More relevant and less hacky, IMO, is that the real numbers aren't algebraically closed - there are polynomials with real coefficients that you cannot factor in

More relevant and less hacky, IMO, is that the real numbers aren't algebraically closed - there are polynomials with real coefficients that you cannot factor into getting real zeros. This is a big problem when you have something like a spring-mass-damper system. The springiness, damping, and mass of the system generate coefficients of a degree-two polynomial, and the zeros of that polynomial correspond to how the system moves over time.