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"商数" is usually translated as "quotient" in both its mathematical and common usage forms (eg in "智商"/intelligence quotient, which is in fact an actual quotient.).

What is the etymology of "商", and why was it used to describe fractional numbers?

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    Just guessing, but commerce and mathematics were always closely related in all parts of the world, as keeping balances and tracking sales provided mathematicians with problems to be solved. It is likely that fractions in ancient China were first used in the context of commerce, hence the name. – Drunken Master Jun 7 '15 at 10:59
  • this is more of 商 = quotient, why? – user3306356 Jun 7 '15 at 12:57
  • might want to check the 算经十书- if it's not in there then it's probably a japanese import – Master Sparkles Jun 7 '15 at 17:39
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    a little confused - couldn't that also mean that the japanese meaning migrated back into chinese relatively recently? – Master Sparkles Jun 7 '15 at 23:56
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    evidence that the use is classical, from 九章算术 at least:"所得商數與左上一相生(乘)"(nknu.edu.tw/~jingxue/download/international2/09.pdf) – Master Sparkles Jun 8 '15 at 0:22
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This is not an easy question. But I think here is an answer.

It originated from the Chinese water clock or clepsydra in the ancient time (刻漏 or 漏壶, http://baike.baidu.com/view/41631.htm). 刻漏 or 漏壶 was a leaky water container, where the water level represents time. 商 was originally the scale plate on this type of water clock. The scale first had 100 grades (later 120 grades). So it roughly meant 'percentage' at some point and then the word 商数 or 商 began to mean 'quotient'. Because its ancient meaning (a scale plate on a water clock) is definitely deprecated, it is hard to see why 商数 means quotient. It has nothing to do with dividend or business numbers. :D

Some reference is below:

《集韵·阳韵》:“商,刻也.”《正字通·口部》:“商乃漏箭所刻之处.”《仪礼·士昏礼》解题引郑玄《目录》:“日入三商为昏.”贾公彦疏:“商为商量,是刻漏之名.故《三光灵曜》亦日入三刻为昏.”准此,可以断定,商之本义是古代滴漏上计时的刻度.商的其他字义皆由此引申而来.《说文》:“漏以铜受水,刻节,昼夜百刻.”《汉书·哀帝纪》:“漏刻以百二十为度.注:“旧漏昼夜其百刻,今增其二十.”可知刻箭上的度数有以百为率.每支刻箭一百度,每一度都是相等的,等于说把一支刻箭用一百来除,得出每一度的长度,这就叫商.算术中除法运算的得数叫商,是由漏箭的刻度引申而来.《广雅·释诂》:“商,常也.”《广韵·阳韵》:“商,常也.”常即经常,恒定不变,其义来自漏箭的刻度是一个恒定的常数.做生意,做买卖叫商,如“商品”、“商业”,此义也由商为漏箭刻度引申而来.《广雅》:“商,度也.”《汉书·沟洫志》:“能商功利”.颜师古注:“商,度也.”商有度量义,商量义,人与人之间的买卖关系要遵守等价交换的原则,交换双方都要度量对方货物的价值与己方货物的价值是否相当,所以,商又用为商业、商品、商人,仍为漏箭刻度的引申义. (http://zuoye.baidu.com/question/6b86ea97bf3190c0c06615f0b9c852cb.html?fr=iks&word=%B3%FD%B7%A8%BD%E1%B9%FB%CE%AA%CA%B2%C3%B4%BD%D0%C9%CC&ie=gbk)

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- is a weighing-machine, a scales, a weigher. Mostly seen in merchants' shops thus "commerce and shop" connotation. Scales were used to weigh matter. Weighed matter was separated from the rest of the bulk (a quotient of 5 kilos for example) and then sold.

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