Vedic Mathematical Sutras
Consider the following three sutras:
1. "All from 9 and the last from 10," and its
corollary: "Whatever the extent of its deficiency, lessen it still
further to that very extent; and also set up the square (of that deficiency)."
2. "By one more than the previous one," and its corollary: "Proportionately."
3. "Vertically and crosswise," and its corollary: "The first
by the first and the last by the last."
The first rather cryptic formula is best understood by way of a simple example: let us multiply 6 by 8.
1. First, assign as the base for our calculations the power
of 10 nearest to the numbers which are to be multiplied. For this example
our base is 10.
2. Write the two numbers to be multiplied on a paper one above the other,
and to the right of each write the remainder when each number is subtracted
from the base 10. The remainders are then connected to the original numbers
with minus signs, signifying that they are less than the base 10.
6-4
8-2
3. The answer to the multiplication is given in two parts. The first digit on the left is in multiples of 10 (i.e. the 4 of the answer 48). Although the answer can be arrived at by four different ways, only one is presented here. Subtract the sum of the two deficiencies (4 + 2 = 6) from the base (10) and obtain 10 - 6 = 4 for the left digit (which in multiples of the base 10 is 40).
6-4
8-2
4
4. Now multiply the two remainder numbers 4 and 2 to obtain the product 8. This is the right hand portion of the answer which when added to the left hand portion 4 (multiples of 10) produces 48.
6-4
8-2
----
4/8
Another method employs cross subtraction. In the current example the 2 is subtracted from 6 (or 4 from 8) to obtain the first digit of the answer and the digits 2 and 4 are multiplied together to give the second digit of the answer. This process has been noted by historians as responsible for the general acceptance of the X mark as the sign of multiplication. The algebraical explanation for the first process is
(x-a)(x-b)=x(x-a-b) + ab
where x is the base 10, a is the remainder 4 and b is the remainder 2 so that
6 = (x-a) = (10-4)
8 = (x-b) = (10-2)
The equivalent process of multiplying 6 by 8 is then
x(x-a-b) + ab or
10(10-4-2) + 2x4 = 40 + 8 = 48
These simple examples can be extended without limitation. Consider the following cases where 100 has been chosen as the base:
97 -
78 - 22
______75/66
In the last example we carry the 100 of the 150 to the left and 23 (signifying 23 hundred) becomes 24 (hundred). Herein the sutra's words "all from 9 and the last from 10" are shown. The rule is that all the digits of the given original numbers are subtracted from 9, except for the last (the righthand-most one) which should be deducted from 10.
Consider the case when the multiplicand and the multiplier
are just above a power of 10. In this case we must cross-add instead of
cross subtract. The algebraic formula for the process is: (x+a)(x+b)
= x(x+a+b) + ab. Further, if one number is above and the other below
a power of 10, we have a combination of subtraction and addition: viz:
108 + 8
_______
105/-24 = 104/(100-24) = 104/76 11/-6 = 10/(10-6)
= 10/4
The Sub-Sutra: "Proportionately" Provides for those cases where we wish to use as our base multiples of the normal base of powers of ten. That is, whenever neither the multiplicand nor the multiplier is sufficiently near a convenient power of 10, which could serve as our base we simply use a multiple of a power of ten as our working base, perform our calculations with this working base and then multiply or divide the result proportionately.
To multiply 48 by 32, for example, we use as our base 50 = 100/2, so we have
Base 50
Note that only the left decimals corresponding to the powers of ten digits (here 100) are to be effected by the proportional division of 2. These examples show how much easier it is to subtract a few numbers, (especially for more complex calculations) rather than memorize long mathematical tables and perform cumbersome calculations the long way.
Squaring Numbers
The algebraic equivalent of the sutra for squaring a number is: (a+-b)2 = a2 +- 2ab + b2 . To square 103 we could write it as (100 + 3 )2 = 10,000 + 600 + 9 = 10,609. This calculation can easily be done mentally. Similarly, to divide 38,982 by 73 we can write the numerator as 38x3 + 9x2 +8x + 2, where x is equal to 10, and the denominator is 7x + 3. It doesn't take much to figure out that the numerator can also be written as 35x3 +36x2 + 37x + 12. Therefore,
38,982/73 = (35x3 + 36x2 +37x + 12)/(7x + 3) = 5x2 + 3x +4 = 534
This is just the algebraic equivalent of the actual method used. The algebraic principle involved in the third sutra, "vertically and crosswise," can be expressed, in one of it's applications, as the multiplication of the two numbers represented by (ax + b) and (cx + d), with the answer acx2 + x(ad + bc) + bd. Differential calculus also is utilized in the Vedic sutras for breaking down a quadratic equation on sight into two simple equations of the first degree. Many additional sutras are given which provide simple mental one or two line methods for division, squaring of numbers, determining square and cube roots, compound additions and subtractions, integrations, differentiations, and integration by partial fractions, factorisation of quadratic equations, solution of simultaneous equations, and many more. For demonstrational purposes, we have only presented simple examples. Bibliography