A geometry theory without irrational numbers?

Question

Is there any theory or theorem of geometry -- whether used in practice or not -- which denies or forbids the use of irrational numbers?

If not, were there any notable attempts at it?

Disclaimer: I am not looking for a proof for the existence of irrational number.

Answer 1

I don't know how helpful you will find it, but there are videos on YouTube by njwildberger on rational trigonometry. The main idea is to avoid taking square roots and deal with squares of lengths and ratios between them. He calls it quadrance.

https://www.youtube.com/watch?v=GGj399xIssQ&list=PL3C58498718451C47

http://www.wildegg.com/intro-rational-trig.html

Trouble is, the irrational approach seems to be working fine so there is no reason to completely overhaul the system.

Answer 2

Have you heard of finite geometry, as in: en.wikipedia.org/wiki/Finite_geometry ? This is geometry where there are only a finite number of points. So you don't even need rationals, natural numbers suffice.