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Multiplication By Nikhilam

 

 

Sutra : " Nikhilam Navatashcharamam Dashataha "

Meaning : All from 9 and the last from 10.

 

Upasutra : "Aanurupyena "

Meaning : Proportionately

 

        This simple sutra teaches us to do multiplication of numbers which are close to a BASE or Aadhara  such as 10, 100, 1000, 10000 etc.

 


 

Consider the example given below

            8 * 13

 

Step 1 : Since 8 and 13 are close to 10 we shall consider 10 as the base.

 

            8 - 10 gives -2  and

            13 - 10 gives +3

 

Step 2 : Write the problem as below

 

            8     -2

          13     +3

 

Step 3 : For the first part of the answer add diagonally.

                i.e. 8 + 3  Or 13 - 2    ==  11

 

        Write it as      8     -2

                             13    +3

                             11      0

( One '0' is added because the base is 101)

 

Step 4 : For the second part multiply the number in the RHS column and put it below zero and total to obtain the final answer.

 

            8     -2

            13    +3

            11     0

                   -6

           10     4

 

Hence the Answer is 104.

 


 

Example 2 :

 

Consider  945 * 1005

 

Here the base can be considered to be 1000

Therefore the problem is written as

 

            945     -055

           1005     +005  

 

By Diagonal Addition

            945 + 005 == 1005 - 055 == 950

 

            945     -055

           1005     +005

            950      000

By linear multiplication of RHS

            -55 * 5 = -275

 

            945     -055

           1005    +005

            950      000

                       -275

            949      725

 

Hence the answer is 949725


 

Aanurupyena

 

An Upasutra or Sub Formula of Nikhilam is Aanurupyena which means 'Proportionately'.

 

Here we consider multiples and sub-multiples of the Main Base as the Working Base.

 

Consider 53 * 57

Let us consider the working base to be 50, which is a sub multiple of the main base 100.

 

Let 'x' be   (Main Base) / (Working Base)

 

Here x = 100/50 = 2

Write the numbers as in normal Nikhilam

 

            53     +3

            57     +7

            60     00

(Two '0' s are added here as the Main Base is 100)

 

Now divide the first answer by x , here 2

 

            53     +3

            57     +7

            30     00

Now do the second part as in normal method

 

            53    +3

            57    +7

            30    00

                   +21

            30    21

 

Hence the Answer is 3021.


 

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