How can you solve square roots by hand, such as \sqrt{5}? I'm looking for a method that doesn't require approximation and then checking back.

Question

How can you solve square roots by hand, such as \sqrt{5}?  I'm looking for a method that doesn't require approximation and then checking back.

anonymous · Answer

you can get it by a recursive method but since this number is irrational you cannot get it using elementary operations.  do you want a recursion?

anonymous · Answer

quickest method is mechanic's rule aka newton's method.  make a guess, say 2, then take 
$$\frac{1}{2}(2+\frac{5}{2})$$
rinse and repeat

anonymous · Answer

i.e. 
$$x_{n+1}=\frac{1}{2}(x_n+\frac{5}{x_n})$$

anonymous · Answer

i did see a "long division" style method once but it is annoying and painfully slow.

anonymous · Answer

I'm not looking for a guess and check...I feel those are better left for calculators.  What is this long division style method you were talking about?