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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 |  1|  1|  1|  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 »
 
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