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This method is called, Nikhilam Sutra. Let us first take up a very easy and simple illustrative example (i.e. the multiplication of single-digit numbers above 5) and see how this can be done without previous knowledge of the higher multiplications of the multiplication-tables. Just now, we state and explain the actual procedure, step by step.
Multiplication using a base of 10.
Example 1. Multiply 9 by 8
Consider the base number as 10 since it is near to both the numbers.
Step 1.Put the two numbers 9 and 8 above and below on the left-hand side.
Step 2. Subtract each of them from the base (10) and write down the remainders (1 and 2) 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 than 10 (as shown below).
Step 3. Now, vertically multiply the two figures (1 and 2). The product is 2. And this is the right-hand-side portion of the answer.
Step 4. Now, the left-hand-side digit (or the answer) can be arrived at in one of 4 ways :--
- Cross-subtract in the converse way (i.e. 1 from 8). And you get 7 as the left-hand side portion of the required answer.
- Cross-subtract Deviation (2) on the second row from the original number (9) in the first row. And you find that you have got (9-2) i.e. 7 again.
- Subtract the sum of the two Deviations (1+2=3) from the base (10). You get the same answer (7) again ; (10-1-2)=7
- Subtract the base 10 from the sum of the given numbers (9 and 7 i.e. 17). And put (17-10) i.e. 7, as the left-hand part of the answer ;
This was easy one and you might also know the answer before start of this example. This method holds good in all cases and is, therefore, capable of infinite application.
Now, time to explain How Nikhilam Sutra Works?
This proves the correctness of the formula. The algebraically explanation for this is very simple :-
(x-a) (x-b) = x (x-a-b)+ab. Where , x is the Base, and a,b are the Deviations respectively. See Proved, below
9 x 8
Lets try another example with different case. A slight difference, however, is noticeable when the vertical multiplication of the right 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 :-
Here, we notice that the required vertical multiplication (of 3 and 4) gives us the product 12 (which consists of two 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 into 4. We thus obtain 42 as the actual product of 7 and 6.
3 /12 = 42
Try this method for solving mathematic problems and beat the clock of Competitive Exams. However, this was easy examples you might know the answers of these. But, In next lessons you will learn how to solve long and complex mathematic calculations within 20 Secs or depends upon your practice.
In the next Lesson, we will learn, How to multiply with base 100