Reading fractions


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?




I suppose we can trace the East Asian system back to China. In the Chinese
text [Nine Chapters on the Mathematical
Art](http://en.wikipedia.org/wiki/The_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:

> 今有 **十八分之十二** 。問、約之得幾何？  
> 答曰、 **三分之二** 。

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](http://www.nri.org.uk/suanshushu.html)
of the [Writings on
Reckoning](http://en.wikipedia.org/wiki/Su%C3%A0n_sh%C3%B9_sh%C5%AB) 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...




I suppose we can trace the East Asian system back to China. In the Chinese
text [Nine Chapters on the Mathematical
Art](http://en.wikipedia.org/wiki/The_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:

> 今有 **十八分之十二** 。問、約之得幾何？  
> 答曰、 **三分之二** 。

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](http://www.nri.org.uk/suanshushu.html)
of the [Writings on
Reckoning](http://en.wikipedia.org/wiki/Su%C3%A0n_sh%C3%B9_sh%C5%AB) 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...

