1. ## calculate root

How the calculator calculate the root of number, e.g. sqrt-2?

3. ## Re: calculate root

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)?

4. ## Re: calculate root

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

5. ## Re: calculate root

O.K.
So why this method works?

6. ## Re: calculate root

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.

7. ## Re: calculate root

Using the binomial theorem, and for appropriate $\displaystyle \ a \ \ and \ \ b,$
$\displaystyle \sqrt{a^2 + b} \ \approx \ a \ + \ \dfrac{b}{2a}$

If $\displaystyle \ a_1 \ = \ 1.4, \ \ then \ \ b \ = \ 0.04, \ \ and \ \ a_2 \ \approx \ 1.414285714$