Originally posted by piderman
Care to explain how it works (and the mechanics behind it)? Say I want to take the square root of 5764801
The mechanics on how it works is not too hard to see once you see the method. It's the method that is hard to type down :p
OK, the process is supposed to look like a division process.
devide the number in parts of two, starting on the right;
5 76 48 01
We start with the 5. The biggest square smaller then 5 is 2*2, so the first digit of the root is a TWO
5-4 = 1 and we take the next two digits of the square making 176
Now we had 2*2, instead of multiplying we add the two two's now, giving a 4. Now we need to find x such that (40+x)*x is as close under 176 as possible.
In this situation x = 4 since 44*4=176. So the second digit of our root is FOUR
176-176=0 and we add the next two digits of our square to make 48
We had 44*4, and now instead of multiplying we add the two numbers to make 44+4=48. Now we need to find x so that (480+x)*x is as close under 48 as possible.
480*0=0 so the third digit of our root is ZERO.
48-0 = 48 and adding the last two digits of our sqaure to it gives 4801.
480+0=480, so now we need to find x so that (4800+x)*x is as close under 4801 as possible. This time x=1 and this leaves remainder 0, meaning we're done. The last digit of our root is 1
The idea behind this method is that you start with a square of size 5, and take a square out of it (here size 4). What remains is an edge consisting of a square and two rectangles. Now you increase the size of the sides by a factor 10 (this is adding the next two digits of our square, as the square we had increases by a factor 100).
Now we try to find x such that 20x + 20x + x^2 approximates the edge as close as possible. above can be rewritten as (40+x)*x wich is the formula we solved.
You keep getting edges as remainder, and you keep approximating it as close as possible, increasing the edges by a factor 10 each time, until you're done.
I hope this made sense, it's much easier to explain on paper