Why can't you divide by zero?
Division asks what you must multiply by to get back where you started, and multiplying by zero destroys that information — so the question either has no answer or has every answer.
Simple intuition
The plain reason, in everyday words
Dividing asks a multiplication question backwards. Six divided by two means: what times two gives six? The answer is three, and it is the only answer. Now try six divided by zero: what times zero gives six? Nothing does, because everything times zero is zero. So there is no answer at all. Then try zero divided by zero: what times zero gives zero? Everything does — one works, five works, a million works. Now there are too many answers and no reason to prefer any of them. So division by zero fails in two different ways: for any other numerator there is no candidate, and for zero there are infinitely many. Either way you cannot name a single value, and arithmetic requires that every operation give exactly one result. The prohibition is not stubbornness; there is simply nothing to put there.
Dividing by zero gives infinity.
Infinity is not a real number, and the quotient does not approach a single value: coming from above it grows without bound, from below it falls without bound. Some extended systems assign an unsigned infinity, but ordinary arithmetic cannot.
Zero divided by zero is one, because anything divided by itself is one.
That rule holds precisely because a non-zero number has a unique inverse. For zero, every number satisfies the defining equation, so no single answer can be selected.
It is just a convention mathematicians agreed on.
Assigning any value lets you derive 1 = 2 in a few lines of ordinary algebra. The prohibition is forced by consistency, not chosen for convenience.
Nothing can be done about it in any system.
The Riemann sphere and the projectively extended real line do define division by zero for non-zero numerators, and are used routinely. They still leave 0/0 undefined, and they give up some ordinary properties in exchange.
It is the cleanest example of a definition being forced rather than chosen: mathematicians did not decide to forbid it, they discovered that permitting it destroys everything else. Recognising the difference between undefined, infinite and indeterminate is also directly useful — it is the distinction between a program that crashes, one that returns infinity and one that returns NaN, and knowing which you are looking at tells you what went wrong.
Who worked it out
Brahmagupta, writing in 628, gave rules for arithmetic with zero and stumbled at division, asserting that zero divided by zero is zero — an answer later rejected.
What problem forced it
Bhāskara II in the twelfth century proposed that a quantity divided by zero is infinite, which captures the limiting behaviour but not a value, and the ambiguity persisted for centuries.
How it changed since
The nineteenth-century development of abstract algebra settled it: once division was defined through multiplicative inverses and fields were axiomatised, zero's lack of an inverse became a proved consequence rather than a matter of opinion. IEEE 754 later made a separate engineering decision for computation.
What indeterminate forms mean in calculus
0/0 as a limit is a genuinely useful expression, and it is a completely different statement from division by zero.
How the Riemann sphere handles infinity
The system where division by zero is defined, and what it gives up to get there.
Written for Curio rather than collected from a forum — it is part of the curated corpus that ships with the platform. The references it draws on are listed under Sources.