Square 2+ Digit Numbers Mentally

Step 3 Mental Squares: The Vedic Way

The technique used here is the same one that appears in the wikipedia page (you did read the wikipedia page, didn't you?). It can be summarized using the following phrase:

"Whatever the extent of the deficiency, lessen it still further by that extent, and add to it the square of the deficiency"

That's a little cryptic, so below is the technique expressed in plain English:
For a given number, choose an arbitrary base. Now, determine the difference between the number and the base (this is the "deficiency"). Add the difference to the number, and multiply this new number by the base. Finally add the square of the difference to the number, which gives you the final answer.

That can be a little hard to digest, so we will apply the technique using the example of 45² from step 2.

1) Given 45², choose 50 to be the base. Now, the deficiency is 45 - 50 = -5.
2) Add the deficiency to the number, so 45 + (-5) = 40.
3) Multiply the new number by the base: 40 * 50 = 4*5*100 = 2000
4) Add to the product the square of the deficiency: 2000 + (-5)² = 2000 + 25 = 2025

This technique, when practiced, allows for rapid mental square computation. More examples in the following steps.