Is it impossible to find the "exact" value (like 3,1415926) of a non exact square root without knowing the nearest exact square root?

Question

Is it impossible to find the "exact" value (like 3,1415926) of a non exact square root without knowing the nearest exact square root?

I looked in a lot of sites and youtube videos, and i can't find any equation for it.

Answer 1

When we approximate a non-exact square root we start with the greatest integer less than the square root and start adding decimal digits to have a better and better approximation. For example for $\sqrt 5$ we start with $2$ and go to $$\{2,2.2,2.23,2.236, 2.2360,....\}$$

Similarly with $\sqrt{27}$ we start with $5$ and go through $$\{5, 5.1,5.19, 5.196,...\}$$

Answer 2

This is probably a ridiculous answer.

You are looking for the square root of a number $a$ and you have no idea about the greatest exact square root below (or above) it. In any manner, it means that you look for the zero of function $$f(x)=x^2-a$$ Assuming $a >1$, we see that $f(a)=a^2-a >0$ and the second derivative is positive. Then, by Darboux theorem, starting Newton method with $x_0=a$ will converge to the solution without any overshoot of the solution. As usual, the iterates will be given by $$x_{n+1}=x_n-\frac{f(x_n) }{f'(x_n) }=\frac{x_n^2+a}{2 x_n}$$

Using your example $a=7039725$, this would give the following iterates $$\left( \begin{array}{cc} n & x_n \\ 0 & 7039725.000 \\ 1 & 3519863.000 \\ 2 & 1759932.500 \\ 3 & 879968.2500 \\ 4 & 439988.1250 \\ 5 & 220002.0624 \\ 6 & 110017.0304 \\ 7 & 55040.50901 \\ 8 & 27584.20490 \\ 9 & 13919.70673 \\ 10 & 7212.722377 \\ 11 & 4094.368672 \\ 12 & 2906.868156 \\ 13 & 2664.312010 \\ 14 & 2653.270982 \\ 15 & 2653.248010 \end{array} \right)$$