Multiplication-Nikhilam method

Multiplication is something which is an essential calculation in all academic subjects and of course in our day-to-day life. Let us see how Vedic Mathematics handles and simplifies the same.

We will start with Nikhilam Method, if you remember, we have used it in “square of number” topic also. It says “all from 9 and last from 10”. This is applied to multiply 2 numbers which are near to the same decimal bases (10,100,1000 etc). The method is –

1. find the deviation of the given numbers from their bases.
2. Multiply the deviation of the 2 given numbers.
3. Add or subtract the deviations (based on whether the number is more or less than the base) from the base.

Let us see with examples to have better understanding, starting with single digit and then move on to higher digits.

8x6

1. 8 is 2 less than 10. 6 is 4 less than 10. So deviations are 2 and 4.
2. Multiply 2 and 4, 2x4=8                 
3. Subtract 2 and 4 from 10. 10-2-4=4

8x6,    Deviations  = 2 and 4
Base+/- deviations	Product of deviations
100-2-4=4	2x4=8
4	8

So the answer is 48.

14x16

I have taken this example particularly to show how to handle a situation if the product of deviations is more than 1 digit (since the base is 10, we can take only one digit from product, anything more is taken as carry to be added to the higher digits)

1. 14 is 4 more than 10. 16 is 6 more than 10. So deviations are 4 and 6.
2. Multiply 4 and 6.                 
3. Add 4 and 6 to 10.

14x16   , Deviations  = 4 and 6
Base+/- deviations	Product of deviations
10+4+6=20 Add the carry 2 from the product.	4x6=24 4 goes to answer and 2 goes as carry.
22	4

So the answer is 224.

Now, what if one number is less and one is more than the reference base. We will see an example for this case.

12x7

12x7  , Deviations  = 2 and 3
Base+/- deviations	Product of deviations
10+2-3=9 Subtract 1 as a borrow given to -6 to get its complement.	2x-3=-6 (positive xnegative is negative) Complement of -6 is 4.
8	4

So the answer is 84

This method in the above calculation may look tricky, but when we apply the same steps to bigger numbers, it will be well appreciated. We will take few examples in 2 and 3 digit numbers.

92x98

We will use the same steps to get the result.

1. Deviations from 100 are 8 and 2
2. Multiply 8 and 2 =16                            
3. Subtract 8 and 2 from 100, 100-8-2=90

92x98, Deviations  = 8 and 2
Base+/- deviations	Product of deviations
100-8-2=90	8x2=16 Since the base is 100, both the digits are taken, no carry is forwarded.
90	16

The answer is 9016

112 x113

1. Deviations from 100 are 12 and 13

2. Multiply 12 and 13.=156
3. Add 12 and 13 to 100, 100+12+13=125

Now, if multiplying 12 and 13 is difficult to do mentally, we can apply nikhilam method again, deviations are 2 and 3, so 2x3=6, 10+2+3 = 15, answer is 156.

112x113, Deviations  = 12 and 13
Base+/- deviations	Product of deviations
100+12+13=125 125+1=126 (add carry)	12x13=156 Carry 1 is forwarded
126	56

The answer is 12656

107x94

1. Deviations from 100 are 7 and 6

2. Multiply 7 and 6 = 42
3. Add 7 and 6 to 100, 100+13=113

107x94, Deviations  = 7 and 6
Base+/- deviations	Product of deviations
100+7-6=101 101-1=100 (subtract 1 to give borrow for complementing)	7x-6=-42 Complement of 42 is 58, as 42 being a negative number
100	58

The answer is 10058

850x995

Let us see some 3 digit numbers.

1. Deviations from 1000 are 150 and 5

2. Multiply 150 and 5 = 750
3. Add 150 and 5 to 1000, 1000+150+5=1155

850x995, Deviations  = 150 and 5
Base+/- deviations	Product of deviations
1000-150-5=845	150x5=750 No carry as base is 1000
845	750

The answer is 845750

1055x1009

1. Deviations from 1000 are 55 and 9

2. Multiply 55 and 9 = 495
3. Add 55 and 9 to 1000, 1000+55+9=1064

1055x1009, Deviations  = 55 and 9
Base+/- deviations	Product of deviations
100+55+9=1064	55x9=495 No carry as base is 1000
1064	495

The answer is 1064495

997x1025

1. Deviations from 1000 are 3 and 25

2. Multiply 3 and 25 = 75
3. Add -3 and 25 to 1000, 1000+25-3=1022.

997x1025, Deviations  = -3 and 25
Base+/- deviations	Product of deviations
1000+25-3=1022 Subtract 1 to borrow for complementing	25x-3=-75 Complement os 75 for base 1000 is 925 (1000-75)
1021	925

The answer is 1021925

This can be extended to any decimal bases, and remember the reference base under consideration to take carry or complement.  The power of VM would have become apparent by now. We will see how we can do the multiplication of any 2 numbers in a different section.