Is every integer a rational number?

Question

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.

Answer 1

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 $\frac{-k}{m}$.

Answer 2

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).

Answer 3

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.