How to compute a lot of digits of $\sqrt{2}$ manually and quickly? After having read the answers to calculating $\pi$ manually, I realised that the two fast methods (Ramanujan and Gauss–Legendre) used $\sqrt{2}$. So, I wondered how to calculate $\sqrt{2}$ manually in an accurate fashion (i.e., how to approximate its value easily).

you can use the formula 
$$\frac{a_{n+1}}{b_{n+1}}=\frac{a_n^2+2b_n^2}{2a_nb_n}$$
if we take an initial value of $\sqrt{2}$ as $\frac{3}{2}$ 
now the new value will become
$$\frac{a}{b}=\frac{3^2+2*2^2}{2*2*3}=\frac{17}{12}$$
the new value of $a=17$ and $b=12$
and then continue
