We now pass on to a systematic exposition of certain salient , interesting , important and necessary formulae of the utmost value and utility in connection with arithmetical calculations beginning with the processes and methods described in the Vedic mathematics Sutras.
The Sutras reads:- Nikiliam Navastascaramam Dasatah, which , literally translated, means :"all from 9 and the last from 10" ! we shall give a detailed explanation, presently, of the meaning and applications of this cryptical - sounding formula. But just now, we state and explain the actual procedure step by step by step.
Suppose we have to multiply 9 by 7
1.) We should take, as base for our calculations, that the power of 10 which is nearest to the numbers to be multiplied. In this case 10 itself is that power:
2.) Put the numbers 9 and 7 above and below on the left -hand side(as shown in the working alongside here on the right-hand side margin);
3.) Subtract each of them from the base (10) and write down the remainders (1 and 3) on the right - hand side with a connecting minus sign (-) between them, to show that the numbers to be multiplied are both of them less 10.
4.) The product will have two parts , one on the left side and one on the right. A vertical dividing line may be drawn for the purpose of demarcation of two parts
5.) Now, the left- hand side digit (of the answer) can be arrived at in one of 4 ways:-
(a) Subtract the base 10 from the sum of the given numbers (9 and 7,i.e;16) and put(16-10) i.e 6; as the left- hand part of the answer.
or (b) Subtract the sum of two deficiencies (1+3) from the base (10). You get the same answer (6) again.
or (c) Cross- subtract deficiency 3 on the second row from the original numbers 9 in the first row. And you find that you have got (9-3) , i.e ; 6 again.
or (d) Cross- subtract in the converse way (.i.e; 1 from 7). And you get 6 again as the left-hand side portion of the required answer
6.) Now ,vertically multiply the two deficit figure (1 and 3). The product is 3. And this is the right-hand side portion of the answer.
7.) Thus
This method holds good in all cases and is, therefore, capable of infinite application. in fact, old historical traditions describe this cross- subtraction process as having been responsible for the acceptance of the 
marks as the sign of multiplication.
marks as the sign of multiplication.As further illustrations of same rule, note the following example:
This proves the correctness of the formula. The algebraical explanation for this is very simple:
(x-a)(x-b) = x (x-a-b) + ab
A slight difference, however, is noticeable when the vertical multiplication of the deficit digits ( for obtaining the right-hand side portion of the answer) yields a product consisting of more than one digit . For example, if and when we have to multiply 6 by 7 and write it down as usual:
We notice that the required vertical multiplication ( of 3 and 4 ) gives us the product 12 (which consists of 2 digits ; but, as our base is 10 and the right - hand - most digit is obviously of units, we are entitled only to one digit (on the right- hand side).
This difficulty, however, is easily surmounted with the usual multiplication rule that the surplus portion on the left should always be "carried" over to the left. Therefore , in the present case, we keep the 2 of the 12 on the right- hand side and '' carry" the 1 over to the left and change the 3 and 4. We thus obtain
42 as the the actual product of 7 and 6.
A similar procedure will naturally be required in respect of others similar multiplication:-
The rule of multiplication by means of cross-subtraction for the left hand portion and of vertical multiplication for the right- hand half being an actual application of the absolute identity
for more:-
