What is the mathematical formula for Square Root? I wasn't at school when we were learning this, and I've forgot how to calculate a square root on paper using a formula?
Can anyone please help me? What is the formula?
I need this to write an algorithm for my college assignment home work. Thanks!
 A: Let $ a \geq 0 $. If you want to find $ \sqrt{a} $, use the following iteration:
\begin{align}
x_{0}   &:= a. \\
x_{n+1} &:= \frac{1}{2} \left( x_{n} + \frac{a}{x_{n}} \right).
\end{align}
Then $ \displaystyle \lim_{n \rightarrow \infty} x_{n} = \sqrt{a} $. This is an example of the Newton-Raphson method, applied to the function $ f(x) = x^{2} - a $. This particular iteration exhibits quadratic convergence, which means that the number of correct digits about doubles in size with each successive step. Hence, on an ordinary scientific calculator, one can obtain a good approximation after just a few steps.
A: Use $$(1+x)^{1/2}=\sum_{n=0} \binom{1/2}{n}x^n$$ for $|x|<1$, where $\binom{a}{n}=\frac{a(a-1)\dots (a-n+1)}{n!}$. 
A: Observe that
$$\sin \left( {\frac{\pi }{4}} \right) = \cos \left( {\frac{\pi }{4}} \right) = \frac{{\sqrt 2 }}{2}$$
so you can use Taylor series for either $\sin (\pi /4)$ or $\cos (\pi /4)$ to approximate ${\sqrt 2 }$ for example.
Or you could just use linear approximation from calculus.
I am not familiar with algorithms, but it seems like linear approximation should be easier.
