DIVISION
In the conventional procedure for division, the process is of the following
form.
Quotient
_______
Divisor ) Dividend
or Divisor ) Dividend ( Quotient
----------
----------
----------
----------
_________
_________
Remainder
Remainder
But in the Vedic process, the format is
Divisor ) Dividend
--------
--------
__________________
Quotient / Remainder
The conventional method is always the same irrespective of the divisor. But Vedic methods are different depending on the nature of the divisor.
Example 1: Consider the division 1235 ÷ 89.
i) Conventional method:
89 ) 1235 ( 13
89
_____
345
267 Thus Q = 13 and R = 78.
_____
78
ii) Nikhilam method:
This method is useful when the divisor is nearer and less than the base. Since for 89, the base is 100 we can apply the method. Let us recall the nikhilam division already dealt.
Step (i):
Write the dividend and divisor as in the conventional method. Obtain the modified divisor (M.D.) applying the Nikhilam formula. Write M.D. just below the actual divisor.
Thus for the divisor 89, the M.D. obtained by using Nikhilam is 11 in the last
from 10 and the rest from 9. Now Step 1 gives
89 ) 1235
__
11
Step (ii):
Bifurcate the dividend by by a slash so that R.H.S of dividend
contains the number of digits equal to that of M.D. Here M.D. contains 2 digits
hence
89 ) 12 / 35
__
11
Step (iii): Multiply the M.D. with first column digit of the dividend. Here it is
1. i.e. 11 x 1 = 11. Write this product place wise under the 2nd and 3rd columns
of the dividend.
89 ) 12 / 35
__
11 1 1
Step (iv):
Add the digits in the 2nd column and multiply the M.D. with that
result i.e. 2+1=3 and 11x3=33. Write the digits of this result column wise
as shown below, under 3rd and 4th columns. i.e.
89 ) 12 / 35
__
11 1 1
33
_______
13 / 78
Now the division process is complete, giving Q = 13 and R = 78.
Example 2: Find Q and R for 121134 ÷ 8988.
Steps (1+2):
8988 ) 12 / 1134
____
1012
Step (3):
8988 ) 12 / 1134
____
1012 1 012
Step(4):
8988 ) 12 / 1134
____
1012 1 012 [ 2
+ 1 = 3 and 3x1012 = 3036 ]
3036
Now final Step
8988 ) 12 / 1134
____
1012 1 012
3036(Column wise addition)
_________
13 / 4290
Thus 121134¸ 8988 gives Q = 13 and R = 4290.
iii) Paravartya method: Recall that this method is suitable when the divisor is nearer but more than the base.
Example 3: 32894 ÷ 1028.
The divisor has 4 digits. So the last 3 digits of the
dividend are set apart for the remainder and the procedure follows.
Now the remainder contains -19, -12 i.e. negative quantities. Observe that 32 is quotient. Take 1 over from the quotient column i.e. 1x1028 = 1028 over to the right side and proceed thus: 32 - 1 = 31 becomes the Q and R = 1028+200 - 190 - 12 =1028-2 =1026.
Thus 3289 ÷ 1028 gives Q = 31 and R = 1026.
The same problem can be presented or thought of in any one of the following
forms.
_
*Converting the divisor 1028 into vinculum number we get 1028 = 1032 Now
__
*Converting dividend into vinculum number 32894 = 33114 and proceeding we get
Now we take another process of division based on the combination of Vedic sutras urdhva-tiryak and Dhvjanka. The word Dhvjanka means " on the top of the flag"
Example 4: 43852 ÷ 54.
Step1: Put down the first digit (5) of the divisor (54) in the divisor column as
operator and the other digit (4) as flag digit. Separate the dividend into two
parts where the right part has one digit. This is because the falg digit is
single digit. The representation is as follows.
4 : 4 3 8 5 : 2
5
Step2: i) Divide 43 by the operator 5. Now Q= 8 and R = 3. Write this Q=8 as the
1st Quotient - digit and prefix R=3, before the next digit i.e. 8 of the
dividend, as shown below. Now 38 becomes the gross-dividend ( G.D. ) for the
next step.
4 : 4 3 8 5 : 2
5 : 3
________________
: 8
ii) Subtract the product of falg digit (4) and first quotient digit (8) from the G.D. (38) i.e. 38-(4X8)=38-32=6. This is the net - dividend (N.D) for the next step.
