"We'd then argue over what shape a circle looked like, but my maths would certainly be easier than yours due to lack of irrationality."

Actually that argument was had and ended a long time ago; you can set pi to 3 and you get a thing called a spherical geometry. See, for instance, http://mathforum.org/library/drmath/view/55021.html . Between the word "spherical geometry" and what you find in that link you'll have the keywords to continue digging if you want to.