Nit: A closer analogy is rationals, more so than reals (as reals have other unrelated properties like uncountability) That is, given your simple rings (the int

Nit: A closer analogy is rationals, more so than reals (as reals have other unrelated properties like uncountability) That is, given your simple rings (the integers) and finite rings (modulo n), you get fields by introducing just division (integers -> rationals; mod-n -> mod-p for prime p).