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.

Answer 1

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?

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

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?