Calculating the square root of a

In this learning project, we learn how to approximate the square root of a number numerically.

Newton's method
Newton's method is a basic method in numerical approximations. The idea of Newton's method is to "follow the tangent line" to try to get a better approximation. For more details, see Newton's method

Step-by-step
Cut and paste the following wikitext into the sandbox, and enter your favourite number and a first (non-zero!) approximation of its square root. See what happen after multiple edit-and-saves: number

Sandbox
17 4.1231056256177 4.1231056256177 4.1231056256177 4.1231056256177 4.1231056256177 4.1231056256177 4.12310562562 4.12310562562 4.12310562562 4.12310562562 4.12310562562 4.12310562562 4.12310562562 4.12310562562 4.12310562562 4.12310562562 4.12310562562 4.12310562562 4.12310562562 4.12310562562 4.12310562562 4.12310562562 4.12310562562 4.12310562562 4.12310877528 4.1282051282 4.33333333333 3

What is going on
Our calculator-template square root implements the formula $$x_{n+1} = (x_n + a/x_n)/2$$, which is an instant of the general formula for Newton's method $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\,\!$$ in the case when the function f(x) is given by $$ f(x)=x^2 - a$$. It solves the equation $$ x^2 - a=0$$ numerically, by successive approximations.