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I'm fuzzy on the deeper details. I think you can do something like this, but you have to be very careful, in ways where you don't have to be so careful with ✓-1.
One of the more obvious ways to consider: plot a graph of y = 1 / x. Note how as x approaches zero from the right, the graph shoots up, asymptotically approaching the y-axis and shooting up to infinity. It's very tempting to say that 1 / 0 is "infinity". "Infinity" is not a real number, but nothing is stopping you from defining a new kind of number to represent this singularity if you want to. But at that point you have left the real numbers. Which is fine, right? Complex numbers aren't real numbers either, after all...
But look at the left side of the graph. You have the same behavior, but the graph shoots down, not up. It suggests that the limit of approaching from the left is "negative infinity". Quite literally the furthest possible imaginable thing from the "infinity" we had to define for the right side. But this is supposed to be the same value, at x = 0. Just by approaching it from different directions, we don't just get two different answers, we get perhaps the most different answers possible.
I think it's not hard to intuit a handwavey answer that this simply represents the curve of y = 1 / x "wrapping around through infinity" or some notion like that. Sure, perhaps that is what's going on. But dancing around a singularity like that mathematically isn't simple. The very nature of mathematical singularities is to give you nonsensical results. Generally, having them at all tends to be a sign that you have the wrong model for something.
You can mostly avoid this problem by snipping off the entire left half of the x-axis. Shrink your input domain to only non-negative numbers. Then, I believe, you can just slap "infinity" on it and run with it and be mostly fine. But that's a condition you have to be upfront about. This becomes a special case solution, not a generalized one.
I haven't looked into it, but I believe this singularity gets even more unweildy if you try to extend it to complex numbers. All the while, complex numbers "just work". You don't need doctor's gloves to handle them. √-1 isn't a mathematical singularity, it's a thing with an answer, the answer just isn't a real number.
Awesome, thank you for taking the time to explore it more, that makes sense to me
Seriously u/pixelscript that’s a really good explanation and I hope it’s right. My understanding of infinities is pretty limited but I’d like to subscribe to your newsletter.