How the calculator calculate the root of number, e.g. sqrt-2?
O.K.
I have my source and I will quote a part from it in my words. O.K?
There are some ways from to calculate root.
One of the way is Taylor Polynomial.
If a function f is very "nice" so can approximate its values by polynomial.
Examples of that "nice" function are: roots, sinus, cosines and another kinds of it.
What the property of the this "nice" function? It can be derived infinitely times.
How I should know if function is derived any time (and example/website, if possible, one can bring on how can it combined by calculating root of two by calculator)?
While using the Taylor polynomial to approximate a given function is very basic, the "CORDIC" method is now more commonly used:https://en.wikipedia.org/wiki/Method...e_roots#CORDIC
I always use either the Babylonian method, for its simplicity or the approximation
$$\sqrt{a} \approx n + \frac{a-n^2}{2n} \qquad n = \lfloor \sqrt{a} \rfloor, \, \text{the greatest integer whose square does not exceed $a$}$$
when accuracy is not so important as speed.
The approximation
$$\sqrt{a} \approx n - \frac{n^2-a}{2n} \qquad n = \lceil \sqrt{a} \rceil, \, \text{the least integer whose square is not smaller than $a$}$$
is similar to the above.
Using the binomial theorem, and for appropriate $\displaystyle \ a \ \ and \ \ b, $
$\displaystyle \sqrt{a^2 + b} \ \approx \ a \ + \ \dfrac{b}{2a}$
If $\displaystyle \ a_1 \ = \ 1.4, \ \ then \ \ b \ = \ 0.04, \ \ and \ \ a_2 \ \approx \ 1.414285714$