# A geometry theory without irrational numbers?

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.

• A geometrically interesting subset of the real numbers are the constructible numbers, you can find some information on that on Wikipedia and read into it from there if interested. However, these also include some irrational numbers (but not all).
– Dirk
Apr 4 '19 at 13:58
• Have you heard of finite geometry, as in: en.wikipedia.org/wiki/Finite_geometry ? This is geometry where there are only a fintie numbre of points, hence you can assign them all natural numbers. Apr 4 '19 at 13:59
• @EyalRoth That is surely a matter of opinion :) Apr 4 '19 at 14:28
• I recall in the book Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity, which was about the mathematicians who first promoted the idea of infinity and set theory, and their religious proclivities, that at a large math conference at the time (around 1880?) one of the great mathematicians proclaimed that all of math would be described using "integer alone". Sorry I can't give you a better reference, but it would be worth reading the whole book on its own, if not only to find the reference. Apr 4 '19 at 18:15
• @EyalRoth The author, Loren Graham, seems to have good credentials: history.mit.edu/people/loren-r-graham Apr 8 '19 at 14:55

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.