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.