Is division purely an arithmetic operator? Is it not the combination of arithmetic operation and relational operation? 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?
 A: 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.
