> Here's a question. How do you calculate the square root of a number with
> pencil and paper only?

If you asked me to do it I'd probably use some sort of guessing method 
for each digit then move onto the next. Surprisingly you can normally 
get to within one or two digits just by looking at the previous answers. 
You better be good at multiplication by hand though :-)

eg Find sqrt(70)
(guess 8)
8^2=64
9^2=81
(guess 8.3)
8.3^2=68.89
8.4^2=70.56
(guess 8.36)
8.36^2=69.8896
8.37^2=70.0569
(guess 8.368)
etc.

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