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"divided by $x$" means "times $x$ inverse". "$x$ inverse" is the number such that $x$ times $x$ inverse = 1. 0 times anything = 0. 0 times nothing = 1. Nothing is 0 inverse. "0 inverse" does not refer to anything. "$x$ times 0 inverse" does not refer to anything. "x divided by 0" does not refer to anything.


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Long division is a useful thing to teach because at some point, polynomial long division is going to be something that someone is going to want to teach them, and it helps if they've seen something similar before. Thus, we don't want to replace it with anything. The question, then, becomes whether we want to add a second algorithm, and I can't see any ...


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Adding another view to the point, one could view a fraction $\dfrac{a}{b}$, as a symbol indicating a change of measure unit. More precisely, talking about integers, at first, what does e.g. $7$ mean? It is $7$ of some unit/unitary quantity. For instance, $7$ cats, or $7$ pies etc. This can lead us to introduce the notion of $\dfrac{7}{1}$, as the symbol that ...


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