[nukkad] Vedic maths - 2

["Hemendra Bhanushali" <>]

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Tip of the day:  Don't worry about the world coming to an end today.
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(A)	We shall start with a method to multiply 2 numbers. It will reveal to
you how innovative Vedic mathematics can be.

Lets deal, to begin with, with numbers that are close to POWERS of 10 i.e.
numbers near to 10, 100, 1000, etc.

Example 1:

Say, we have to multiply 98 and 97.
(First, try using conventional method and find out how much time it takes!)

Now we go for the Vedic method.
Write down the numbers in this manner:

98	-2
97	-3


In the above representation, -2 and –3 stand for the difference of 98 and 97
from 100.
We may call these differences –2 and –3 as OFFSETS of 98 and 97 from 100. We
may call 100 as the ‘BASE’ as differences are taken from 100.

Step 1) First, we multiply the offsets –2 and –3
   	We get 6.
Since our base is 100, which has 2 zeros, the product of offsets must also
have 2 digits. Hence we write 6 as 06 and write it down as last 2 digits of
our answer.

      98       -2
      97       –3
-------------------------
               06

Step 2) Now for the previous digits, just add any 2 numbers in the above
figure     crosswise i.e. either (98-3) or (97-2)
	We get 95. This is written before 06 as follows: -

      98       -2
      97       –3
---------------------------
      95   /   06

That gives us 9506. That’s our answer.
Hence 98 x 97  = 9506

Aint it simple!!
Example 2:
Once again, we solve 25 x 98.

     25	  -75
     98        -2
---------------------------
Here, once again, our base is 100. -75 and –2 are offsets of 25 and 98 from
100.

Step 1)	-75  x  –2 gives 150

Since our base (100) has 2 zeros, the product of offsets must have 2 digits.
Hence we write down last 2 digits of 150 i.e. 50 as last 2 digits of our
answer.
1 would be treated as carry.


     25	  -75
     98       -2
---------------------------
              50            ( 1 carry )

Step 2) Add the numbers crosswise. i.e. ( 25-2 ) or ( 98 – 75 ).
        We get 23.
        Add the carry from previous step.
        We get 23 + 1 = 24
        These are written as first 2 digits of our answer.


     25	  -75
     98       -2
---------------------------
    (23+1)  /  50




i.e.
     25	  -75
     98       -2
---------------------------
     24   /  50

 Answer:  25 x 98 = 2450

Example 3:
Now, lets try for 108 x 109

Let us write down the numbers as:

108	8
109	9
---------------------------
and carry out the same method as above.

Note that 8 and 9 are offsets of 108 and 109 from base 100.

Step 1) Product of offsets 8 x 9 = 72
        Write down 72 as last digits of our answer.

108	8
109	9
--------------------------
      72
Step 2) Add any 2 numbers cross wise  ( 108 + 9 ) or ( 109 + 8 )
        We get 108 + 9 = 117
        These are the front digits of the answer.

108       8
109	    9
---------------------------
117	/  72

Hence 108 x 109 = 11772





Example 4:

Lets try 98 x 104.

98 	-2
104	4
---------------------------

Step 1 ) Product of offsets –2 x 4  = -8
	  We write it down as,
98 	   -2
104       4
---------------------------
         -8

Step 2 ) Now, 98 + 4 or 104 –2 gives 102.
These are the earlier digits of our answer.

98       -2
104       4
---------------------------
102	/  -8

Step 3) Since the number after “/ ” is as negative number we have to
decrease the number to the left of “/” by 1. ( i.e. 102 becomes 101 )
	Next, the number 8 (which has a negative sign) has to be deducted from the
number base times 1 i.e. deduct 8 from 100 x 1.
	i.e. 100 – 8 = 92
	92 is written in the place of –8. That makes our result as: -

98       -2
104       4
--------------------------
101	  /   92

Hence 104  x  98  = 10192.

Lets see how it simplifies our calculation for huge numbers:





Example 5:

Multiply 888 by 998: -


888	-112
998	-002
---------------------------
886    /    224

Here, the base is 1000 i.e. offsets are differences from 1000

-112  x  – 002   =    224 ( Simple enough )

And	 888  –  002     =   886

Hence, 888  x  998  =  886224.
Example 6:

Multiply 9997 by 9998:-


9997	-3	( base 10,000 )
9998	-2
------------------------------
 9997 – 2 )   /  0006

( Here, base is 10,000, which has 4 zeros. Hence the product of offsets –2
and –3 is padded with zeros to make it a 4-digit number )

Hence,

9997	-3
	9998	-2
------------------------------
         9995   /  0006                 =       99950006  (Answer!!!)

Example 7:

Multiply 999979 by 999998: -

999979	-21	(base 1,00,000)
999998	-2
--------------------------------
          ( 999979 – 2 )  /  000042

Here, -21 x –2  = 42.
Base has 6 zeros. Hence pad 42 with 4 zeros to make 000042.
Then, 999979 – 2 = 999977. Those are the earlier digits of our answer.

999979	-21
999998	-2
------------------------------
999977    /   000042

Hence,   999979  x  999998 =  999977000042.

Ever wondered multiplying such big numbers could be that simple!!!

The methods discussed above are extremely effective for multiplying numbers
near the powers of 10. Similar methods can be applied to find the product of
ANY 2 numbers (irrespective of whether they are near powers of 10) using ANY
MULTIPLE of 10 as base. They have a slight variation, though, which shall be
discussed in the forthcoming article.

There are other interesting methods of multiplication too, which shall be
discussed in the next mail.




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