I think the physical world is rational - we have number systems that are internally consistent and which can be used to compute values for the structure of machines, compute orbits and trajectories for space flight, build the Hoover Dam, and a myriad of other uses in science, medicine, and the arts.

There are many examples of approximation of an ideal; the many-sided polygon is one. Whether it is a good enough representation depends on the use of the computer-generated polygon. What you call a "real" circle is only the idea of a circle, which does not exist in the physical world. But we don't need n to the infinite power accuracy in the real world. Do automobile wheel and tire manufacturers have a perfect circle to build their products? No, but they work fine in the real world.