Wikipedia defines "quotient" as:

In arithmetic, a quotient (from Latin: quotiens "how many times", pronounced /ˈkwoʊʃənt/) is the quantity produced by the division of two numbers.[1]

Does the character 「商」 have the meaning of "how many times" from its origin?
Why is "quotient" translated as 「商」?


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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.

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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」.


  • "Measure" is actually a more apropos reference than one might think - and the relevant cognition appears in other cultures as well: e.g. if one looks to the classic Greek text by Euclid, the Elements, division of one number by another is called "measuring" the one by the other. A quotient, $a/b$, can be interpreted as "given a quantity $a$, measure it out in units of size $b$, and give its value in terms of those units". And this is also how all physical measurements work, too - in effect, you're taking the "quotient" of one physical object by another, reference Jan 30 '20 at 7:18
  • object (the unit), to yield a number. Jan 30 '20 at 7:19
  • Hence 商量, "measured quantity", is pretty much perfect: "the measured quantity [商量] formed by measuring a in units of b", is a/b. Jan 30 '20 at 7:20

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