8. Trick - Squaring numbers near powers of 10

In this article, we shall learn about squaring numbers that are near to powers of 10 like 10, 100, 1000 etc.

The sutra for this is "Yavdunam Jaavdunikritya Varga Cha Yojayet" meaning whatever the extent of its deficiency, lessen it still further to that very extent; and also set up the square of that deficiency.

To understand the sutra, let's take an example. We would square 96. One way is to do it by squaring 104. Even shorter is this method.

96 is closer to 100 (we need to take power of 10). 100 - 96 = 4.. We further reduce 4 from 96 to get 92. Square of 4 is 16. So, the answer is 9216. Since 100 has two zeroes, the square must be of two significant digits.

Let's take a few more examples:

982 = (98 - 2)|(22) = 96|04 = 9604.
1022 = (102 + 2)|(22) = 104|04 = 10404

Now, we would examine is algebraically. Let the number be 100x - y.

(100x - y)2
   = 10000x2 - 2*100xy + y2
   = 100(100x - 2y) + y2
   = 100(100x - y - y) + y2

Now, (100x - y) is our original number from which we again subtract the dificiency y.

What if y² extends to 3 digits? We add the carry over to the front. Here's an example:

892 = (100 - 11)2
    = (89 - 11)|112
    = 78|121
    = (78 + 1)|21
    = 7921

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