in arabic sites which is interested in maths , i find many topics like ,here is a proof that 0=2 .
and we answer that the proof is wrong as we can't divide by zero .
but i really wonder , why can't we divide by zero ?
i think the reason that mathematicians refused dividing by zero is that make us into a contradictions like $1=2=3=...$ and things like this , but those are facts about the physical world , why should mathematicians obey the outside world ?
i also have read a news in BBC about a new theorem which find special cases where we can divide by zero , but not details was mentioned i think , have any one had any idea about this ?
$\endgroup$ 73 Answers
$\begingroup$By definition, for any $x,y$ in a field, $\frac{x}{y}$ is the unique field element $z$ (if it exists) such that $zy=x$. If $y=0$ and $x\ne 0,$ then no such $z$ exists. If $x=0$ and $y=0,$ then we don't have uniqueness.
That's why we don't/can't define division by $0$.
Now, if we expand the definition of a field to allow $0=1$, then it is proved easily that $0$ is the only element of the field--making such a field a supremely uninteresting structure.
$\endgroup$ 4 $\begingroup$You are correct that division by zero results in statements such as $1 = 2 = 3 =\cdots$, but that is not just a statement about facts in the real world (or "physical world"): mathematics as we know it would fall apart if this is allowed. What, after all, can be said, mathematically, if we have that $1 = 2 = 3 = \cdots$?
Just consider the implications, as they are vast, if we division by zero defined, and hence "meaningful"...
How would you define division if you allow division by zero? See, e.g., this answer regarding division by zero, with respect to how we define division, and division as we know it would fail if we were permit division by zero.
Indeed, how would we define $0^{-1}$ so that $0\cdot 0^{-1} = 1$?
These questions are simple prompts to suggest that to allow/define division by zero would entail having to redefine all axioms of arithmetic, and the field axioms, and much more. I.e., any successful redefining and reconstruction of a consistent system which is also consistent with allowing division by zero would yield, as first suggested in the comments, a system which would be exceedingly uninteresting, even perhaps meaningless.
$\endgroup$ 9 $\begingroup$Please note that as you imply, mathematics in general is actually quite a free field in which where able to invent systems in which we can do pretty much anything we want so you could come up with a system in which you can divide by zero. However, usually these systems all have their own axioms and generally for good reason and the reason we can't normally divide by zero is, in a sense because of the axioms we use for normal arithmetic.
Really, division is just the opposite of multiplication. When I divide 6 by 2, I'm really just looking for the number that I multiply by 2 to get 6 (which is 3). However, say I wanted to divide 6 by 0. What number can I multiply by 0 to get 6???
I'd also like to take you back to how you would have first learnt about division (presumably) which is "how many lots of one number can I get into another?". When I ask what is 6 divided by 2, I'm asking how many 2's go into 6 and since $6=2+2+2$, the answer is 3. However, how many 0's go into 6? Even if I keep adding 0 to itself infinitely many times, I'm never going to get to 6, so the answer is undefined.
In short, yes we are free to make new systems with new rules that allow us to divide by zero, but in our current system it is just not sensible.
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