TITLE: Algorithms for approximating $\sqrt{2}$
QUESTION [13 upvotes]: Well, "Solving" is the wrong term since I am speaking about irrational numbers. I just don't know which word is the correct word... So that can be part $1$ of my question... what is the correct word since you obviously can't "solve" an irrational number because it goes forever.
Part $2$ (my real question) are there algorithms for figuring out the answer to a problem like the square root of $2$ other than guess-and-checking your way to infinity? Again, I'm obviously not asking for an algorithm to give me the never ending answer because that's crazy... but for example if I wanted to know what the $15^{th}$ decimal place of the square root of $2$ was, is there an algorithm for that?
Thank you! (I'm new here and know nothing about how to format math questions so any help or links would be appreciated as well, thanks!)

REPLY [3 votes]: $\newcommand{\+}{^{\dagger}}%
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It's best to evaluate $\sqrt{2} = 2\sqrt{1\over 2}$ and the guess $3/4$ for $\sqrt{1 \over 2}$: It yields $38$ exact decimal places in $5$ iterations !!!.
$$
x_{n + 1} = \half\,\pars{x_{n} + {1 \over 2x_{n}}}\quad\mbox{with}\,\ n \geq 0\,,\quad x_{0} = {3 \over 4}\ \mbox{and}\ \root{2} = 2\lim_{n \to \infty}x_{n}
$$

1.500000000000000000000000000000000000000 -> 2.250000000000000000000000000000000000000

1.416666666666666666666666666666666666667 -> 2.006944444444444444444444444444444444444

1.414215686274509803921568627450980392157 -> 2.000006007304882737408688965782391387928

1.414213562374689910626295578890134910117 -> 2.000000000004510950444942772099280764361

1.414213562373095048801689623502530243615 -> 2.000000000000000000000002543584239585437

1.414213562373095048801688724209698078570 -> 2.000000000000000000000000000000000000000

1.414213562373095048801688724209698078570 -> 2.000000000000000000000000000000000000000

1.414213562373095048801688724209698078570 -> 2.000000000000000000000000000000000000000

1.414213562373095048801688724209698078570 -> 2.000000000000000000000000000000000000000

1.414213562373095048801688724209698078570 -> 2.000000000000000000000000000000000000000