"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?
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:
商 - 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.