<story><title>The Mystery of 355/113</title><url>http://davidbau.com/archives/2010/03/14/the_mystery_of_355113.html</url></story><parent_chain></parent_chain><comment><author>CoreDumpling</author><text>In Chinese the fraction 355/113 has been known as the 密率 (&quot;detailed ratio&quot;) and 22/7 as the 约率 (&quot;approximate ratio&quot;).  It is documented in the Book of Sui, volume 16:&lt;p&gt;宋末，南徐州從事史祖沖之，更開密法，以圓徑一億為一丈，圓周盈數三丈一尺四寸一分五厘九毫二秒七忽，朒數三丈一尺四寸一分五厘九毫二秒六忽，正數在盈朒二限之間。密率，圓徑一百一十三，圓周三百五十五。約率，圓徑七，週二十二。 [1]&lt;p&gt;While it suggests that the value was calculated around the end of the Liu Song Dynasty (~479 CE) by Zu Chongzhi to have an upper bound of 3.1415927 and lower bound of 3.1415926, the 密法 method is not explained.&lt;p&gt;It has been thought that Zu may have used a &quot;method of averaging days&quot; to calculate an intermediate value between two prior known approximations of 22/7 and 157/50 [2].  Not a very satisfying result for anyone who wants to find something special in the numbers 355 and 113.&lt;p&gt;[1] &lt;a href=&quot;https://zh.wikisource.org/wiki/%E9%9A%8B%E6%9B%B8/%E5%8D%B716&quot; rel=&quot;nofollow&quot;&gt;https://zh.wikisource.org/wiki/%E9%9A%8B%E6%9B%B8/%E5%8D%B71...&lt;/a&gt;&lt;p&gt;[2] &lt;a href=&quot;http://books.google.com/books?id=QlbzjN_5pDoC&amp;#38;lpg=PA34&amp;#38;ots=bLTzciZHz4&amp;#38;dq=zu%20chongzhi&amp;#38;pg=PA30#v=onepage&amp;#38;q&amp;#38;f=false&quot; rel=&quot;nofollow&quot;&gt;http://books.google.com/books?id=QlbzjN_5pDoC&amp;#38;lpg=PA34&amp;#...&lt;/a&gt;</text></comment>
<story><title>The Mystery of 355/113</title><url>http://davidbau.com/archives/2010/03/14/the_mystery_of_355113.html</url></story><parent_chain></parent_chain><comment><author>StefanKarpinski</author><text>There is no mystery here. If you pick a random real number and find the best quality approximation of it — as calculated by log_b 1/|x-a/b| — then the median quality of the best such approximations within a tolerance of 1/1024 is around 3.19. So with respect to the measure this article is talking about, pi is completely normal and 355/113 is a completely unexceptional best approximation with a denominator of at most nine binary digits (about 3 decimal digits). 355/113 isn&apos;t even the best quality approximation of pi by the given criterion — that would be 22/7, which, as mentioned in the article, has a quality measure of 3.429288337281781.&lt;p&gt;Julia code to figure this out can be found here:&lt;p&gt;&lt;a href=&quot;https://gist.github.com/3170899&quot; rel=&quot;nofollow&quot;&gt;https://gist.github.com/3170899&lt;/a&gt;</text></comment>