I have seen multiple answers on the web, but I can't get my mind around why division by zero outputs an error and not zero. Can anyone explain this in laymen's terms?
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4I bet this is a duplicate. – Nov 15 '18 at 18:48
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On that note, see this post, this one, another one and one more – Nov 15 '18 at 18:51
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@Cody Rutscher Basically, without going into all of the details, division by zero doesn't make sense. It is a question that doesn't have an answer. What is the color of the wind? Undefined. – J. Moeller Nov 15 '18 at 19:00
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Why on earth would it be $0$??? If $\frac ab = x$ then that means by definition that $bx = a$. So if $\frac 50 = x$ would mean by definition that $0x = 5$. If $x = 0$ you get $00 = 5$ which is not true so $\frac 50 \ne 0$. You'll have to try something else. If $0*x = 5$ what is $x$. If you can answer that you are done. Can you answer that. – fleablood Nov 15 '18 at 19:03
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Well, let's try it. Let's divide $x$ by $0$ and say it's equal to some number $y$. Then
$$\frac x0=y.$$
By definition, we may multiply both sides by $0$, and see $x=y\cdot0=0$. This isn't true; we didn't assume $x$ to be zero. However, this tells us the only number we can divide by $0$ is $0$ itself; indeed, $\tfrac00=y$ doesn't result in a contradiction.
The problem with saying $\tfrac00$ is defined, is that it is everything at once. It doesn't have only one value. $\tfrac00=1$ wouldn't reach a contradiction (not by multiplying by $0$, that is) and neither would $\tfrac00=0$. In mathematics, we like things to only represent one thing and one thing only. If we would let $\tfrac00$ mean both $0$ and $1$ at the same time, then we wouldn't be able to use $=$ like we want; if we did, then
$$0=\frac00=1\implies 0=1$$
So, that would be a hassle for just letting this thing that is everything at once exist. Hence, we just agree to not divide by $0$, so we can keep all our rules, axioms, and theorems we know.