Vedic Maths - Series 3

Vedic Mathematics Series Part 3

We shall now move on to the next formula Nikhilam Navatas’caramam Dasatah which means “all from 9 and the last from 10”. This is a very powerful formula which can be used primarily for multiplying numbers near multiples of 10 like 10, 100, 1000 etc and the method can be extended to solve other calculations as well as will be shown. We also can do division using the same, which will also be taken up later in the series. In MBA exam parlance it can be called very efficient since it achieves the twin purpose of time and space saving which are of upmost importance here.

For multiplication of numbers closer to powers of 10

There are two steps involved in this technique. Initially we will need to write it down but soon enough we should be able to do mentally. In the initial stage we need to convert both the numbers into a representation that is conductive to the technique. For this we need to write the number and then its deviation from the base as show below.



Some rules for the deviation

•    The deviation can be found for numbers less than the base by Nikhilam formula by subtracting each digit from 9 and last digit form 10. For numbers which are greater than the base, It is simply the last n-1 digits

•    Sign of the deviation should reflect whether the number is bigger or smaller than the base

•    The number of digits in the deviation should be equal to the power of 10 in the base (number of 0’s in the base). If they are not sufficient, 0’s need to be padded.

•    For both the numbers the bases need to be same for the applying this technique.

Now for multiplication there are 3 separate cases that must be considered which we shall illustrate one by one explaining the steps with examples.

a) When both the numbers are less than the base

To illustrate this concept let us try the multiplication of 97 x 94

1.    Find the numbers and their deviation
                                                                     __
For 97, base = 100, deviation = -03 or 03
                                                                     __
For 94, base = 100, deviation = - 06 or 06

2.    Start by writing the numbers and their deviation one after the other.
        __
97    03
        __
94    06
--------------

3.    Now the answer will have 2 parts LHS and RHS. They need to be separately found out. A ‘/’ sign  may be used to separate the two parts
        __
97    03
        __
94    06
--------------
      /
--------------

4.    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 3x6 = 18. If it were say, 1x 7 it would have been 07 etc. (zero padding to match the length)
        __
97    03
        __
94    06
--------------
  /   18
--------------

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

i) Cross-subtract deviation 6 on the second row from the original number 97 in the first row i.e., 97-6 = 91.

ii) Cross–subtract deviation 3 on the first row from the original number 94 in the second row (converse way of (i)) i.e., 94 - 3 = 91

iii) Subtract the base 10 from the sum of the given numbers.  i.e., (97 +94) – 100 = 91
         __
97    03
        __
94    06
--------------
 91    /   18
--------------

The answer is hence 91/18 which is 9118.

Some examples

[1]    996 X 989

Base is 1000
          ___
996    004
          ___
989    011
--------------
 985    /   044
--------------

Hence 996 X 989 = 9, 85,044

[2]    800 X 997

Base is 1000
          ___
800    200
          ___
997    003
--------------
 797    / 600
--------------

800 X 997 = 7, 97,600.

An important thing to note here is that one of the numbers is not really very near the base but still we have applied the formula since the ensuing calculations are simple. Hence the formula is applicable even for numbers not close to the base and it’s the ease of the calculations involved that decide whether to apply the formula or not

b) Both the numbers are higher than the base.

To illustrate this case let us try the multiplication of 107 x 104

1.    Find the numbers and their deviation

For 107, base = 100, deviation = 07

For 104, base = 100, deviation = 04

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

107    07
104    04
--------------
      /
-------------
3.    As before the answer will have 2 parts LHS and RHS and they need to be separately found out. A ‘/’ sign  may be used to separate the two parts

107    07
104    04
--------------
      /
-------------

4.    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 07x04 = 28. If it were say, 1x 7 it would have been 07 etc.

107    07
104    04
--------------
  /   28
--------------

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

i) Cross-add deviation 04 on the second row with the original number 107 in the first row i.e., 107 + 04 = 111.

ii) Cross–add deviation 07 on the first row with the original number 104 in the second row (converse way of (i)) i.e., 104 - 07 = 111

iii) Subtract the base 10 from the sum of the given numbers.  i.e., (107 +104) – 100 = 111

107    07
104    04
--------------
 111    /   28
--------------

The answer is hence 111/28 which is 11128.

The some examples illustrating some exceptional cases, the 3rd case of set of numbers and finally some effective ways to use the method for objective questions will be illustrated in the next article in the series.