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सूत्र ४. परावर्त्य योजयेत्

(Sūtra 4. parāvartya yojayet) - Transpose and adjust.
 

The Sūtra: parāvartya yojayet is use for division (x ÷ y) of two numbers, especially when a remainder is required. This technique, also sometimes referred as Base-Division, is a generic method that supercedes the techniques for division used by Sūtra: nikhilaṃ navataścaramaṃ daśataḥ and Sūtra: ūrdhva tiryagbhyāṃ, which is the reason that the technique is not discussed in the afore-mentioned Sūtras.
 
As an illustration, let us use this Sūtra for:
97 ÷ 12
  Steps 97 ÷ 12
1. Find a Base, that is near the Divisor - that is a power of 10, that is 10n. In this case, the close power of 10 that is nearest to 12 is 10, which is the Base.
2. Find the Transposition of the Divisor - as (Base - Divisor), and represent as Vinculum, if required. Make sure that the Transposition has the same number of digits, as the Zeroes of the Base, 10n In this case, the transposition is:
10 - 12 = -2 = 2
3. Arrange the numbers in a manner, that the Divisor is slightly raised, from its Transposition, and the Dividend on the right-hand side. In this case,
12 2 ) 97
4. Leave some space between the digits of the Dividend, and partition it into 2 parts:
1st Part is for Quotient, and the 2nd Part is for Remainder, such that the digits for the Remainder should be same - as the Zeroes of the Base, 10n.
In this case,
12 2 ) 9  |  7
          |
     ---------
5. Drop the first digit (or the cummulative sum of any other number in the same column) of the Dividend, and multiply it (the 'Dropped' digit)with each digits of the Transposition - placing them in rows, next to the place of the 'Dropped' digit. In this case, the first digit is dropped:
12 2 ) 9  |  7
          |
     ---------
       9  |
And, Multiplying the dropped digit, with each digits of the Transposition:
12 2 ) 9  |  7
          | 18
     ---------
       9  |
 
Note: 9 × 2 = 9 × -2 = -18 = 18
5. Drop the next digit (or the cummulative sum of any other number in the same column) of the Dividend, and repeat above step (Step 4) till all the places for every digit of the Dividend is exhausted. In this case, the next digit is dropped:
12 2 ) 9  |  7
          | 18
     ---------
       9  | 11
All places for the digits in the Dividend is exhausted, so we stop.
 
Note: 7 + (-18) = -11 = 11
6. Finalize the Quotient and Remainder parts as:
1. Remove any viniculum number from Quotient;
2. Remove any viniculum number from Remainder;
3. If the Remainder is negative, take appropriate numbers of Quotient to make it positive;
4. If the Remainder is not less than the Dividor, give appropriate numbers to Quotient to make it lesser.
In this case,
The Quotient part contains: 9
And, the Remainder part contains: 11 = -11
 
The Remainder is negative. So, we take one from the Quotient: 9 - 1 = 8
And give it to Remainder: 1 × 12 - 11 = 1
 
So, we get: Quotient = 8, and Remainder = 1, which is the answer!
Note that, the process uses Rekhanks & Vinculum Numbers (discussed here »), and when 'giving' to Remainder, from Quotient: 'a' from Quotient goes as (a × Divisor) to the Remainder - in accordance to the concept of Division. So, for a practitioner of Vedic Mathematics, for something like:
10406 ÷ 1212
  The Base is 1000 and, 1000 - 1212 = -212 = 212
So, we arrange as:
1212 212 ) 1   0   |   4   0   6
                   |
                   |
          ----------------------
                   |
 
Step 1 - First Digit:
After dropping the first digit, and multiplying it with all the digits of the Transposition, we get:
1212 212 ) 1   0   |   4   0   6
               2   |   1   2
                   |
          ----------------------
           1       |
Note above, that 1 × 2 = 2, 1 × 1 = 1, and 1 × 2 = 2. And, it is arranged in next row, leaving the digit that is dropped.
 
Step 2 - Next digit:
We drop 0 + 2 = 0 + (-2) = -2 = 2, and multiplying it with all the digits of the Transposition, we get:
1212 212 ) 1   0   |   4   0   6
               2   |   1   2
                   |   4   2   4
          ----------------------
           1   2   |
Note above, 2 × 2 = 4, 2 × 1 = 2, and 2 × 2 = 4. And, it is arranged in next row, leaving the digit that is dropped.
 
