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You are watching: Is every integer a rational number

Why is this the case? $0$ is an integer and it can"t be separated by $0$...

It"s on mine textbook, together it claims

We conclude that every essence is a rational number, and so the reasonable numbers type an expansion of the integers.

Why would you need to have the ability to divide 0 by itself? The defining characteristic of a rational number is commonly taken to be that it deserve to be stood for as a ratio of two integers, and zero can absolutely be represented this means (for example, as 0/1).

By her comments you room confusing "Every rational number have the right to be created as $\frac ab$ where $a$ and also $b$ are integers" (which is true) with "Every $\frac ab$ where $a$ and also $b$ are integers, is rational" (not true; $b$ deserve to never be same to $0$).

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

But $\frac 00$ is not rational due to the fact that it is meaningless garbage. $\frac 00$ (which is not the exact same thing as $0$; not even close come the very same thing together $0$) is not a number or anything in ~ all. That is undefined. It is meaningless garbage.

P.S. Every integers are rational due to the fact that for any integer $k \in \invernessgangshow.netbb Z$ climate $k = \frac k1$.

A text with a much more careful an interpretation might state the to be reasonable it should be expressible together $\frac ab$ where $a$ is an integer and also $b$ is a natural number. This not only rules the end $\frac k0$ but likewise avoids ambiguities an problems of $\frac k-m$ vs $\frac-km$.

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edited Dec 22 "19 at 0:26
answered Dec 21 "19 in ~ 23:08

fleabloodfleablood
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Zero end zero is sometimes referred to as an indeterminate form, specifically when taking care of limits, and also it"s not necessarily garbage. Yep, it usually is, but depending on context, you have the right to use it come do beneficial calculation if you"re careful and understand what you"re doing.

See more: How Many Square Bales In A Round Bale S To Make A Large Round Bale

For example,

$$\frac00 = \lim_x \to 0 \fracx^2x = 0$$

is legitimate as I understand it.

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answered Dec 21 "19 in ~ 23:26

TrevorTrevor
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