 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.