As for extending the rationals by 0⁻¹ and giving up properties, a·0 = 0 can still hold for all a ≠ 0⁻¹. To get an idea of what operations do and don't make sen

As for extending the rationals by 0⁻¹ and giving up properties, a·0 = 0 can still hold for all a ≠ 0⁻¹. To get an idea of what operations do and don't make sense when extending the rationals in the way you propose, define ∞ as our 0⁻¹ and refer to https://en.wikipedia.org/wiki/Projectively_extended_real_line#Arithmetic_operations https://en.wikipedia.org/wiki/Projectively_extended_real_lin... substituting ℚ and ℚ̂ for ℝ and ℝ̂. For a similar construction that maintains ordering properties, see https://en.wikipedia.org/wiki/Extended_real_number_line#Arithmetic_operations https://en.wikipedia.org/wiki/Extended_real_number_line#Arit... Finally, note that the complex number equivalent to your construction is really interesting and useful: https://en.wikipedia.org/wiki/Riemann_sphere https://en.wikipedia.org/wiki/Riemann_sphere https://www-users.cse.umn.edu/~arnold/moebius/ https://www-users.cse.umn.edu/~arnold/moebius/