Why is dividing by zero undefined rather than infinity? Nobody ever explained why infinity is wrong

The clearest way in is to remember what division is asking.

Ten divided by two asks: what number multiplied by two gives ten? Five. Division is defined as the inverse of multiplication, and the answer is whatever fills that blank.

So ten divided by zero asks: what number multiplied by zero gives ten?

And there is no such number, because anything multiplied by zero is zero. Not a hard number to find, there is nothing that satisfies the question. That is precisely what undefined means: the question has no answer, so the expression names nothing.

Now try zero divided by zero: what number multiplied by zero gives zero? Every number does. So this one has no unique answer, which is a different failure and why it gets its own name.

That is the whole thing, and notice it needs no limits, no infinity and no advanced machinery. The operation is defined by a question, and for zero the question either has no answers or has all of them.

Your infinity intuition is not silly, it is describing something real about limits, which is the next answer: but infinity is not a number that can go in that blank, so it cannot be the answer to this question.

Undefined means no answer exists, not that the answer is too big to write down. Those get confused constantly.

A calculator saying error is reporting the same thing, just rudely.

On why infinity specifically fails, your intuition is worth taking seriously because it is half right, and the half that is wrong is easy to see.

You divided ten by smaller and smaller positive numbers and the result grew without bound. True.

Now do it from the other side. Divide ten by numbers approaching zero from below, minus one, minus a tenth, minus a hundredth. The results are minus ten, minus a hundred, minus a thousand. It grows without bound in the negative direction.

So approaching zero from one side heads to positive infinity and from the other to negative infinity. There is no single value it is heading towards, which means no consistent answer exists even in the limiting sense.

That is the specific reason infinity is rejected: not that it is too big, but that the two sides disagree.

This is also why the distinction between undefined and infinite matters. Something like one divided by zero squared does head to positive infinity from both sides, and mathematicians are comfortable saying the limit is infinite there - while still saying the expression itself is undefined, because infinity is not a number you can do arithmetic with.

The other reason, which is the one that settles it: if you define it anyway, ordinary arithmetic collapses.

Suppose we declare that one divided by zero equals some value, call it k. Then by the definition of division, k times zero must equal one. But k times zero is zero for every number k. So zero equals one.

And once zero equals one, everything else follows - you can prove any number equals any other, and arithmetic stops distinguishing anything. That is not an inconvenience, it is the whole structure failing.

So the prohibition is not squeamishness. It is that the alternative destroys the system.

Worth knowing that mathematicians have built structures where something like this is allowed - the extended number line, and the projective line where there is a single point at infinity that both directions reach. Those are consistent and useful, particularly in geometry and complex analysis.

What they give up is arithmetic. In those systems certain expressions remain undefined, and the ordinary rules do not all hold. So you can have division by zero, or you can have arithmetic that works, and for everyday purposes the second is worth far more.

That is the honest answer to why nobody explained it: the real reason is that it is a design decision with a cost, not a fact about zero.