[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.

[asolove]

These are more or less the first people to think about geometry rigorously as an abstract system. Anyone previous would have just pointed to the hypotenuse and said “it’s that length right there” and not asked a further question.

[bandrami]

OK but at the time it was literally an open research question: given two reals A, B is there always a rational Q such that QA=B? Number theory as such was still in its infancy but I think it's impressive that this was exactly the right question to ask and they understood how important it was.

[InitialLastName]

A lot of early math was done using geometry tools rather than symbolic representation.

If you are drawing a diagram for a building and you need a distance equal to the diagonal of a square, you set your compass to the two points and use that distance. No need to determine that it can't be represented by a comfortable multiple of the sides.

[moioci]

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