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An algorithm for computing the ratio of two positive numbers is the following: determine the number of times that the divisor can be subtracted from the numerator, until the accumulated effect of the subtraction operations produces a value less than the denominator. So, to obtain the ratio (i.e. quotient), we perform an arithmetical operation (specifically, subtraction) and also a comparison (to find whether the net result of subtraction is less than the divisor or not).

In this sense, is the division algorithm that was described above purely arithmetical, or is it a combination of an arithmetical operation and a comparison operation?

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    $\begingroup$ This has nothing whatsoever to do with division-algebras. Please read the tag descriptions before using them. If the tag description is all Greek to you, don't use it. It is guaranteed not to be appropriate. $\endgroup$ – Jyrki Lahtonen Nov 16 '19 at 8:02
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    $\begingroup$ @Jyrki Lahtonen: If the tag description is all Greek to you, don't use it. It is guaranteed not to be appropriate. --- I see this type of inappropriate tag assignment very often and I've had an opinion what was behind it that is exactly what you said, but I never managed to verbalize my feeling anywhere close to how well you just now verbalized my feeling. The same thing happens in mathoverflow. For example, someone asks a question about how to get a common denominator of some two- or three-digit integers (explicitly given), (continued) $\endgroup$ – Dave L. Renfro Nov 16 '19 at 8:26
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    $\begingroup$ and I'm left wondering how they thought their question might have been appropriate when it was obvious to me that essentially everything asked there would have been totally Greek to them. $\endgroup$ – Dave L. Renfro Nov 16 '19 at 8:26
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I think most people would consider division a "basic arithmetic operation" in its own right. However, let's say we don't: what distinctions can we draw?

One precise sense in which division is not "arithmetically built from" addition, multiplication, and subtraction (say) is that there is no way to compose those functions to produce division. This leads into the idea of distinguishing between kinds of definability, which is a fundamental concept in logic and universal algebra. In the relevant jargon we could say for example that in $(\mathbb{R}; +,\times,-)$ division is definable but not term definable.

(Note that in the above I'm talking about exact division of real numbers, in contrast with your OP, but the result holds whichever notion we use.)

Another sense in which division isn't "purely arithmetic" is that (again, in the context of $\mathbb{R}$) it is partial: division by zero isn't allowed (while by contrast addition, subtraction, and multiplication are total). This winds up having universal-algebraic consequences: fields do not form a variety - roughly, this means that the basic axioms of division can't be boiled down to simple equations.

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