By this line of reasoning, isn’t 7/7 also zero? If you’re using the word “add” to talk about division you’re thinking about it wrong. Division is the inverse of multiplication, it has nothing to do with addition. I’ll give my standard answer for questions about 0/0: Division, as the inverse of multiplication, can be thought of as a question. That question is “what do I need to multiply this by to get that?”. For example, when you say “8/2”, you are asking what number you need to multiply 2 by to get 8, the answer of course being 4. With 0/0, you are asking the question “what number do I need to multiply 0 by to get 0”? 0 would work. But so would 1, 2, π, any other number, because (almost by definition) any number multiplied by 0 is 0. 0/0 is the only “division question” with multiple answers, all just as good as each other without further context. And in contrast when you ask the question “1/0” (or 2/0, etc), there is no answer. No number yields 1 when multiplied by 0. More on reddit.com

Several years ago I was bored and so for amusement I wrote out a proof that $\dfrac00$ does not equal $1$. I began by assuming that $\dfrac00$ does equal $1$ and then was eventually able to deduce ... More on math.stackexchange.com

0 goes into 0 1 time, so 0 divided by 0 is 1, not 0. Discuss. Debate. Ignore. I really don't care. More on tinychan.net

It's is totally okay to divide any number except zero by zero. This is done all the time by mathematicians and you get x/0=infinity, for any nonzero x. To do this properly, you have to link the +infinity and -infinity in calculus into one thing, this wraps up the number line into a circle. The resulting object is the Projective Real Line.

What you can't do is 0/0. This is what is undefined. There are many reasons for this. For instance, if we investigate things that do exist but seem to get into a 0/0 form, we find that this should be different values. For instance, if f(x)=x/x, then you can't evaluate this at zero. But every other point, except zero, results in 1. If we extrapolate to x=0, you would deduce that you "should" get 0/0=1. On the other hand, if we play the same game with g(x)=2x/x, then we'll find that our extrapolations say that we "should" get 0/0=2. So if 0/0 were to have a value, it would have to simultaneously 1 and 2. In fact, it would have to simultaneously equal every value, finite and infinite. This can't happen, because we know that 1 is, in fact, not equal to 2.

Furthermore, there are a lot of false proofs that 1=2. They generally have one thing in common, at some point you get an equation like 2x(x-x)=x(x-x), then you divide through by x-x to get 2x=x, or 2=1. But x-x=0, so canceling out the x-x is the same as saying that 0/0=1. In fact, if 0/0 were any finite value, you could prove the 1=2. So 0/0 can't have a value. Therefore 0/0 is undefined. Note that this proof isn't incorrect because you divide through by zero, it's incorrect because you implicitly assign a value to 0/0. You can divide through by zero, but you can't use division by zero to cancel out multiplication by zero.

So, division by zero is totally fine, as long as what you're dividing is not zero. It is zero divided by zero that is undefined and where problems arise.

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