Comment on Why is division so much more complex than other arithmetic operations?parentComments−jasomill3yAs 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 tohttps://en.wikipedia.org/wiki/Projectively_extended_real_lin...substituting ℚ and ℚ̂ for ℝ and ℝ̂.For a similar construction that maintains ordering properties, seehttps://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_spherehttps://www-users.cse.umn.edu/~arnold/moebius/−JonChesterfield3yThe Riemann sphere has always existed beyond my grasp of mathematics. I remember the maths undergraduates being excited about it. Thank you for the references, it looks like I need to learn more before proceeding with certainty.
Comments
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_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:
https://en.wikipedia.org/wiki/Riemann_sphere
https://www-users.cse.umn.edu/~arnold/moebius/
The Riemann sphere has always existed beyond my grasp of mathematics. I remember the maths undergraduates being excited about it. Thank you for the references, it looks like I need to learn more before proceeding with certainty.