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?

Solution 1:

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.

Solution 2:

No. The definition of an irrational number is a number which is not a rational number, namely it is not the ratio between two integers.

If a real number is not rational, then by definition it is irrational.

However, if you think about algebraic numbers, which are rational numbers and irrational numbers which can be expressed as roots of polynomials with integer coefficients (like $\sqrt2$ or $\sqrt[4]{12}-\frac1{\sqrt3}$), then there are irrational numbers which are not algebraic. These are called transcendental numbers.