Remainders from roots

I described how multisets can be divided to get a quotient and a remainder. This generalizes the process of division of a number, which is actually equivalent to dividing a equal multiset. The process of dividing a multiset can be generalized to other multisets though, which produces a different process. The first immediate thought is that this can be applied to prime factorizations. So for example if we prime factorize 24 we get {2,2,2,3} and if we divide it 2 we get {2} with remainder {2,3}. This division of the prime factorization multiset is essentially the same as taking roots.

So if we take the square root of 24 we get 2 with a remainder of 6, and it is expressed as $2\sqrt{6}$. I particularly like the number 360 because of its unique minimal prime factorization {2,2,2,3,3,5} which forms a progression multiset. If we divide this by two we get $6$ with a remainder of 10 so it is $6\sqrt{10}$. In the case of a cube root we get $2$ with a remainder of $45$ so $2 \sqrt[3]{45}$. In any case, the remainder is the object still in the root symbol and the quotient is the part which is not.

Ordinary division is essentially additive division so when computing division the remainder is added to the quotient as a fraction and the introduction of fractions is what distinguishes the result from the ordinary integers. Roots are multiplicative division so the remainder is multiplied by the quotient, rather then added to it. The remainder is then the algebraic part that distinguishes it from the other rational part.

I have seen the remainder be used to refer to the fractional part of a root, so the square root of 24 would then be 4 with a remainder of 0.898979... going on infinitely in a non-periodic manner. It is useful to consider 4 as the square root of the smallest square number less then the number, but it is wrong to consider 0.898979... to be the remainder. Instead this is the fractional part of the root. This is largely an issue of terminology, but it is interesting nonetheless.