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\section*{Are there gaps in the rationals?}
There is a sense in which there is no empty space in the rational numbers: no matter how close together two rationals are, there are infinitely many other rationals between them. In this
sense, there are no gaps in the rationals.
However, there is a sense in which there \emph{is} empty space in the rational numbers. Let \[
A=\left\{ r\in\mathbb{Q}:r>0 \text{ and } r^{2}\leq2\right\} \hspace{1em}\text{and}\hspace{1em}B=\left\{ s\in\mathbb{Q}:s>0 \text{ and } s^{2}\geq2\right\}. \]
Clearly, $A$ is less than $B$. Nonetheless, there is no element $q\in\mathbb{Q}$
such that $q$ is between $A$ and $B$. Thus it is not always
possible to find a rational number between any two \emph{sets }of rationals. In this sense, there are gaps in the rationals.
To make matters geometrically
explicit, consider the graph of $y=2-x^{2}$ using only rational numbers. Notice that there's no $x\in\mathbb{Q}$ that makes $2-x^{2}=0$. Thus, the graph never crosses the $x$-axis, despite the fact that the graph is above the $x$-axis at $x=0$ and below
the $x$-axis at $x=3$. This demonstrates
the existence of a more devios type of gap in the rationals, and
the failure of $\mathbb{Q}$ to model our intuition about the physical
world.
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