 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)