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उपसूत्र ७. यावदूनं तावदूनीकृत्य वर्गं च योजयेत्

(Upasūtra 7. yāvadūnam tāvadūnīkṛtya vargañca yojayet) - Lessen by the deficiency, and set up square of the deficiency.
 

The Upasūtra: yāvadūnam tāvadūnīkṛtya vargañca yojayet (Lessen by the deficiency, and set up square of the deficiency) is used for squaring (x2) a number, that is close to a power of ten (10n). The technique followed by this Upasūtra uses Rekhanks & Vinculum Numbers (discussed here »).
 
As an illustration, let us use this Upasūtra for:
962
  Steps 962
1. Consider the nearest power of 10 as the Base In this case, the Base, closest to 96, is 100.
2. Find the deficiency, and represent as Rekhank for negative deficiency. In this case, the Deficiency =
100 - 96 = 4
3. Subtract the deficiency from that number In this case,
96 - 4 = 92
4. Set-up as many Zeroes, as the Base In this case, 100 has two Zeroes.
So, we get 9200
5. Square the Deficiency, and add to get the answer In this case, 42 = 16
And, 9200 + 16 = 9216, which is the answer!
Let us take another example, for something like:
1032
  The Base is 100 and, the Deficiency is: 100 - 103 = -3 = 3
And, 32 = 9
 
Also, 103 - 3 = 103 + 3 = 106
And, 10600 + 9 = 10609, which is the answer!
So, for a practitioner of Vedic Mathematics, for something like:
9912
  Clearly, the Deficiency is 9
And, 92 = 81
 
Also, 991 - 9 = 982
And, 982,000 + 81 = 982,081
 
Thus, 9912 = 982,081
Again, for something like:
100062
  Clearly, the Deficiency is 6
And, 62 = 36
 
Also, 10006 - 6 = 10006 + 6 = 10012
And, 10012,0000 + 36 = 10012,0036
 
Thus, 100062 = 100,120,036
Similarly, one take take any (n-digit) number and execute the steps above to obtain the desired calculated value.
 
But, why does it work? For the above Upasūtra (yāvadūnam tāvadūnīkṛtya vargañca yojayet), let us consider the following:
  Assuming N is a number close (and less) to a power of 10, then N = a - b
'a' being the power of 10, and 'b' being the Deficiency
 
Now, N2 = (a - b)2
= a2 - 2ab + b2
= a (a - 2b) + b2
But, N = a - b. So, substituting a = N + b
N2 = a ([N + b] - 2b) + b2
= a (N - b) + b2
 
This is exactly what this Upasūtra 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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