How to calc the square root of a number without calculator? How can I find the square root of a number without using a calculator?
 A: Newton's method says that
$$
x_{k+1}=\frac{n+x_k^2}{2x_k}
$$
converges to $\sqrt{n}$.
A: One approach: find a perfect square close to your number.  E.g. for $23$, we could take $25$.  Now write (for this example) $\sqrt{23} = 5 \sqrt{23/25} = 5 \sqrt{1 - 2/25}.$  Now if we want to compute $\sqrt{1 - x}$ where $|x| < 1$, there is the binomial series:
$$\sqrt{1 - x} = 1 - \dfrac{1}{2} x - \dfrac{1}{8} x^2 + \cdots.$$
In our examples this gives $\sqrt{1 - 2/25} = 1 - \dfrac{1}{25} - \dfrac{1}{1250} + \cdots,$ 
giving $\sqrt{23} = 5 - \dfrac{1}{5} - \dfrac{1}{250} + \cdots \sim 4.796,$
which isn't a bad approximation.

Another approach is to define the sequence $x_{n+1} = (x_n + A/x_n)/2$, where $A$ is the number whose square root you are trying to compute.  (Here you start with $x_0$ some reasonable ball-park estimate to $\sqrt{A}$.)  Then $x_n$ will converge to $\sqrt{A}$.  (This is a special case of Newton's method, applied here to find a root of $x^2 - A = 0$.)
In the case of $A = 23$, we might take $x_0 = 5$, then $x_1 = 4.8$, and $x_2 \sim 4.796$ again.
