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Does not $0\mid 0$ contradicts that divisibility by zero is undefined?

Could anyone clarify this for me please?

Bill Dubuque
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Emptymind
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    Where in the definition of the symbol "|" do you see a quotient? –  Sep 26 '17 at 01:54
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    No. In some sense it is just a slightly unfortunate choice of words. – Qiaochu Yuan Sep 26 '17 at 01:57
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    Divisibility by zero is well-defined, and $0$ is the only integer divisible by zero. Division by zero is undefined. –  Sep 26 '17 at 01:58

2 Answers2

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In general, we have $x|y$ when there is some integer $z$ such that $x \cdot z = y$. If there is exactly one such number $z$, then we can define the quotient $\frac{y}{x}$ as $z$.

We have $0|0$ because there does exist an integer $z$ such that $0 \cdot z = 0$. However, there isn't just one such integer, meaning that $\frac{0}{0}$ is undefined (I think that's what you meant by 'divisibility by zero is undefined').

Also, there is no integer $z$ such that $0 \cdot z = x$ for any $x \not = 0$, and so $\frac{x}{0}$ for any non-zero $x$ is also undefined.

Bram28
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Given integers $a$ and $b$, we say that $a$ divides $b$ if there is an integer $c$ with $b=ac$. This is purely multiplicative, no division around.

So, by our definition, is it true that $0$ divides $0$? Yes, because $0=0 \cdot 1$. Done.

Again, none of this was phrased with division.

Randall
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  • But we say that 1 divide 0 is undefined for the students when we teach them, are we using wrong words? – Emptymind Sep 26 '17 at 04:15
  • No, we say that you cannot actively divide 1 by 0. It is a subtle but important difference in the language. (Also, 0 does not divide 1 so there is no problem here.) – Randall Sep 26 '17 at 11:39