while extracting square root of numbers, why we double the quotent for next step?

Answer 1

Wow, I haven't heard about manually finding square roots since... 1662. Anyway, the entire square root algorithim is based on the following observation:

(10a+b)²=100a²+20ab+b²

The first step of the manual square root algorithm is to find the first digit of the square root, which is easily done by brute force. Once we have done this, we know that the square of this digit is less than the first (one or two) digits of the number we are trying to find the root of. How much less? Our remainder. Let the part of the root we have already found be a. Now, if we consider the remainder after bringing down the next two digits, we have the difference between the first (three or four) digits and 100a². 100a²=(10a)², so in other words, this is the difference we would find between the square of the first two digits of our quotient and the number we're trying to find the root of, if the next digit were zero. If the next digit is not zero, then it will increase the current square by 20ab+b², where b is the next digit. As such, to find the next digit we need to find the largest value of b such that 20ab+b² doesn't exceed the remainder. Now, a fast way to do that is to note that 20ab+b²=(2*10a+b)*b, thus if we double the current quotient, multiply it by ten, add b, and multiply our result by b, we get 20ab+b², and can test to see whether it exceeds the remainder. This is, you will note, exactly what the square root algorithm asks you to do, and in fact is really the entirety of the algorithm. Thus the reason you have to double the quotient at each step is because (10a+b)²=100a²+20ab+b², and not 100a²+10ab+b².

Of course, when taking square roots myself, I prefer to use the time-honered method of using a pocket calculator.

Answer 2

You can check my answer here http://kv.krishnaprasad.googlepages.com/yahoo_answer

(a+b)^2 = a^2 + 2ab + b^2
= a^2 + b(2a + b)

See the figure in my link and you may get an idea. Before I give a detailed explanation, Pascal has given a detailed one. So mine will be reduntant.

Answer 3

becauze its the basic