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 w:Newton's method

Try your hand on 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:

{{subst:square root|number|first approximation}}


Sandbox

{{subst:square root|17|4.1231056256177}}
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 ${\displaystyle x_{n+1}=(x_{n}+a/x_{n})/2}$ , which is an instant of the general formula for Newton's method ${\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}\,\!}$  in the case when the function f(x) is given by ${\displaystyle f(x)=x^{2}-a}$ . It solves the equation ${\displaystyle x^{2}-a=0}$  numerically, by successive approximations.