Though the cardinality of the set of natural numbers is the same as the cardinality of the set of rational numbers, when one looks at a number line this fact seems counterintuitive, since between every rational number there exist an infinite number of more rational numbers. Of course, it can be shown that $\mathbb{Q}$ is countable by simply ordering them on an infinite array (the "zig-zag" method). Thus there are the same "number" of rationals as there are natural numbers (they have the same size). 

My question is as follows: if we "organize" the rational numbers *in a certain way* on a line - 

$
\\
\frac{1}{1}, -\frac{1}{1}, \frac{1}{2}, -\frac{1}{2}, \ldots
\\
\frac{2}{1}, -\frac{2}{1}, \frac{2}{2}, -\frac{2}{2}, \ldots
\\
\vdots
\\
\frac{n}{1}, -\frac{n}{1}, \frac{n}{2}, -\frac{n}{2}, \ldots
\\
\vdots
$

where the "line" would be the "zig-zag" path ($0/1$, $1/1$, $-1/1$, $2/1$, $3/1$...; note that this line would not be straight geometrically, but assume it has been "straigtened" for sake of argument) the intervals between these numbers would have *nothing in between them* - they would be very similar to the natural numbers on a normal line. However, if we consider the rational numbers on a normal number line, there are an infinite number of rationals in between two arbitrary rational numbers, and hence the layman is mislead into thinking that rationals outnumber naturals. Why are these two "lines" so different? Why does the normal number line appear to be more "densely" populated with rationals than the "reorganized" number line? Lastly, I am curious whether or not, since $\mathbb{Q}$ and $\mathbb{N}$ are the same size, can the natural numbers be organized such that there are an infinite number of natural numbers in between two natural numbers, as can be done with rationals?