NIKHILAM NAVATAS’CHARAMAM DASATAH

The formula simply means : “all from 9 and the last from 10”

The formula can be very effectively applied in multiplication of numbers, which are nearer to bases like 10, 100, 1000 i.e., to the powers of 10(eg: 96 x 98 or 102 x 104). The procedure of multiplication using the Nikhilam involves minimum number of steps, space, time saving and only mental calculation. The numbers taken can be either less or more than the base considered.

Case (i) : when both the numbers are lower than the base.
Find 97 X 94. Here base is 100. Now following the rules, the working is as follows: 97 is 3 less than the nearest base 100. and 94 is 6 less than the same nearest base 100. Hence 3 and 6 are called deviations from the base. Always the base should be same for the two numbers.
=91/18=9118
Note: Here '/' signifies just a seperation and has nothing to do with division.
In genreal, let N1 and N2 be two numbers near to a given base in powers of 10, and D1 and D2 are their respective deviations from the base. Then N1 X N2 can be represented as:

Case ( ii) : When both the numbers are higher than the base.
The method and rules follow as they are. The only difference is the positive deviation. Instead of cross – subtract, we follow cross – add.104 X 102. Base is 100.

Note: We are considering 04x02=08 and appending '08' and not just '4x2=8'. This is done because, we need to consider two digits in deviation as it the base 100 has two zeros. If the deviation is near 1000 then we need to consider 3 digits in the deviation (eg, 004 and not just 4).

Case ( iii ): One number is more and the other is less than the base.
In this situation one deviation is positive and the other is negative. So the product of deviations becomes negative. So the right hand side of the answer obtained will therefore have to be subtracted. To have a clear representation and understanding a vinculum is used. It proceeds into normalization. 13 X 7. Base is 10

10/(-9)=91 how?

10/(-9) should be read as 'one zero, nine bar'. Here 'one' and 'zero' are in normal form. 'nine' is in complement form. so, when we bring a carry from normal form to complment form, '10' becomes '9' and '9-bar' becomes '1'( 10's complement of 9). Hence 10/(-9)-91.

Another example would be 94/(-3)=93/7=937.

Find the following products by Nikhilam formula.
1) 7 X 4           2) 93 X 85           3) 875 X 994           4) 1234 X 1002


5) 1003 X 997           6) 11112 X 9998           7) 1234 X 1002           8) 118 X 105