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since I can remember, I've been told that $0 / 0$ is not allowed because division by zero is not defined, but $0|0$ is okay because it means there exist a $k$ such a that $0 = k$ * $0$. and $k$ can be any number. but now, in a new book, I've read that $0|0$ is not allowed for the same reason of $ 0 / 0$

finally, is $0|0$ allowed or not???

PM.
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Fatemeh Karimi
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    It is only a matter of notation. It depends entirely on the author's wish. – ajotatxe Apr 21 '17 at 15:55
  • There is no supreme body specifying what is and is not allowed in mathematics. There are many things that most people agree on but in other cases, you need to check what the author means. You have discovered something that not all agree on. There are many more. – badjohn Apr 21 '17 at 15:57
  • There are subtle differences. The sentences "There is an integer k so that a$$k =b" and "$\frac ba$ is an integer" "a is a factor of b" and "there is a unique integer k so the a$$k=b" are different and have different consequences but could each be a definition of a|b. I personally prefer the first and 0|0, every a|0 and 0 divides no b but 0. – fleablood Apr 21 '17 at 16:38

3 Answers3

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Allowed by whom? The math cops?

You get to pick the definitions that are most useful to you for the work you are trying to do. Some authors might make different decisions than others. Certainly it is reasonable to define divisibility by: $$a\vert b \Leftrightarrow \exists c\, b = ac$$ in which case $0\vert 0$. You could also change "there exists" to "there exists a unique," if for some reason you wanted to forbid this case.

Are there any reasons to prefer forbidding $0\vert 0$? I suppose that zero dividing zero complicates the cancelation law, $$ac\vert bc \Rightarrow a\vert b.$$ But it's not that important of a decision, in the grand scheme of things.

user7530
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I think what you're really touching on is the difference between $0\mid 0$ and $\frac{0}{0}.$

The key distinction is that $0\mid 0$ because $0=k0,$ meaning that you can write $0$ as a product of $0$ and some number $k$, and that is the definition of divisible.

On the other hand $\frac{0}{0}$ is undefined, in the traditional sense, because if you allowed this to be a fraction, then you make every ratio $\frac{a}{b}=0.$ Proof: Since $$\frac{a}{b}=\frac{c}{d}\iff ad=bc,$$ then it follows that $\frac{a}{b}=\frac{0}{0},$ since $0a=0b.$ So we only need to show that $\frac{0}{0}=0,$ but this is clearly true, since $0^2=1\cdot 0,$ and so $\frac{0}{1}=\frac{0}{0}=0,$ and we have proved the claim.

Chickenmancer
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Yes, $0 \mid 0$. You even have $\gcd(0,0) = 0$.

Aside: note that the "greatest" part of "greatest common divisor" means greatest in terms of divisibility, not greatest in terms of magnitude.

The only sense in which $0 \mid 0$ is not "allowed" would be if, for example, the subject of discussion was the divisibility among positive integers. Since $0$ is not a positive integer, it doesn't belong in such a discussion. (at least, not until the subject of discussion gets expanded to the more general context)