Eh, the half-size relationship is hardly an innovation of the A-series of paper: the US uses letter, tabloid/ledger (double-letter-size), and half and quarter letter size. US printed matter and blank books mostly end up in that same set of sizes, because they're all made by cutting and folding the starting stock in half, modulo some trimming of the edges to square them up after binding.
The innovation of the A-series line was using the golden ratio, so that the half-sized paper has the same aspect ratio; half-sized paper always had half the area.
That is among the plausible definitions of "half-sized paper". LOL. But the A-series innovation makes it easier to imagine scaling content/resources because you don't have to "retypeset" in your brain.
But, mostly, my fascination with A-series paper as an American is rooted in my utter disappointment that we refuse to use the metric system (even though every single American does without knowing it).
The innovation of the A-series line was using the golden ratio, so that the half-sized paper has the same aspect ratio; half-sized paper always had half the area.