*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 9^{2}

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 9** ^{2}** = 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 102** ^{2}**
= 10404

If we have multiples or sub multiples of a base, we employ the same technique
as in 'Aanurupyena'. * (See Nikhilam Multiplication)*

Consider 28^{2}

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

28** ^{2}** = 72+6 | 4 == 784

Some More Examples

36** ^{2}** = 128 |

522 = 27 | 04 =
2704
*(Base = 50)*

9972 = 994 | 009 = 994009 *
(Base = 1000)*

10042 = 1008 | 016 = 1008016 *(Base =1000)*