# उपसूत्र १४. ध्वजांक

## (Upasūtra 14. dhvajāṅka) - On the flag. The Upasūtra: dhvajāṅka (On the flag) is used for division (x ÷ y) of two numbers. It is also sometimes referred as Straight-Division, because with some practice, a practitioner of Vedic Mathematics can reach 'straight' to the calculated figure - without any intermediate steps (which means the intermediate steps are executed mentally).

As an illustration, let us use this Upasūtra for:
75 ÷ 25
 Steps 75 ÷ 25 1. Arrange the numbers in a manner that the Divisor is in the left-hand side, with the last digit slightly raised. Leave some space between the digits of the Dividend.This slightly-raised last digit of the Divisor is the Flag. In this case, 25 ) 7  5 2. Check: How many times the first part of the Divisor 'goes' into the first digit of the Dividend. If it does not 'go', put a Zero, and move on to the next digit of the Dividend.Lookup: The next digit of the Dividend should not be negative. In case, it is negative, keep reducing the Quotient till it is Zero, or Positive. In this case, 25 ) 7  5           -----      3 3. Find the next digit of the Dividend by placing the Remainder of the above operation (Step 2) before the next digit of Dividend (slightly lowered) and substracting the (Quotient of the above operation, multiplied by the Flag) In this case, 25 ) 7 15       -15       ---         0     -----      3 4. Repeat the above steps (from Step 2) for every digit of the Dividend. On reaching the last digit of the Dividend, if the (Last digit to be operated) is Zero, stop operating. If the (Last digit to be operated) is non-Zero but perfectly divisible, execute the operation and stop operating. If the (Last digit to be operated) is non-Zero but not divisible, execute the operation, and put a decimal point before the last digit of the Quetient, and stop operating. In this case, we get a Zero. So, we are left with 3, which is the answer!
Since, Division is the most complex operation for starters (next to Cube-Roots), we will take up a few more examples - especially to deal with all the cases of above. Let us take an example of the simplest form:
25 ÷ 25
 25 ) 2 05       - 5       ---         0     -----      1  So, 1 is the answer!  Note, we stop operating because the Next Digit to be operated is Zero.
Now, let us take another example:
225 ÷ 25
 25 ) 2 22  45       - 0 -45       --- ---        22   0     ---------      0  9  So, 9 is the answer!  Notes: 1. 2 'goes' into 2 - once. But that would mean the Next Digit as (02)-(1 × 5) = -8. As per 'Lookup' in Step 2, we reduce it to Zero. 2. Moving on to the next digit, 22: 2 'goes' into 22 - 11 times. But that would mean the Next Digit as (05)-(11 × 5), which is negative. As per 'Lookup' in Step 2, we reduce it to 10 which gives us (25)-(10 × 5), which is again negative. So, we further reduce it to 9. 3. We stop operating because the Next Digit to be operated is Zero.
Let us take another example:
235 ÷ 25
 25 ) 2 23  55       - 0 -45       --- ---        23  10     ---------      0  9  .4  So, 9.4 is the answer!  Notes: 1. Proceeding as in the above example, we get to a Last Digit of 10. 2. 2 'goes' into 10 - 5 times. But, considering an imaginary Zero at the end, that would mean the Next Digit as (00)-(5 × 5), which is negative. As per 'Lookup' in Step 2, we reduce it to 4. 3. And, finally we place a Decimal Point, before 4.
The point that should be noted above is, when we reach the last digit, which is non-Zero, we assume an imaginary Zero - and introduce a decimal point. This is in-line with:
x ÷ y = 10x ÷ 10y = (10x ÷ y)/10

For that matter, dhvajāṅka doesn't really stop at one decimal point because:
x ÷ y = 10nx ÷ 10ny = (10nx ÷ y)/10n
Taking this ahead for a more realistic example:
98374 ÷ 87
 87 ) 9 18  33  27  64  80  70  90  70  40  30  80  90 100  70  70       - 7 - 7 -21 - 0 -49 -21 -35 -42 -21 -14 - 7 -56 -21 -63 - 0       --- --- --- --- --- --- --- --- --- --- --- --- --- --- ---        11  26   6  64  31  49  55  28  19  16  73  34  79   7  70     -----------------      1  1   3   0  .7   3   5   6   3   2   1   8   3   9   0   8  So, 1130.735632183908... is the answer!  Notes: 1. Note that the imaginary Zeroes are greyed out. 2. Also, we stop at the last digit voluntarily, and can be continued for more decimal digits.
Similarly, a smaller number may be divided by a bigger number - by assuming imaginary Zeroes and adjusting the decimal point accordingly.

So, for a practitioner of Vedic Mathematics, for something like:
785 ÷ 21
 21 ) 7 18  15       - 3 - 7       --- ---        15   8     ---------      3  7  .3  Thus, 785 ÷ 21 = 37.3 (Ignoring more decimal places)
The entire intermediate function can be done mentally, with some practice, to obtain the desired calculated value. Taking it ahead, for something like:
8547 ÷ 24
 24 ) 8 25  34  27       -12 -20 -24       --- --- ---        13  14   3     -------------      3  5   6  .1  Thus, 8547 ÷ 24 = 356.1 (Ignoring more decimal places)

Again, for something like:
20104 ÷ 103
 103 ) 2 20 101  80  34        - 0 - 3 -27 -15        --- --- --- ---         10  98  53  19      -----------------       0  1   9   5  .1  Thus, 20104 ÷ 103 = 195.1 (Ignoring more decimal places)
And, for something like:
85781547 ÷ 126
 126 ) 8 85 137  58 101  55  74 147        - 0 -36 -48 - 0 -48 - 0 -30        --- --- --- --- --- --- ---         85 101  10 101   7  74 117      -----------------------------       0  6   8   0   8   0   5  .9  Thus, 85781547 ÷ 126 = 680805.9 (Ignoring more decimal places)
One can take any (n-digit) Divisor and iterate the steps above to obtain the desired calculated value.

But, why does it work? For this Upasūtra (dhvajāṅka), let us consider the following:
 This division method is self-explanatory, because it is only a variation of the 'long division' method, where the Flag is used as a graphic mnemonic device.  However, the point that should be worth observing is that the remainder of previous digit is added in front of the next digit.That is, (10 x Remainder) is added to the next digit. And, tactically (Quotient of the previous digit, multiplied with the Flag - which is the Unit's digit of the Divisor) is subtracted from it - to obtain the next digit to be operated upon.  Not only does this tactic provide a marked advantage of dividing the digits by (n-1) digits instead of n-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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