Squaring by Yaavadunam
Sutra : Yaavadunam Taavaduunikruthya vargam cha yogayet
Meaning : Whatever the extent of its deficiency, lessen it further to that very extent; and also set up the square of that deficiency.
This sutra is a corollary of the Nikhilam sutra.
Consider a simple example 92
Step 1 : Consider the nearest base (here 10).
Step 2 : As 9 has a deficiency of 1 (10 - 9 = 1), we should decrease it further by 1, and set down our LHS of the Answer as '8'.
Step 3 : On the RHS put the square of the deficiency (here 1).
we get 92 = 81.
Now Consider 102
1) Base is 100
2) Deficiency is '-2' (100 - 102 = -2)
Therefore we subtract '-2' from 102
102 - (-2) = 104
This is our RHS
3) Our LHS now becomes (-2)2 which is 4
Since the base is 100 we write it as '04', so that we get 1022 = 10404
If we have multiples or sub multiples of a base, we employ the same technique as in 'Aanurupyena'. (See Nikhilam Multiplication)
Consider 282
1) Let 20 be the Working Base and 10 as the Main Base.
Therefore x = (Main Base)/(Working Base) = 10/20 = 1/2
2) Here the deficiency = 20 - 28 = -8
Therefore RHS = 28 - (-8) = 36
Divide by x i.e. by (1/2).
We get 36/(1/2) = 72. This is the required RHS.
3) LHS = (-8)2 = 64
Since Main Base is 10, we put only '4' on the LHS and carry over '6' to the RHS
Therefore we get
282 = 72+6 | 4 == 784
Some More Examples
362 = 128 |1 6 = 1296 (Base = 40)
522 = 27 | 04 = 2704 (Base = 50)
9972 = 994 | 009 = 994009 (Base = 1000)
10042 = 1008 | 016 = 1008016 (Base =1000)