# उपसूत्र १५. शुद्ध

## (Upasūtra 15. śūddha) - Purification. The Upasūtra: śūddha (Purification) is for finding the addition (x + y) and subtraction (x - y) of two numbers. It is similar to the conventional methods, but conducts a column-wise addition or subtraction - followed by a process of purification, śūddhikaran.

Addition of two (or more) numbers using Upasūtra 15. śūddha:
As an illustration, let us use this Upasūtra for:
28 + 165
 Steps 28 + 165 1. Equate the digits of both the numbers by prefixing imaginary Zeroes in front, and separate the columns for each place. In this case,   0 | 2 | 8 + 1 | 6 | 5 ------------ 2. Sum the digits of each column, even if the the result is aśūddha (impure, to mean more than one digit holding a position). In this case,   0 | 2 | 8 + 1 | 6 | 5 ------------   1 | 8 | 13 3. Conduct a śūddhikaran to purify the number, so that every place is held by only one positive digit. In this case, [1,8,13] has 13 in the One's place. So, only 3 can remain, and the 1 needs to be added to the next column, containing 8. Thus, [1,8,13] = 193, which is the answer!
Note that, the detailed explanation for the process of śūddhikaran is discussed here ».

So, for a practitioner of Vedic Mathematics, for something like:
234 + 403 + 564 + 721
 2  |  3  |  4     4  |  0  |  3     5  |  6  |  4 +   7  |  2  |  1 ------------------     18 |  11 |  12 By śūddhikaran, [18,11,12] = 1922  Thus, 234 + 403 + 564 + 721 = 1922
The process of śūddhikaran may be executed during the calculations by retaining the last digit and marking the 'carry-over' digits. As in the next example, for something like:
78924 + 27272 + 72684
 7  |  8  |  9  |  2  |  4     2  |  7  |  2  |  7  |  2 +   7  |  2  |  6  |  8  |  4 ------------------------------     17 |  18 |  18 |  18 |  10  Thus, 78924 + 27272 + 72684 = 178880  Note that, the 'carry-over' digit is taken into consideration for each columns.
Although, it does not provide a visible marked difference, this technique clarifies the number system at a conceptual level, and easy to explain to children. Additionally, this technique finds implementation throughout many Vedic methods.

And, this Upasūtra obviously works because it is only a variation of the conventional method of addition, by dealing with the 'carry-over' digits in a different manner.

Subtraction of numbers using Upasūtra 15. śūddha:
As an illustration, let us use this Upasūtra for:
34 - 18
 Steps 34 - 18 1. Equate the digits of both the numbers by prefixing imaginary Zeroes in front, and separate the columns for each place. In this case,   3  |  4 - 1  |  8 ---------- 2. Convert the digits as Rekhanks, and sum the digits of each column, even if the the result is aśūddha (impure, to mean more than one digit holding a position) In this case,   3  |  4 + 1  |  8 ----------   2  |  4 3. Conduct a śūddhikaran to purify the number, so that every place is held by only one positive digit. In this case, [2,4] has 4 in the One's place. So, 1 needs to be taken from the previous digit and given to the next digit, which becomes (10 + 4) = (10 - 4) = 6. Thus, [2,4] = 16, which is the answer!
So, for a practitioner of Vedic Mathematics, for something like:
5124 - 3608
 5  |  1  |  2  |  4 + 3  |  6  |  0  |  8 ----------------------   2  |  5  |  2  |  4 By śūddhikaran, [2,5,2,4] = 1516  Thus, 5124 - 3608 = 1516
Again, the process of śūddhikaran may be executed during the calculations by retaining the last digit and marking the 'carry-over' digits, for something like:
78924 - 27272
 7  |  8  |  9  |  2  |  4 + 2  |  7  |  2  |  7  |  2 ----------------------------   5  |  1  |  6  | 15  |  2  Thus, 78924 - 27272 = 51652  Note that, the 'carry-over' digit is taken into consideration for each columns. Also, for negative results, it is replaced with the relative complement from the nearest m × 10 and m is carried-over, for consideration in next column.For example above, 2 + 7 = 5 and, 10 + 5 = 5. Hence, carry-over of 1.
Although this method can only subtract smaller numbers from bigger numbers, it can also be used in subtracting bigger numbers from smaller ones because x - y = -(y - x), as for something like:
234 - 7381
 7  |  3  |  8  |  1 + 0  |  2  |  3  |  4 ----------------------   7  |  1  |  4  | 17  Thus, 234 - 7381 = -(7381 - 234) = -7147

And, this Upasūtra obviously works because it is only a variation of the conventional method of subtraction, by dealing with the 'carry-over' digits in a different manner.

Mixed Addition and Subtraction of more than two numbers using Upasūtra 15. śūddha:
The discussion of śūddha cannot be completed without mentioning its marked advantage in dealing with mixed addition and subtraction (x - y + z) of more than two numbers. Unlike the conventional method ((x + z) - y), this Upasūtra allows us to operate with the help of Rekhank (discussed here »)

As an illustration, let us consider something like:
6371 - 2647 + 8096 - 7381 + 1234
 6  |  3  |  7  |  1    2  |  6  |  4  |  7    8  |  0  |  9  |  6    7  |  3  |  8  |  1 +  1  |  2  |  3  |  4 -----------------------    5  | 16  |  7  |  3  Thus, 6371 - 2647 + 8096 - 7381 + 1234 = 5673

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 »

 « Upasūtra 14 Upasūtra 16 »