[xigoi]

This is assuming that Θ interacts with arithmetic operations the usual way (that is, ℝ ∪ {Θ} is a field), which the person you're replying to did not say.

[afiori]

True, but the point of giving a "value" to 0/0 is to use it somehow.

For example in the context of limits you define a whole lot of number like values like 0+ or 0- that are useful wrt operations on limits.

I was trying to give an example of how ℝ ∪ {Θ} has almost no advantages compared to just ℝ

[tshaddox]

Sure, but the whole "problem" we were trying to solve was that zero doesn't interact with arithmetic operations the usual way.