  This paper introduces Haros graphs, a construction which provides a
graph-theoretical representation of real numbers in the unit interval reached
via paths in the Farey binary tree. We show how the topological structure of
Haros graphs yields a natural classification of the reals numbers into a
hierarchy of families. To unveil such classification, we introduce an entropic
functional on these graphs and show that it can be expressed, thanks to its
fractal nature, in terms of a generalised de Rham curve. We show that this
entropy reaches a global maximum at the reciprocal of the Golden number and
otherwise displays a rich hierarchy of local maxima and minima that relate to
specific families of irrationals (noble numbers) and rationals, overall
providing an exotic classification and representation of the reals numbers
according to entropic principles. We close the paper with a number of
conjectures and outline a research programme on Haros graphs.
