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Why is this the case? $0$ is an integer and it can't be divided by $0$...

It's on my textbook, as it says

We conclude that every integer is a rational number, and so the rational numbers form an extension of the integers.

ViHdzP
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Eric R.
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  • What is the issue about not being possible to divide by $0$? – Célio Augusto Dec 21 '19 at 22:48
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    Well, for every integer $n$, we can write $n$ as a quotient of integers - namely $n/1$. $0$, in particular, can be written as $0/1$. – csch2 Dec 21 '19 at 22:48
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    $0$ can be divided by $1$. $0 = \frac 0 1$. Why did you think $0$ would have to be divisible by $0$. Nothing in the definition of rational, tells you what the denominator has to be. – fleablood Dec 21 '19 at 23:02
  • OP, you're not understanding the definition. We can express 0 as two integers $a \over b$. Take $0 \over b$ where $b$ is any integer, as others have pointed out in this thread. – dan Dec 22 '19 at 03:50
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    @DanielMishan: It is $\frac{0}{b}$ where $b$ is any integer except zero. The definition of a rational number does specify that the denominator shouldn't be zero. – Axion004 Dec 24 '19 at 01:08
  • My apologies, @Axion004, you are correct. Except zero. – dan Dec 24 '19 at 06:58

3 Answers3

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Why would you need to be able to divide 0 by itself? The defining characteristic of a rational number is typically taken to be that it can be represented as a ratio of two integers, and zero can certainly be represented this way (for example, as 0/1).

starfish
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By your comments you are confusing "Every rational number can be written as $\frac ab$ where $a$ and $b$ are integers" (which is true) with "Every $\frac ab$ where $a$ and $b$ are integers, is rational" (not true; $b$ can never be equal to $0$).

$0$ is rational because $0 = \frac ab$ where $a = 0$ and $b = 1$.

But $\frac 00$ is not rational because it is meaningless garbage. $\frac 00$ (which is not the same thing as $0$; not even close to the same thing as $0$) is not a number or anything at all. It is undefined. It is meaningless garbage.

P.S. All integers are rational because for any integer $k \in \mathbb Z$ then $k = \frac k1$.

A text with a more careful definition might state that to be rational it must be expressible as $\frac ab$ where $a$ is an integer and $b$ is a natural number. This not only rules out $\frac k0$ but also avoids ambiguities an problems of $\frac {k}{-m}$ vs $\frac{-k}{m}$.

fleablood
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  • Restricting the denominator to be a natural number doesn't solve the problem of multiple representations of the same number. There are always reducible fractions. – Matt Samuel Dec 22 '19 at 02:26
  • "Restricting the denominator to be a natural number doesn't solve the problem of multiple representations of the same number. " Did anyone ever say it did? – fleablood Dec 22 '19 at 17:12
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Zero over zero is sometimes called an indeterminate form, especially when dealing with limits, and it's not necessarily garbage. Yep, it usually is, but depending on context, you can use it to do useful calculation if you're careful and understand what you're doing.

For example,

$$\frac{0}{0} = \lim_{x \to 0} \frac{x^2}{x} = 0$$

is legitimate as I understand it.

Trevor
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  • $\frac00$ is always meaningless except as suggesting the form of a limit, not as actually being a fraction. Your equation is nonsense. Only the rightmost equals sign makes sense. – Matt Samuel Dec 22 '19 at 02:29
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    You understand wrong. $\lim_{x\to 0}\frac {x^2}x -=0$, and $\lim_{x\to 0}\frac x{x^2} =\infty$, But It is not acceptable to refer to either of those as $\frac 00$. – fleablood Dec 22 '19 at 17:15