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सूत्र १०. यावदूनम्

(Sūtra 10. yāvadūnāṃ) - By the deficiency.
 

The Sūtra: yāvadūnāṃ (By the deficiency) is used for multiplication (x × y) of two numbers, that are close to a power of ten (10n. The technique followed by this Sūtra uses Rekhanks & Vinculum Numbers (discussed here »).
 
As an illustration, let us use this Sūtra for:
99 × 97
  Steps 99 × 97
1. Consider the nearest power of 10 as the Base In this case, the Base, closest to 97 and 99, is 100.
2. Find the deficiencies of both the numbers, and represent as Rekhank for negative deficiencies. In this case, the two deficiencies are:
100 - 99 = 1 and 100 - 97 = 3
3. Subtract the (Sum of deficiencies) from the Base In this case,
100 - (1 + 3) = 100 - 4 = 96
4. Set-up as many Zeroes, as the Base In this case, 100 has two Zeroes.
So, we get 9600
5. Add the product of the deficiencies, to get the answer. In this case, the product of the deficiencies is: 1 × 3 = 3
And, 9600 + 3 = 9603, which is the answer!
Let us take another example, for something like:
98 × 103
  The Base is 100 and, the deficiencies are: 100 - 98 = 2 and 100 - 103 = -3 = 3
 
So, sum of deficienies = 2 + 3 = 1
And, product of deficiencies = 2 × 3 = 6
 
Thus, 100 - 1 = 100 + 1 = 101
And, 101,00 + 6 = 10100 - 6 = 10094, which is the answer!
So, for a practitioner of Vedic Mathematics, for something like:
989 × 1005
  The Base is 1000 and, 11 × 5 = 55
Also, 1000 - (11 + 5) = 994
And, 994000 + 55 = 993945
 
Thus, 989 × 1005 = 993,945
Again, for something like:
100012 × 100004
  The Base is 100000 and, 12 × 4 = 48
Also, 100000 - (12 + 4) = 100016
And, 100016,00000 + 48 = 100016,00048
 
Thus, 100012 × 100004 = 10,001,600,048
Similarly, one take take any (n-digit) numbers and execute the steps above to obtain the desired calculated value.
 
But, why does it work? For the above Sūtra (yāvadūnāṃ), let us consider the following:
  Assuming two numbers (10n - a) and (10n - b)
Then, (10n - a) × (10n - b)
= 10n × 10n - a10n - b10n + ab
= 10n × 10n - 10n(a + b) + ab
= 10n (10n - (a + b)) + ab
 
This is exactly what this Sūtra makes us do.
 
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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