Editing Square root of 2 (section)

==Decimal value==
===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>

===Rational approximations===
A simple rational approximation {{sfrac|99|70}} (≈ '''1.4142'''857) is sometimes used. Despite having a denominator of only 70, it differs from the correct value by less than {{sfrac|1|10,000}} (approx. {{val|+0.72e-4}}). 

The next two better rational approximations are {{sfrac|140|99}} (≈ '''1.414'''1414...) with a marginally smaller error (approx. {{val|-0.72e-4}}), and {{sfrac|239|169}} (≈ '''1.4142'''012) with an error of approx {{val|-0.12e-4}}.

The rational approximation of the square root of two derived from four iterations of the Babylonian method after starting with {{math|''a''<sub>0</sub> {{=}} 1}} ({{sfrac|665,857|470,832}}) is too large by about {{val|1.6e-12}}; its square is ≈&thinsp;{{val|2.0000000000045}}.

===Records in computation===
In 1997, the value of <math>\sqrt{2}</math> was calculated to 137,438,953,444 decimal places by [[Yasumasa Kanada]]'s team. In February 2006, the record for the calculation of <math>\sqrt{2}</math> was eclipsed with the use of a home computer. Shigeru Kondo calculated one [[Trillion (short scale)|trillion]] decimal places in 2010.<ref>{{citation |url=http://numbers.computation.free.fr/Constants/Miscellaneous/Records.html |title=Constants and Records of Computation |publisher=Numbers.computation.free.fr |date=2010-08-12 |access-date=2012-09-07 |url-status=live |archive-url=https://web.archive.org/web/20120301190937/http://numbers.computation.free.fr/Constants/Miscellaneous/Records.html |archive-date=2012-03-01 }}</ref> Other [[mathematical constant]]s whose decimal expansions have been calculated to similarly high precision include [[pi|{{pi}}]], [[e (mathematical constant)|{{mvar|e}}]], and the [[golden ratio]].<ref name="y-cruncher">{{citation |url=http://www.numberworld.org/y-cruncher/records.html |title=Records set by y-cruncher |access-date=2022-04-07 |url-status=live |archive-url=https://web.archive.org/web/20220407052022/http://www.numberworld.org/y-cruncher/records.html |archive-date=2022-04-07 }}</ref> Such computations provide empirical evidence of whether these numbers are [[normal number|normal]].

This is a table of recent records in calculating the digits of <math>\sqrt{2}</math>.<ref name="y-cruncher" />
{| class="wikitable sortable"
|-
! data-sort-type="usLongDate" | Date !! Name !! data-sort-type="number" | Number of digits
|-
|style="text-align:right;" | 4 April 2025 || data-sort-value="H" | Teck Por Lim || style="text-align:right;" | {{val|24000000000000}}
|-
|style="text-align:right;" | 26 December 2023 || data-sort-value="H" | Jordan Ranous || style="text-align:right;" | {{val|20000000000000}}
|-
|style="text-align:right;" | 5 January 2022 || data-sort-value="H" | Tizian Hanselmann || style="text-align:right;" | {{val|10000000001000}}
|-
|style="text-align:right;" | 28 June 2016 || data-sort-value="W" | Ron Watkins || style="text-align:right;" | {{val|10000000000000}}
|-
|style="text-align:right;" | 3 April 2016 || data-sort-value="W" | Ron Watkins || style="text-align:right;" | {{val|5000000000000}}
|-
|style="text-align:right;" | 20 January 2016 || data-sort-value="W" | Ron Watkins || style="text-align:right;" | {{val|2000000000100}}
|-
|style="text-align:right;" | 9 February 2012 || data-sort-value="Y" | Alexander Yee || style="text-align:right;" | {{val|2000000000050}}
|-
|style="text-align:right;" | 22 March 2010 || data-sort-value="K" | Shigeru Kondo || style="text-align:right;" | {{val|1000000000000}}
|}