[moioci]

That would seem to imply that for A =/= 0, C/A is irrational. This seems counterintuitive.

[asolove]

What is counter-intuitive? In a triangle with sides A=1, B=1, then C=root(2), so C/A is irrational. That's what was so impactful about the discovery.

Imagine not knowing about irrational numbers. You assume all numbers are just integers and fractional ratios between integers. It would be weird (terrifying?) that something as simple as a right triangle would require a whole category of numbers you can't express.

[Ekaros]

For some reason that feels so weird that it would be that late "discovery"... Once you define a square(sides same length) the length of diagonal is one of the first questions. And this being very weird number is something I believe someone must have thought about long before that point of time.

[moioci]

My bad. I was thinking A, B, and C were integers.

[hinkley]

The 3/4/5 triangle is rational. The unit right triangle is not. You’ve dropped the “some” from the parent.

[bandrami]

For "most" right triangles, yes, C/A is irrational. In fact the triangles for which C/A is rational are vanishingly rare (though Pythagoras proved many important things about them[1])

But before Pythagoras, it was still an open question if for any two reals A, B there might be a rational Q such that QA = B. Whereas we now know that for "most" reals there is no such Q, thanks to Pythagoras.

1: https://en.wikipedia.org/wiki/Pythagorean_triple