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_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#Arit...
Finally, note that the complex number equivalent to your construction is really interesting and useful:
