# Why Is Every Integer A Rational Number Give Reason? Every Integer Is A Rational Number

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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

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

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 $b = 1$.

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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

adechworld.combb 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 $frac{-k}{m}$.

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edited Dec 22 “19 at 0:26

answered Dec 21 “19 at 23:08

fleabloodfleablood

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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).

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answered Dec 21 “19 at 22:48

starfishstarfish

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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.

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For example,

$$frac{0}{0} = lim_{x o 0} frac{x^2}{x} = 0$$

is legitimate as I understand it.

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answered Dec 21 “19 at 23:26

TrevorTrevor

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