Horner's method and related
(aka, Hoerner's method)

This just says to compute the roots of a polynomial by finding the
integer part of a solution, say $r$, subtracting that off
(replacing $x$ by $x+r$,
 which computationally yields $f(x+r) = f(r) + xf'(r) + \frac{1}{2}x^2
 f''(r) + \dotsb$)
and then replacing $x$ by $x/10$
  (which means $x^2+ax+b$ goes to $x^2+10ax+100b$,
   after you multiply through by $10^n$)
and repeating to get decimal places of accuracy.

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Similarly, you can use this compute square roots.

A cuter method to compute square roots (of $A$) is to have a starting
number $x_0$, then divide, and let $x_1$ be the average of $x_0$ and
$A/x_0$, and repeat.
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There's also Newton's method, which I think converges pretty fast but
requires fancier computations which aren't so good -manually-.
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Note that emphasis on decimal degrees of accuracy, while good
manually, obscure the math that's going on.
