Jose wrote:
Because, unlike headings, runways are always rounded off to the
nearest ten degrees.


I'd say that the fact that headings are equal to their value, whereas
runway designations are not (they are equal to roughly ten percent of
their value, with exceptions) is closer to the reason. Runways are not
merely "rounded off", rather, the last digit of their magnetic heading
is ruthlessly ripped away after the roundoff process has taken place.
And even then, some runways are given different names ("Sorry, 9 is
already taken, how about 8?") for convenience. After all, it's better
than having runway "nine all the way left", runway "nine sort of left",
runway "nine right, looks like rightmost but isn't" and "nine so far to
the right it looks like another airport tucked in the corner".

No, headings are just rounded to a different level. A heading can
certainly can be a real number, it doesn't fundamentally have to the
ones digit.

Matt