Step3: Now N.D Operator gives Q and R as follows. 6 ÷ 5, Q = 1, R = 1. So Q = 1,
the second quotient-digit and R - 1, the prefix for the next digit (5) of the
dividend.
4 : 4 3 8 5 : 2
5 : 3 1
________________
: 8 1
Step4: Now G.D = 15; product of flag-digit (4) and 2nd quotient - digit (1) is 4X1=4 Hence N.D=15-4=11 divide N.D by 5 to get 11 ÷ 5, Q = 2, R= 1. The
representation is
4 : 4 3 8 5 : 2
5
: 3 1 :1
________________
: 8 1 2 :
Step5: Now the R.H.S part has to be considered. The final remainder is obtained by subtracting the product of falg-digit (4)and third quotient digit (2) form 12 i.e., 12:
Final remainder = 12 - (4 X 2) = 12 - 8 = 4. Thus the division ends into
4 : 4 3 8 5 : 2
5
: 3 1 :1
________________
: 8 1 2 : 4
Thus 43852 ÷ 54 gives Q = 812 and R = 4.
Consider the algebraic proof for the above problem. The divisor 54 can be represented by 5x+4, where x=10
The dividend 43852 can be written algebraically as 43x3 + 8x2 + 5x + 2
since x3 = 103 = 1000, x2 = 102 = 100.
Now the division is as follows.
5x + 4 ) 43x3
+ 8x2 + 5x + 2 ( 8x2 + x + 2
43x3+ 32x2
_________________
3x3 – 24x2
= 6x2 + 5x (
3x3 = 3 . x . x2
5x2 + 4x
= 3 . 10x2 = 30 x2)
_________________
x2 + x
= 11x + 2 (
x2 = x . x = 10x )
10x + 8
__________________
x – 6
= 10 – 6
= 4.
Observe the following steps:
1. 43x3 ÷ 5x gives first quotient term 8x2 , remainder = 3x3 - 24x2 which really mean 30x2 + 8x2 - 32x2 = 6x2.
Thus in step 2 of the problem 43852 ÷ 54, we get Q= 8 and N.D = 6.
2. 6x2 ÷ 5x gives second quotient term x, remainder = x2 + x which really mean 10x + x = 11x.
Thus in step 3 & Step 4, we get Q=1and N.D =11.
3. 11x ÷ 5x gives third quotient term 2, remainder = x - 6 , which really mean the final remainder 10-6=4.
Example 5: Divide 237963 ÷ 524
Step1: We take the divisor 524 as 5, the operator and 24, the flag-digit and proceed as in the above example. We now seperate the dividend into two parts where the RHS part contains two digits for Remainder.
Thus
24 : 2 3 7 9
: 63
5
Step2:
i) 23÷5 gives Q = 4 and R = 3, G.D = 37.
ii) N.D is obtained as
= 37 – ( 8 + 0)
= 29.
Representation
24 : 2 3 7 9 : 63
5 3
_________________
: 4
Step3:
i) N.D ÷ Operator = 29 ÷ 5 gives Q = 5, R = 4 and G.D = 49.
ii) N.D is obtained as
= 49 – (10 + 16)
= 49 – 26
= 23.
i.e.,
24 : 2 3 7 9 : 63
5 : 3 4 :
_________________
: 4 5 :
Step 4:
i) N.D ÷ Operator = 23 ÷ 5 gives Q = 4, R = 3 and G.D = 363.
Note that we have reached the remainder part thus 363 is total sub–remainder.
24 : 2 3 7 9 : 63
5 : 3 4 :3
_________________
: 4 5 4 :
Step 5: We find the final remainder as follows. Subtract the cross-product of the two, falg-digits and two last quotient-digits and then vertical product of last flag-digit with last quotient-digit from the total sub-remainder.
i.e.,,
Note that 2, 4 are two falg digits: 5, 4 are two last quotient digits:
represents the last flag - digit and last quotient digit.
Thus the division 237963 ÷ 524 gives Q = 454 and R = 67.
Thus the Vedic process of division which is also called as Straight division is a simple application of urdhva-tiryak together with dhvajanka. This process has many uses along with the one-line presentation of the answer.