Are there real numbers that are neither rational nor irrational? I wouldn't have asked this question if I hadn't seen this image:

From this image it seems like there are reals that are neither rational nor irrational (dark blue), but is it so or is that illustration incorrect?

A real number is irrational if and only if it is not rational. By definition any real number is either rational or irrational.
I suppose the creator of this image chose this representation to show that rational and irrational numbers are both part of the bigger set of real numbers. The dark blue area is actually the empty set.

This is my take on a better representation:

Feel free to edit and improve this representation to your liking. I've oploaded the SVG sourcecode to pastebin.