Step 3 - Next digit:
All the places of the Dividend is exhausted, so we stop - after totalling the columns:
1212 212 ) 1   0   |   4   0   6
               2   |   1   2
                   |   4   2   4
          ----------------------
           1   2   |   7   0  10
Note above, (4 + 1 + 4) = 4 - 1 + 4 = 7, (0 + 2 + 2) = 0 - 2 + 2 = 0, and (6 + 4) = 10
 
So, we get:
Quotient = [1,2] = 10 - 2 = 8
And, Remainder = [7,0,10] = 710
Since the Remainder is positive and less than the Quotient, no further finalizing is required.
 
Thus, 10406 ÷ 1212 = 8, and Remainder: 710
Again, for something like:
131690 ÷ 1013
  The Base is 1000 and, 1000 - 1013 = -13 = 013
(The number of digits of the Transposition, should be the number of Zeroes of the Base)
 
Working ahead, we get:
1013 013 ) 1   3   1   |   6   9   0
               0   1   |   3
                   0   |   3   9
                       |   0   0   0
          --------------------------
           1   3   0   |   0   0   0
 
So, we get:
Quotient = [1,3,0] = 130
And, Remainder = [0,0,0] = 0
Since the Remainder is Zero, 131690 is divisible by 1013.
 
Thus, 131690 ÷ 1013 = 130
Note that, for all the above examples, the Divisor is greater than the Base. For a Divisor, which is less than the Base, the method used is also commonly known as the 'Nikhilam Method of Division' - but it is exactly the same as this Sūtra's method, and thus, discussed here. For something like: 28237 ÷ 98
  The Base is 100 and, 100 - 98 = 2 = 02
(The number of digits of the Transposition, should be the number of Zeroes of the Base)
 
Working ahead, we get:
98 02 ) 2   8   2   |   3   7
            0   4   |
                0   |  16
                    |   0  12
       ----------------------
        2   8   6   |  19  19
 
So, we get:
Quotient = [2,8,6] = 286
And, Remainder = [19,19] = 209
 
But 209 is greater than 98.
If we give one to Quotient, the Remainder becomes 209 - 98 = 111
This is still greater than 98, so we give another to Quotient, and the Remainder becomes 111 - 98 = 13
Now, we have our final Quotient = 286 + 2 = 288, and Remainder = 13
 
Thus, 28237 ÷ 98 = 288, and Remainder: 13
 
Note that, not even a single change needs to be incorporated for a number less than the Base.
Also, this Sūtra may also be used to derive decimal figures, by adding imaginary Zeroes after the Remainder and further dividing by the Divisor. This is because x ÷ y = 10nx ÷ 10ny = (10nx ÷ y)/10n.
 
Let us rework the above example for decimal figures:
  28237 ÷ 98 = 288, and Remainder: 13
(We have achieved this, in the above example)
 
Working ahead with the Remainder, we get:
98 02 ) 1   3   0   |   0   0
            0   2   |
                0   |   6
                    |   0   4
       ----------------------
        1   3   2   |   6   4
 
Note above, that the imaginary Zeroes are greyed-out.
 
So, we get 13/98 as:
Quotient = [1,3,2] = 132
And, Remainder = [6,4] = 64
 
But, we suffixed 3 Zeroes, so the Quotient has to be divided by 1000, which becomes 0.132
 
Thus, 28237 ÷ 98 = 288.132...
 
Note that, more decimal point may be derived by suffixing more imaginary Zeroes.
Also, the method remains fairly simple if the Remainder is obtained, and then later reworked for decimal points - rather than trying to obtain the Quotient and decimal figures at the same time.
Similarly, a smaller number may be divided by a bigger number - by assuming imaginary Zeroes and adjusting the decimal point accordingly.
 
However, is should be discussed herein, that the method may get very cumbersome for complicated differences (not necessarily, bigger) from the Base. For example,
20067 ÷ 899 is not cumbersome, because the Transposition is 211
But, the same number divided by 931 may get cumbersome, because the Transposition is 079.
This is the reason that the Upasūtra: dhvajāṅka (discussed here »), also popularly known as Straight-Division, remains the most popular method of Division amongst practitioners of Vedic Mathematics.
 
But, why does it work? For this Sūtra (parāvartya yojayet), let us consider the following:
  This division method is self-explanatory, because it is only a variation of the 'polynomial long division' method, where the Transposition is only a representation of (x - r), where x is Base and r is the Divisor.
 
Not only does this tactic provide a marked advantage by reducing the division to multiplications of digits - it is also a generalized tactic that will work with any n-digit Divisor.
 
Lastly, please remember that, as in any other form of mathematics, the mastery of Vedic Mathematics require practice and the judgement of applying the optimal method for a given scenario - a guideline of which, is presented in Applications »
 
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