Is there a dimension which extends to negative or even irrational numbers?
See this discussion: https://physics.stackexchange.com/questions/52176/could-negative-dimension-ever-make-sense
Also, reals can include irrationals! So Hausdorff-Besicovitch already allows irrationals. And Mandelbrot himself has an interesting take on negative fractal dimension, where he says it measures the '"emptiness" of certain empty sets': http://users.math.yale.edu/~bbm3/web_pdfs/123negativeFractalDimensions.pdf