Are there real numbers that are neither rational nor irrational?
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.
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.
Of course, the "traditional" answer is no, there are no real numbers that are not rational nor irrational. However, being the contrarian that I am, allow me to provide an alternative interpretation which gives a different answer.
What if you are using intuitionistic logic? – PyRulez
In intuitionistic logic, where the law of excluded middle (LEM) $P\vee\lnot P$ is rejected, things become slightly more complicated. Let $x\in \Bbb Q$ mean that there are two integers $p,q$ with $x=p/q$. Then the traditional interpretation of "$x$ is irrational" is $\lnot(x\in\Bbb Q)$, but we're going to call this "$x$ is not rational" instead. The statement "$x$ is not not rational", which is $\lnot\lnot(x\in\Bbb Q)$, is implied by $x\in\Bbb Q$ but not equivalent to it.
Consider the equation $0<|x-p/q|<q^{-\mu}$ where $x$ is the real number being approximated and $p/q$ is the rational approximation, and $\mu$ is a positive real constant. We measure the accuracy of the approximation by $|x-p/q|$, but don't let the denominator (and hence also the numerator, since $p/q$ is near $x$) be too large by demanding that the approximation be within a power of $q$. The larger $\mu$ is, the fewer pairs $(p,q)$ satisfy the equation, so we can find the least upper bound of $\mu$ such that there are infinitely many coprime solutions $(p,q)$ to the equation, and this defines the irrationality measure $\mu(x)$. There is a nice theorem from number theory that says that the irrationality measure of any irrational algebraic number is $2$, and the irrationality measure of a transcendental number is $\ge2$, while the irrationality measure of any rational number is $1$.
Thus there is a measurable gap between the irrationality measures of rational and irrational numbers, and this yields an alternative "constructive" definition of irrational: let $x\in\Bbb I$, read "$x$ is irrational", if $|x-p/q|<q^{-2}$ has infinitely many coprime solutions. Then $x\in\Bbb I\to x\notin\Bbb Q$, i.e. an irrational number is not rational, and in classical logic $x\in\Bbb I\leftrightarrow x\notin\Bbb Q$, so this is equivalent to the usual definition of irrational. This is viewed as a more constructive definition because rather than asserting a negative (that $x=p/q$ yields a contradiction), it instead gives an infinite sequence of good approximations which verifies the irrationality of the number.
This approach is also similar to the continued fraction method: irrational numbers have infinite simple continued fraction representations, while rational numbers have finite ones, so given an infinite continued fraction representation you automatically know that the limit cannot be rational.
The bad news is that because intuitionistic or constructive logic is strictly weaker than classical logic, it does not prove anything that classical logic cannot prove. Since classical logic proves that every number is rational or irrational, it does not prove that there is a non-rational non-irrational number (assuming consistency), so intuitionistic logic also cannot prove the existence of a non-rational non-irrational number. It just can't prove that this is impossible (it might be true, for some sense of "might"). On the other hand, there should be a model of the reals with constructive logic + $\lnot$LEM, such that there is a non-rational non-irrational number, and I invite any constructive analysts to supply such examples in the comments.