Nikhilam Division

Let’s learn about the art of performing quick divisions.

We will start with simple division of 2 digits by 9 i.e. the dividend is a 2-digit number and the divisor is 9.

13 ÷ 9 The quotient (Q) is 1, Remainder (R) is 4.

Since,

This is the usual way of performing division. Let’s learn the Vedic way of doing this:

i) Split each dividend into a left-hand part for the Quotient and right -hand part for the remainder by a slant line or slash.

E.g. 13 as 1 / 3,

34 as 3 / 4,

80 as 8 / 0.

ii) Leave some space below such representation, draw a horizontal line.

E.g.

1 / 3

______ ,

3 / 4

______ ,

8 / 0

______

iii) Most important step

Take 1st digit – ‘1’ down as it is and multiply it with the deficiency 10-9 =1 and put below the right-hand part and add them column wise. i.e., sum of the digits of the numbers is the remainder.

E.g.

Now the problem is over. i.e.,

13 ÷ 9 gives Q = 1, R = 4

34 ÷ 9 gives Q = 3, R = 7

80 ÷ 9 gives Q = 8, R = 8

The examples given so far convey that in the division of two-digit numbers by 9, we can mechanically take the first digit down for the quotient – column and that, by adding the quotient to the second digit, we can get the remainder.

But in case where there is a slight change. We add the product of the 2 digits in the quotient part with the deficiency (10 – 9 =1) and place them under the remainder column.

Now in the case of 3-digit numbers, let us proceed as follows:

104 ÷ 9 =?

212 ÷ 9 =?

212 ÷ 9 =?

401 ÷ 9 =?

Note that the remainder is the sum of the digits of the dividend part (i.e. left-hand part). The first digit of the dividend from left is added mechanically to the second digit of the dividend to obtain the second digit of the quotient. This digit added to the third digit sets the remainder. The first digit of the dividend remains as the first digit of the quotient.

Consider 511 ÷ 9

Add the first digit 5 to second digit 1 getting 5 + 1 = 6. Hence Quotient is 56. Now second digit of 56 i.e., 6 is added to third digit 1 of dividend to get the remainder

i.e., 1 + 6 = 7

Note this division rule is only applicable for division by 9. So, we will generalise this concept later on.

Extending the same principle even to bigger numbers of still more digits, we can get the results.

E.g.: 1204 ÷ 9

i) Split Dividend in 2 parts (Quotient & Remainder) in such a way Remainder to have same number of digits as that of Divisor (9). In this case its 1 digit.

ii) Take 1st digit – 1 down as it is.

iii)
Multiply the above deficiency (10 – 9=1) with the digits in the quotient part and put them below the remainder (4) part (as shown). Then add them up to obtain the remainder.

ii) Add first digit 1 to the second digit 2. 1 + 2 = 3

iii) Add the second digit of quotient 13. i.e., 3 to third digit ‘0’ and obtain the Quotient. 3 + 0 = 3, 133

In symbolic form

Another example: 132101 ÷ 9

Getting things messy? Don’t worry see below generalisation to get a hold of it.

Now consider the divisors of two or more digits whose last digit is 9, when divisor is 89.

Representing in the previous form of procedure, we have:

113 ÷ 89 =?

10015 ÷ 89 =?

But how to get these? What is the procedure?

Now Nikhilam rule comes to our rescue. The Nikhilam states “all from 9 and the last from 10”. Now if you want to find 113 ÷ 89, 10015 ÷ 89, you have to apply Nikhilam formula on 89 and get the complement 11. Further, while carrying the added numbers to the place below the next digit, we have to multiply by this 11.

In the second case:

A short Summary:

Those who can’t still figure out what’s short here? See the gist of it below.

Division by 9:

1. 103 ÷ 9 =?
   

1. 12031 ÷ 9 =?
   

1. 23 ÷ 8 =?
   

1. 12 ÷ 7 =?
   

Let’s try some bigger Divisions:

1. 111÷ 89 =?
   

1. 1234 ÷ 888 =?
   

   
   

2. 12345 ÷ 7999 =?
   

   
   

3. 210012 ÷ 8997 =?
   

   
   

4. 11111111 ÷ 99979 =?
   

Points to be remembered during Nikhilam Division:

1. Do not fail to consider 0/0’s after subtraction in initial case of Divisor. (1000-996 = 004)
   
2. Split Dividend in 2 parts (Quotient & Remainder) in such a way that the Remainder have the same number of digits as that in Divisor.
   

More posts related to Multiplication and Division in general will be discussed in the upcoming posts of Urdhava Tiryagbhyam.