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WikiProject Mathematics (Rated Start-class, Mid-priority)
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 Field:  History of mathematics
WikiProject China / History (Rated Start-class, Mid-importance)
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Name of the ratios[edit]

The Chinese names '約率' means 'a rough/coarse ratio' and '密率' means 'an accurate/precise ratio' (but not detailed ratio). (talk) 06:10, 28 October 2015 (UTC)


The first sentence in this article is a Garden path sentence. -- 19:46, 31 January 2007 (UTC)

Are you sure? Can we get a senior Wikipedian's opinion on this? I mean..I just want to be sure you're right before we change anything, you know? -- 19:48, 31 January 2007 (UTC)

I agree. Shouldn't the word Milü apear in the first sentence? I would make it "Milü is an approximation of pi, and is given by 355⁄113."Fegor 21:59, 30 May 2007 (UTC)

Warning Possible Sock-Puppets[edit]

The two posters above me posted in a highly convenient fashion which leads me to suspect they are actually the same person, in direct contravention of WP:SOCK. Using a simple IP Proxy, these "two" people who posted the comments above me could in fact be ONE! Paging an admin to this page! -- 02:03, 1 February 2007 (UTC)

Incorrect math?[edit]

Not sure about that 10399333102. Isn't 5216316604 a better approximation than 335113 with five digits instead of six, and 10434833215 a better approximation than 10399333102 with equal number of digits? Both Excel and Matlab think so. Piet | Talk 11:03, 22 July 2008 (UTC)

According to my calculator, the 3 digit one is marginally better if you calculate it in terms of the ratio of the approximation to the actual value. Were you doing it in terms of the difference? I think ratio is better. --Tango (talk) 16:56, 22 July 2008 (UTC)
It doesn't matter whether you look at the absolute difference or the ratio of the approximation. Piet is correct - the next "best rational approximation" to π after 335113 is 5216316604. The mistake in this article has arisen from a misunderstanding of the relationship between continued fraction convergents and best rational approximations. The continued fraction convergents of a real number will all be best rational approximations (i.e. there is no better rational approximation with a smaller denominator). However, there may also be other intermediate best rational approximations i.e. the continued fraction convergents do not exhaust the best rational convergents. In particular, if there is an even term in the continued fraction expansion, then the rational number obtained by truncating the continued fraction expansion at this term and then halving the final term may be a best rational approximation that is not a continued fraction convergent. This is what happens with π. We have 335113 = [3; 7, 15, 1], 10399333102 = [3; 7, 15, 1, 292] and 10434833215 = [3; 7, 15, 1, 292, 1] which are successive continued fraction convergents for π, but we also have 5216316604 = [3; 7, 15, 1, 146] (note that 146 is half of 292), which is a best rational approximation but not a continued fraction convergent. Thus:

Other best rational approximations which are not c.f. convergents include 5251816717 = [3; 7, 15, 1, 147], 5287316830 = [3; 7, 15, 1, 148] etc. - see this list. I have corrected the article. Gandalf61 (talk) 11:24, 23 July 2008 (UTC)
Thanks to Gandalf61 for fixing this. JRSpriggs (talk) 00:33, 24 July 2008 (UTC)


It wouldn't hurt to know whether the Japanese mathematician referred to here is a currently living professor under the age of 25 or someone who lived centuries ago. This article is apparently only about history, yet it neglect such an obvious point. Michael Hardy (talk) 04:12, 10 December 2008 (UTC)

I'd also like to know the reconstructed pronunciation of 密率 in Zŭ's lifetime. —Tamfang (talk) 22:51, 10 June 2014 (UTC)


Yeah, animation it includes 47/15 and others! Preceding unsign comment: ( )