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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.

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    $\begingroup$ 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). $\endgroup$
    – Dirk
    Apr 4 '19 at 13:58
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    $\begingroup$ 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. $\endgroup$
    – quarague
    Apr 4 '19 at 13:59
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    $\begingroup$ @EyalRoth That is surely a matter of opinion :) $\endgroup$ Apr 4 '19 at 14:28
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    $\begingroup$ 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. $\endgroup$
    – user151841
    Apr 4 '19 at 18:15
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    $\begingroup$ @EyalRoth The author, Loren Graham, seems to have good credentials: history.mit.edu/people/loren-r-graham $\endgroup$
    – user151841
    Apr 8 '19 at 14:55
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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.

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    $\begingroup$ It should also be mentioned, however, the njwildberger is considered a bit of a contrarian on the fringes and that one should be ready with a grain of salt when consuming his material. If you (eyal roth, the original poster) do not have a lot of mathematical maturity, his message might be more confusing/distracting than informative. I'm far from an expert on his subject area though, and maybe some of it stands up better than the negative parts I have heard about. $\endgroup$
    – rschwieb
    Apr 4 '19 at 14:52
  • $\begingroup$ @rschwieb Thanks for the warning. I'm quite agnostic in nature, so I tend to employ a lot of critical thinking and try to figure out things on my own before I accept a proposition. $\endgroup$
    – Eyal Roth
    Apr 4 '19 at 14:58
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    $\begingroup$ @EyalRoth That's good, but even so, keep an eye on your watch as you budget time to sink into that material. $\endgroup$
    – rschwieb
    Apr 4 '19 at 15:00
  • $\begingroup$ I agree, he is somewhat eccentric, but I can see the rationale behind some of his objections. I think the rational trig idea is more that he thinks it would be easier to teach because it is more intuitive and teaches you a geometry closer to the Greek's understanding. But for someone who has learned the existing system, it is like trying to learn to write with your other hand. $\endgroup$ Apr 4 '19 at 15:05
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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.

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  • $\begingroup$ Well, the natural numbers "sort of" suffice. The things that are being used as coordinates in finite geometries aren't really like natural numbers either (there's no order, for example.) . But in terms of there only being finitely many things in the field, yeah, you wouldn't need "as many" things in your system of numbers. $\endgroup$
    – rschwieb
    Apr 4 '19 at 14:56

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