In Japanese (and other East Asian languages), the denominator of a fraction is read as a part of the modifier to the numerator:

3分の5
'five that was divided by three'
'five thirds'

This indicates that East Asian languages interpret fractions as the result of division operations: "five divided by three" (which is a "modified five" rather than a "modified three").

On the other hand, most/many Western languages interpret fractions not in that way, but count how many unit fractions there are (which I remember reading in a book when I was a junior high-school student that it dates back to the ancient Egyptians):

five thirds
'there are five unit fractions whose denominator is three. I.e., (1/3) * 5'

How did these different groups of languages come to adopt these different thoughts in expressing fractions?

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I am interested, too. Maybe more suitable on linguistics.stackexchange.com? – Tsuyoshi Ito Dec 28 '11 at 12:38
@TsuyoshiIto Right. It may be off topic. If you vote for so, that is fine. – user458 Dec 28 '11 at 19:29
I am not sure if it is off topic, but I just thought that you may have better chance getting answers on linguistics.stackexchange.com because it seems to me that the knowledge about many different languages is needed to answer the question. – Tsuyoshi Ito Dec 28 '11 at 23:40

I suppose we can trace the East Asian system back to China. In the Chinese text Nine Chapters on the Mathematical Art 九章算術 (dating to 2nd C BCE at the latest if I understand correctly), we see this question and answer:

This looks like a likely precursor for the Japanese X分のY notation, with 之 simply translated の (which is of course extremely common). But I am not convinced that it actually says "twelve divided by eighteen" and "two divided by three". I would interpret it rather as "Twelve of eighteen parts" and "two of three parts" respectively.

In Christopher Cullen's translation of the Writings on Reckoning 筭數書, another Chinese text of similar vintage, his notes indicate that he agrees:

The topic of division naturally leads into that of fractions. When faced with the standard form of expression sān fēn zhī yī 三分之一, literally ‘one of three parts’ I have decided simply to write ⅓. It does seem reasonable to take the solidus line as relating 1 and 3 just as fēn zhī 分之 ‘of [ ] parts’ relates yī 一 and sān 三 in the reverse order. The term fēn 分 has been translated as ‘part’ rather than ‘fraction’; this seems closer to the usage of the text.

So I propose that your premise is incorrect: Egypt and China used essentially the same method to represent fractions. Your example is not "Five that was divided by three" but "five of three parts" -- which is obviously not an intuitive concept when you express it in terms of "parts" of an implicit (single) whole, since the result is more than one, but you can see how it could develop as an extension from terminology like "one of three parts".

One interesting way in which I could be incorrect is that all of the above is correct in terms of where the X分のY terminology came from, but most modern Japanese speakers (represented by you and Tsuyoshi) nevertheless do in fact interpret it as "Y divided by X" rather than "Y of X parts [of a whole]". In that case, the question would be, how did this reanalysis of the phrase arise in Japan? (And how about Korea and other nations in China's sphere of influence?) I bet the answer would be something to do with word order, in that case...

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Finally, I got an answer. Your historical consideration is convincing. I think you are correct. So, sometime during the history of Japanese (and perhaps other East Asian languages as well) reanalysis must have happened. – user458 Jan 27 '12 at 5:38
@Matt +1 for tracing it back to Chinese. Although I don't know much Chinese, I was able to look up that 分 can also be a verb meaning divide. Since I believe 之 is an old version of the relative-clause-ender 的, 三分之二 could be taken to mean 'the 2 that 3 divides', i.e. '2 divided by 3'. At this point, we're really outside the scope of this forum, but I'd still be interested in the opinion of any Chinese speakers. – dainichi Jan 27 '12 at 6:02
Yes, the 之 worries me too, but I figure if Cullen analyses it the same way it must at least be a possible interpretation. Maybe someone should venture out to Chinese.stackexchange.com and see if they have any classicists there. – Matt Jan 27 '12 at 6:15
I've always thought of 三分之一 as meaning "one out of three (parts)", but then again I have never quite internalised the reading of numbers in foreign languages... – Zhen Lin Jan 27 '12 at 7:52
@ZhenLin That interpretation cannot be applied beyond proper fractions. – user458 Jan 30 '12 at 21:09