Vedic Math - Squaring a number Less Than Base
Many a times you can save time in an exam, if you can calculate squares of numbers mentally. It is easy to calculate squares of numbers ending in 0. For example square of 100 is 10,000. But what if we have to find the square of say, 92. Let us use elementary algebra to arrive at an answer. We can write 92 as 100 - 8. If we square, we get three terms, = 100^2 - 2 *100*8 + 64. Now we have the last two digits of the square as 64, because the other two terms have got 2 or 4 zeroes in them. For the remaining terms 10,000 - 1600, which can also be written as 9200 - 800 = 8400. Now these will be the first two digits of the answer because of the two zeroes present. So the answer comes out to be 8400 + 64 = 8464.
While squaring a number, you need a base. All those numbers can be taken as bases, which have a 1 and the rest number of zeroes with them (i.e. the complete round numbers like 100, 1000, 10000 etc.). So the method that we can develop is start by taking the nearest complete base, in this case 100. The difference between the base and the number given is 8. The square of this difference is 64, which will become the right side. Because it is already having two digits, so it would be simply placed on the right side. Now the difference of 8 is subtracted from the number given i.e. 92 – 8 = 84 and it will become the left side. Therefore the square of 92 is 8464.
f the square of the difference is having lesser digits then required, then in order to have the needed number of digits on the right side, 0's can be put with the square. e.g. if you square a number like 97, difference is 3. The right side in this case would become 09, because 9 is a single digit number and you'll have to put a '0' before it to make it a two-digit right hand side. The left side would be 97 - 3 = 94. The square is 9409.
In case, the number of digits is more than needed, then the extra digits are carried to the left side. e.g. take 86. The difference is 14 and the square of the difference is 196, which is a 3-digit number, so the 3rd extra digit 1 would be carried to the left side. The left side is 86 – 14 = 72 + 1 (carried over) = 73. So the square of the number is 7396.
The square of a number will have two parts, the left part and the right part. There is no limit for the left side, but the right side must have as many digits as the number of zeroes in the base i.e. if 100 is taken as base there should be 2 zeroes on the right side and if 1000 is taken as base then the number of digits on RHS should be 3.
To conclude, to get the square of a number less than a base, find the difference. The important thing is the left hand digits. These will be the number minus the base, and the right hand digits will be the square of the difference.