Forgive me if this question is very basic, I don't know much about the properties of square roots.
My goal is to compute the square root of a very large number (around 1024 bits). While toying around with existing algorithms, I came across a weird property to guess the initial value.
Suppose that you have a large integers 43212215 that you square (1867295525206220)
If you compute the square root of the first digits from the results, sometimes it gives the most significant digits of the square root of the whole numbers, sometimes it does not.
sqrt(186) == 13.638
sqrt(1867) == 43.208
sqrt(18672) == 136.64
sqrt(186729) == 432.121
So I have three questions now :
1-What makes the square root behave like this and how can I predict the results?
2-When using an odd number of digits, is there some information to be extracted from the result (13xxx)?
3-Does knowing the first digits of the square root simplifies the computation for the square root of the whole number in any other way than the initial guess?
In other words for 3, knowing that 43200000 * 43200000 == 1,866,240,000,000,000 can I use this product to reduce the number of digits I will have to work on?
Hoping that I properly understood the first part related to the square roots, you consider a number $x$ and you compute $$y_n=\sqrt{\left\lfloor \frac{x}{10^n} \right\rfloor }$$ $n$ being the number of digits you want to consider. I suppose that this is sufficient to explain why $\frac{y_{n-1}}{y_n}\approx \sqrt{10}$ and why $\frac{y_{n-2}}{y_n}\approx 10$.