Why is "quotient" translated as 「商」 in mathematics?
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「商」 (to measure > calculate, approximate, discuss > (1) quotient, (2) commerce) was originally comprised of 「辛」 (picture of a chisel or torture weapon > hardship, difficulty) and 「丙」 (picture of a workshop table), indicating trade professions involving measurement (e.g. of raw materials and dimensions). This is one of the earliest and most literal meanings of the word 「商量」.

「口」 was later added as a proper noun mark, referring to the name of the Shang Dynasty.

商金商婦甗集成867戰國・楚金曾侯乙鐘集成290秦簡日甲145睡虎地秦簡今楷 　


  Compare the similar addition of 「口」 (or its derivatives 「甘」 and 「曰」) in the characters 「周」 (Zhou Dynasty), 「曹」 (State of Cao), 「曾」 (State of Zeng), etc.


To measure was later extended to mean approximation or calculation, and then specialised to mean quotient, derived from its use as a kind of jargon in rod calculus.


  See a rod calculus tutorial on computing the square root of 234567. The top row in this calculation layout is called 「商」.




References:


季旭昇《說文新證》
小學堂
國學大師


郭沫若《甲骨文合集》
中國社會科學院考古研究所《殷周金文集成》