MATH or MAGIC!

The Sanskrit sutras offer several techniques for division. Some of them work spectacularly to reduce to a single line certain cases of division that may take dozens of lines of calculation in the conventional method of long division taught in schools.

While there are special techniques reserved for special cases, we highlight here a general-purpose long division technique that will work for both small and large numbers. It is basically applying in reverse the Urdhva Tiryagbhyam (Vertically and Crosswise) principle used to simplify multiplication. Such is the power and elegance of this technique that we can easily take a large number and make short work of it by any divisor of two or more digits.

Take for example, 716769 ÷ 54. Yes, you too CAN work it out manually -- and in one line -- without having to reach for the calculator! You will instead rely on the calculator inside your head and the Dhvajanka sutra, meaning "on top of the flag". The trick is to reduce the divisor to a mentally manageable value by putting its other digits "on top of the flag". In this example, the divisor will be reduced to 5 (instead of 54) by pushing the 4 up the flagpost, as shown below. Corresponding to the number of digits flagged on top (in this case, one), the rightmost part of the number to be divided is split to mark the placeholder of the decimal point or the remainder portion.


Now observe carefully as we walk through the steps of this example:


716769 ÷ 54 = 13273.5

1. 7 ÷ 5 = 1 remainder 2. Put the quotient 1, the first digit of the solution, in the first box of the bottom row and carry over the remainder 2
   

2. The product of the flagged number (4) and the previous quotient (1) must be subtracted from the next number (21) before the division can proceed. 21 - 4 x 1 = 17
   
   17 ÷ 5 = 3 remainder 2. Put down the 3 and carry over the 2
   

3. Again subtract the product of the flagged number (4) and the previous quotient (3), 26 - 4 x 3 = 14
   
   14 ÷ 5 = 2 remainder 4. Put down the 2 and carry over the 4
   

4. 47 - 4 x 2 = 39
   
   39 ÷ 5 = 7 remainder 4. Put down the 7 and carry over the 4
   

5. 46 - 4 x 7 = 18
   
   18 ÷ 5 = 3 remainder 3. Put down the 3 and carry over the 3
   

6. 39 - 4 x 3 = 27. Since the decimal point is reached here, 27 is the raw remainder. If decimal places are required, the division can proceed as before, filling the original number with zeros after the decimal point
   
   27 ÷ 5 = 5 remainder 2. Put down the 5 (after the decimal point) and carry over the 2
   

7. 20 - 4 x 5 = 0. There is nothing left to divide, so this cleanly completes the division

While this example is easily solved, there are finer details in the application of technique that will be highlighted in subsequent examples later.