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@jason
The "sensible" meaning came from the religious significance that the Pythagoreans gave them in their cosmology.

@spacehobo did the Pythagoreans know about irrational numbers?

@spacehobo Ok I googled this and the answer seems to be that they didn't, they thought irrational numbers didn't exist. Euclid proved they did exist three centuries later when there weren't so many Pythagoreans around, but a century or two after that the philosophy was revived as Neopythagoreanism. There is an apocryphal story of a Pythagorean-era philosopher Hippassus proving that sqrt(2) was irrational and dying for it.

@spacehobo According to Wikipedia, the Pythagoreans "... revered the pentagram, as each diagonal divides the two others at the golden ratio" which is pretty funny since the golden ratio is itself irrational.