"q the self-contradicting infinite number" is so useful that it sits at the foundation of the whole subject of projective geometry.

Which we can use for such lovely practical applications as rendering the CGI baddies we want to blast in 3D shooter games.

This "number" is also of fundamental importance to complex analysis. See https://en.wikipedia.org/wiki/Riemann_sphere https://en.wikipedia.org/wiki/Zeros_and_poles https://en.wikipedia.org/wiki/Meromorphic_function