Whether you are referring strictly to the representational and quantifying function which, after all, is the purest and ultimate purpose of mathematics - the language of abstract measurement, or the hypothesizing that it can describe some physical or tangible event, division by zero is an impossibility.
In both worlds, it is at best, a redundancy. Division by zero means, rather literally, not dividing. Whereas division by one is a validating operation - it results in and identifies the thing in question, division by zero leaves the function and/or item "in limbo". It may SEEM like dividing by zero, being essentially, NOT dividing, would give the same results as, and therefore be identical to division by 1, but it is precisely that very consequence which must invalidate one of them. For, certainly, to let both numerical concepts "zero" and "one" mean the same thing would mean to assign them the same value. It would make them identical.
It would render them equal!
In mathematics, there exists the concept of rational and irrational numbers. Rational numbers are those that can be expressed exactly. All decimals are rational. But, being thus, it is ironically impossible to represent all that is with decimals alone. Fractions can represent every possibility of existing and not existing, and the entire numerical and measured representation of all proportions thereof.
Were zero to equal one, and one to equal zero, the world - the whole vastness of all that is, was, and *never* will be, in both space AND time would thus be undefined in the literary sense first, and oh yes, in the abstract and measuring function as well.
If zero were to equal one, how would we differentiate being from NOT being? That right there is the main property that in fact defines both. No math is needed at this purest level to measure this. In this instance, I am refering to a difference that is not quantitative - but QUALITATIVE! One expresses being, the other expresses NOT being!
That was the birth of the controversy surrounding "zero" at ITS birth. When, in the oldest of human and mathematical worlds, the need arose for expressing and measuring the concept of "there's nothing here" - zero showed up, to make sure that, literally, "nothing" *could* exist.
In the pre-historic world (before the time of marking events in records or language - in *representational* and, theoretically, permanent form) there was no need to express nothingness. There was no past that could be discerned or detected by the human mind of that time. There was only now. Thus, what *is* is all that matters. But, as our modern minds are now prone to take for granted, you must have SOME way to remember and record and HOLD ON to things of life. In fact, our rather materialistic nature speaks to this need to want to hold onto a thing and thus subconsciously validate that it once did and will again not exist. It is this futile grasp for certainty and permanence that compels this behavior. Such behavior, and the mindset behind it, is irrational!
Referring back to the concept of fractions, we have a way of expressing and representing all proportions and exponential relationships that are the "verbs" of life. The stuff are the nouns. The "stuff" is represented by digits. The action is represented by function. Fractions are in fact, an action. They are division. One half is one over two, or, literally, one divided by two.
Division is an action. And you have to exist to perform an action - to DO! Zero represents the concept of "nothing here". This is why it was so controversial in human history. Ironically, in pre-historic history, it was literally, invisible. It was the 800 pound gorilla in the room that had no shape or form. It could be ignored. It was purely abstract.
But wait a minute! How much does this gorilla weigh? 800 pounds. How would I express that quantity without zero? Let me try. Does 8 pounds describe the weight of a gorilla? Nope. Does 811 accurately represent eight hundred? Nope. This is the best illustration of the need for this thing called a "place holder" in math - a way to represent that this spot contains nothing, but that the spot itself still exists.
We NEED to be able to express nothingness. And to fully understand and appreciate what that means - the humungous revelation that is this idea, we must let it mean "not there" in every instance. Something either exists or it doesn't, right? In fact, let me right now assert one of the very important principles in the language of measurement, that is, boundaries, by offering a pre-emptive warning: I of course am not intending to express an "absolutism" that can apply to the intangibles of humanity and life itself. This "exists or doesn't exist" property surely cannot be applied to love and relationships and politics and humanity and good and bad.
Not all of life, however, can be so simply reduced to either existing or not existing. That would render equal meaning to quality and quantity. How boring. And how unfathomable. Thus, let me go back to the realization that fractions are division, and that divisioni is ACTION. You cannot perform an action if you exist not! Again - that would mean that 1 and 0 are the same! This is why something divided by one and something divided by zero MUST show and have different results. Dividing by one is validating. It results in the whole item or concept as the "answer" or result. Then there's dividing by 2 and three and four and so on, resulting in more but smaller pieces of the whole the further out you go. These all exist.
But in order for that to mean anything, existing must be defined in order to BE SOMETHING. What then is "undefined". What does "something divided by nothing" mean if "something" divided by one" results in "something"?
Something divided by nothing means nothing? Of course not. It is not divided, thus it exists. And it is not 1, because that result comes from dividing a thing by itself. What, then, is the answer?
Something is. It is a thing. It exists. It is ONE. What then, isn't? Zero! That is its definition, which necessarily entails all of the properties contained IN that definition. That is precisely my point. This clear definition HELPS us live. It HELPS us understand because it allows us to give form to IDEAS! In order for ideas to exist, there needs to be a way to draw lines that separate this from that. We are free NOT to draw lines if that is our preference (or error!), but the ability is vital to expressing and understanding everything.
The only way to do that is to acknowledge the existence of nothing.