purposes of the Ancients.
To project modern knowledge and ideas on to the scriptures effectively becomes a very damaging act to such a study.
Accordingly, a modern mathematician may justifiably be skeptical about the statements found in the book “Vedic Mathematics” which claim that a large part of our mathematical knowledge is included in the Vedas. On the other hand, the sixteen Sutras are very short and cryptic and they are not found all together in a chapter concerning mathematics, but scattered here and there as has already been mentioned. It is difficult then, even for a modern mathematician, to collect them all and then to explain such a large part of our Mathematics based only on these Sutras.
In effect, this work is not only one of the rarest that effectively help people in performing a vast variety of mathematical operations mentally, but it recognizes the profound structure of several modern theorems in the sixteen Sutras.
That is why the work of the author of “Vedic Mathematics” is formidable.