You can prove additional facts about the world with those values, that was the point of my usage of ‘i’ as one of the examples.

For h = ^{1}⁄_{0} you can upgrade R to the projectively extended real line. If I’m not mistaken, one needs to do this in order to satisfy additional proofs in real analysis (or upgrade to this one instead).

You seem to be asking whether or not doing so in every conceivable case would prove to be useful. I’m saying that we’d be likely to know beforehand about whether or not it would be. Like finding polynomial roots, one might be inclined to wish that all polynomials with coefficients in the reals had roots, therefore, upgrading the space to the complex numbers allows one to get their wish.

You

canprove additional facts about the world with those values, that was the point of my usage of ‘i’ as one of the examples.For h =

^{1}⁄_{0}you can upgrade R to the projectively extended real line. If I’m not mistaken, one needs to do this in order to satisfy additional proofs in real analysis (or upgrade to this one instead).You seem to be asking whether or not doing so in every conceivable case would prove to be useful. I’m saying that we’d be likely to know beforehand about whether or not it would be. Like finding polynomial roots, one might be inclined to wish that all polynomials with coefficients in the reals had roots, therefore, upgrading the space to the complex numbers allows one to get their wish.