Here we are talking about the sutra "Nikhilam" which would help us solve the mathematical multiplication in our mind without the use of calculator and even without having to write anything down! Let us understand the method with the help of examples.
Example 1:
605 x 504 = ?
We first need to select a base number (preferably a multiplier of 100 in case of three digit numbers to make it easier). In this case the base would be 500.
Now, define the distance of multipliers from the base as given below
605 -500 = +105 (the sign is very important)
504 -500 = +004
Now lets reproduce these numbers as:
605 + 105
504 + 004
From this, we derive two numbers and their names would be as follows:
LHS (is the result of multiplying the deficiencies) = +105 x +104 = +420
RHS (by adding diagonally on any one side) = 105 + 504 = 609 (remember, you can select any diagonal side, the answer will remain the same)
Now because our base is not a 10 series number (i.e. 10, 100, 1000, 10000, etc.) we need to establish the relation of the base with next highest 10 series number and the relation is 1000 / 2 = 500
Therefore, we have to express the LHS and RHS as follows:
(609 /2) | (+420)
304.5 + 420
304.5 | 420
Now see that the L.H.S has a decimal point ,so carry out that half to the R.H.S side from L.H.S.
Now R.H.S side becomes 420 + 500 = 920
304 | 920
Therefore ,605 x 504 = 304920
Example 2
485 x 475 = ?
The nearest base for the above two numbers are 500.So we fix this 500 as our base.Now, see that the above both numbers are less than the base.
How much deficient are are the above numbers are from base ?
485 -500 = -015
475 -500 = -025
485 - 015
475- 025
--------------
460 | ( +375) -->the RHS is got by adding diagonally on any one side ,the LHS is the result of multiplying the deficiencies .
230 | 375 ---- (Half of 460 is 230)
230 | 375
Therefore 485 x 475 = 230375
Example 3
614 x 495 =?.
The nearest base for the above two numbers are 500.So we fix this 500 as our base.Now, see that the above numbers .One is greater than the base and one is less than the base.
How much deficient are are the above numbers are from base ?
614 - 500 = +114
495 - 500 = -005
614 + 114
495 - 005
------------
609 | (-570)--->-->the RHS is got by adding diagonally on any one side ,the LHS is the result of multiplying the deficiencies .
(609 /2) | (-570)
304.5 | (-570) ----- > the minus sign is removed by 1000 -570 =430 and remove one from the L.H.S side
303.5 | 430
303| (430+500) ----> the 0.5 when carried to R.H.S, becomes 500 and add it to R.H.S
303| 930
Therefore ,614 x 495= 303930
These are general rules. There would not be issues when multiplications are above or higher, but when one is greater and one is lesser, follow the rules in the last example. All in all, the rules are:
The (0.5) on L.H.S is carried as 500 and added to the R.H.S every time you fall into a decimal (1/2).
The Minus signs on R.H.S is removed by adding 1000 to it.
It would seem very easy once you get in this mode of calculations.
Example 1:
605 x 504 = ?
We first need to select a base number (preferably a multiplier of 100 in case of three digit numbers to make it easier). In this case the base would be 500.
Now, define the distance of multipliers from the base as given below
605 -500 = +105 (the sign is very important)
504 -500 = +004
Now lets reproduce these numbers as:
605 + 105
504 + 004
From this, we derive two numbers and their names would be as follows:
LHS (is the result of multiplying the deficiencies) = +105 x +104 = +420
RHS (by adding diagonally on any one side) = 105 + 504 = 609 (remember, you can select any diagonal side, the answer will remain the same)
Now because our base is not a 10 series number (i.e. 10, 100, 1000, 10000, etc.) we need to establish the relation of the base with next highest 10 series number and the relation is 1000 / 2 = 500
Therefore, we have to express the LHS and RHS as follows:
(609 /2) | (+420)
304.5 + 420
304.5 | 420
Now see that the L.H.S has a decimal point ,so carry out that half to the R.H.S side from L.H.S.
Now R.H.S side becomes 420 + 500 = 920
304 | 920
Therefore ,605 x 504 = 304920
Example 2
485 x 475 = ?
The nearest base for the above two numbers are 500.So we fix this 500 as our base.Now, see that the above both numbers are less than the base.
How much deficient are are the above numbers are from base ?
485 -500 = -015
475 -500 = -025
485 - 015
475- 025
--------------
460 | ( +375) -->the RHS is got by adding diagonally on any one side ,the LHS is the result of multiplying the deficiencies .
230 | 375 ---- (Half of 460 is 230)
230 | 375
Therefore 485 x 475 = 230375
Example 3
614 x 495 =?.
The nearest base for the above two numbers are 500.So we fix this 500 as our base.Now, see that the above numbers .One is greater than the base and one is less than the base.
How much deficient are are the above numbers are from base ?
614 - 500 = +114
495 - 500 = -005
614 + 114
495 - 005
------------
609 | (-570)--->-->the RHS is got by adding diagonally on any one side ,the LHS is the result of multiplying the deficiencies .
(609 /2) | (-570)
304.5 | (-570) ----- > the minus sign is removed by 1000 -570 =430 and remove one from the L.H.S side
303.5 | 430
303| (430+500) ----> the 0.5 when carried to R.H.S, becomes 500 and add it to R.H.S
303| 930
Therefore ,614 x 495= 303930
These are general rules. There would not be issues when multiplications are above or higher, but when one is greater and one is lesser, follow the rules in the last example. All in all, the rules are:
The (0.5) on L.H.S is carried as 500 and added to the R.H.S every time you fall into a decimal (1/2).
The Minus signs on R.H.S is removed by adding 1000 to it.
It would seem very easy once you get in this mode of calculations.
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