Vedic Maths - Series 4

Vedic Mathematics Series Part 4

In this series we shall deal with the rest of the rules/ cases involved in using the Nikhilam Navatas’caramam Dasatah sutra (formula) in case of multiplication.

As discussed in the earlier article there are 3 cases of pairs of numbers where the rules change.  In the 2nd case discussed in the earlier part are some examples that illustrate some exceptions.

Some examples (of exceptions when both the numbers are higher than the base)

[1]    111 * 125

Base is 100

111    11
125    25
--------------
 136    /   2’75
--------------
Note:  see that the RHS has exceeded the length that it can accommodate which was 2 digits hence the “2” in the 275 carries over to the LHS making the answer 13 (6+2) 75 = 13,875

Hence 111* 125 = 13875

[2]    1250 x 1005

Base is 1000

1250 x 1005

1250    250
1005    005
--------------
 1255   /   1’250
--------------

Hence 1250 x 1005 = 12, 56,250

c) When one number is more and the other is less than the base.

In this case one of the deviations becomes positive and the other negative hence the number that is obtained on the right hand side as the product of the deviations has a negative sign and needs to be subtracted.  This subtraction can complicate matters and hence

it can be done by converting it into the vinculum of the number.

To illustrate this concept let us try the multiplication of 107 x 98

1.    Find the numbers and their deviation

For 107, base = 100, deviation = 07
                                                         __
For 98, base = 100, deviation = 02

2.    Start by writing the numbers and their deviation one after the other.

107    07

98    02
--------------

3.    RHS of the answer is the product of both the deviations and it is as long as the number of zeros in the base. In this case it is 07x-02 = -14. If it were say, 1x 7 it would have been 07 etc.
      
107       07
            __
98        02
--------------
             __
       /   14
--------------

4.    L.H.S is a cross addition/subtraction operation which can be arrived at in 3 different ways

i) Cross-subtract deviation 02 on the second row with the original number 107 in the first row i.e., 107 - 02 = 105

ii) Cross–add deviation 07 on the first row with the original number 98 in the second row (converse way of (i)) i.e., 98 + 07 = 105

iii) Subtract the base 10 from the sum of the given numbers.  i.e., (107 +98) – 100 = 105,

This makes

107       07
            __
98        02
 --------------
               __
105     /   14
 --------------

This is equivalent to 10500 - 14 = 10486. This can be done faster by converting 14 into a Vinculum. Vinculum numbers are numbers which by presentation contains both positive and negative digits. More on this will be covered in the later part of the series. Here
__                       _
14 = 100-86 = 186 (the bar is on top of only 1)
This makes the subtraction an addition
10,500
          _
     186
---------
10,486
---------

This conversion helps conduct this subtraction mentally as a simple addition. Hence 107 x 98 = 10486
Similarly
              ___
 998       002
1025        025
 --------------
              ___
1023 /   050
--------------
        ___                           ___
Now 050 = 1000- 950 = 1950 
                                                 _
Hence 998 x 1025 = 1023 / 1950 = 1022950

So we have now covered how to use this formula to perform certain kinds of multiplications very fast by considering the 3 broad cases of number pairs where this method will be useful

Some tricks for objective questions

•    Since it splits the answer into two parts, RHS AND LHS, it helps us get each part pretty much independent of the other part. This can be used for eliminating answers in an objective exam by looking at just part of the options RHS or LHS

•    You also get a good idea of the number of digits of the final answers without actually doing the actual calculations which can again be used for eliminating the options.

•    Even when the numbers are not near the base, either part of the calculation for LHS (A simple addition/subtraction) or RHS (A simple multiplication) might be easy helping in eliminating some solutions. This is particularly important in DI questions where apparently the skill tested is number crunching where as actually what they are testing is the ability to eliminate and make calculated guesses by giving such tricky options.

For example for 650 * 990

1) 64350 2) 65350 3) 653500 4) 643500

Option 1 and 2 are eliminated since you know it is going to have 6 digits in the answer where as 4 can be chosen by doing just looking at 350*10
              ___
 650       350
              ___
 990    010
--------------
(No need /   3,500
      to do)
--------------

Now in the next part of the series we shall look at using the Nikhilam for division.