[syzarian]

Division is multiplication by the multiplicative inverse. Subtraction is addition by the additive inverse. Both division and subtraction undo their corresponding operation. Multiplying by a (provided it’s not zero) is undone by dividing by a. Adding a is undone by subtracting a.

In a ring the elements form a group under addition and thus every element has an additive inverse. The additive identity element, let’s call it e, has the property that ea = e and ae = e. For this reason we use 0 instead of e. In a nontrivial ring 0 can’t have a multiplicative inverse because if it did then every element would be equal to the multiplicative identity (which is unique).

[iamerroragent]

Okay so if you can get to a ring without a multiplicative inverse and then applying that operation to the ring forms it into a field then wouldn't it be fair to say that division is not really the opposite of multiplication the same way that subtraction absolutely is for addition?

[syzarian]

The definition of division is multiplication by the multiplicative inverse. It may be the case that some elements don’t have such an inverse but the definition is analogous to that of subtraction. The analogy is not perfect because every element has an additive inverse while not every element had a multiplicative inverse.

[topaz0]

What you're saying is that the analogy between subtraction and division is good as far as it goes. So why should "as far as it goes" end at zero not having an inverse, rather than division by zero producing something other than the multiplicative inverse of zero? The two choices end up having different structure, and so they end up being applicable to different things, but there is nothing wrong with either choice.

[syzarian]

The word division means something in mathematics. There is general agreement in what that word ought to mean. You can define a binary operation in such a way that it doesn’t look like what we normally think of as division and label your operation division. In the same way you can define the symbol duck to refer to what most people call a chair. You won’t get anyone else agreeing with your new definition though.

[topaz0]

We redefine multiplication for new contexts every day in math, I don't see why division should be any different. See also: https://en.wikipedia.org/wiki/Division_(mathematics)#Divisio...

[syzarian]

I think I understand better where you are coming from. In computer science I don’t know what they typically mean when they say “division”. I’ll be more precise. In abstract algebra division means multiplying by the inverse. All of the notions of division mentioned in the Wikipedia page come from this idea. Computers can’t work with within the realm of the entire real number system. There they have notions of type. They like to extend common operators like “/“ to things that normally it doesn’t apply to. A computer language will sometimes return a value of int or some other type when the integer 5 is divided by 3. Depending on how the language designer wanted things to work. This isn’t division in a mathematical sense though.