Editing Square root of 2 (section)

===Computation algorithms===
{{Further|Methods of computing square roots}}
There are many [[algorithm]]s for approximating <math>\sqrt{2}</math> as a ratio of [[integer]]s or as a decimal. The most common algorithm for this, which is used as a basis in many computers and calculators, is the [[Babylonian method]]<ref>Although the term "Babylonian method" is common in modern usage, there is no direct evidence showing how the Babylonians computed the approximation of <math>\sqrt{2}</math> seen on tablet YBC&nbsp;7289. Fowler and Robson offer informed and detailed conjectures.<br />Fowler and Robson, p. 376. Flannery, p. 32, 158.</ref> for computing square roots, an example of [[Newton's method]] for computing roots of arbitrary functions. It goes as follows:

First, pick a guess, <math>a_0 > 0</math>; the value of the guess affects only how many iterations are required to reach an approximation of a certain accuracy. Then, using that guess, iterate through the following [[recursion|recursive]] computation:

:<math>a_{n+1} = \frac12\left(a_n + \dfrac{2}{a_n}\right)=\frac{a_n}{2}+\frac{1}{a_n}. </math>

Each iteration improves the approximation, roughly doubling the number of correct digits. Starting with <math>a_0=1</math>, the subsequent iterations yield:

:<math>\begin{alignat}{3}
a_1 &= \tfrac{3}{2} &&= \mathbf{1}.5, \\
a_2 &= \tfrac{17}{12} &&= \mathbf{1.41}6\ldots, \\
a_3 &= \tfrac{577}{408} &&= \mathbf{1.41421}5\ldots, \\
a_4 &= \tfrac{665857}{470832} &&= \mathbf{1.41421356237}46\ldots, \\
    &\qquad \vdots
\end{alignat}</math>