system
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human
ELI5: How can infinity be used in mathematical equations?

I did a quick search for ELI5 questions on infinity and didn't find exactly what I was looking for. 

I understand that infinity is "important to physics", and frequently used in a lot of advanced mathematics, but I can't wrap my head around that. For instance, when a calculator uses **π** in a calculation, it obviously can't use an infinite version of **π** - at some point it just has to round it off (which is probably good enough for most uses of **π**). Otherwise, it would require an infinite amount of time to "realize" the number before it could do anything with it.

How is **∞**, or any form of infinite number, actually used in any equation or calculation by mathematicians or physicists without some form of rounding or treating it merely as an abstraction/variable? (Abstraction/variable like, if I make up a symbol "&&&", I could say "(&&&+9)-(&&&+7)=(&&&+2)" but that ignores whatever "&&&" might mean. I don't actually "calculate" the symbol "&&&")
gpt
I think you're combining a couple of related concepts, which might be leading to the confusion.

Pi isn't infinite, but it does have an infinite number of digits, which are two different things. Like you guessed, your calculator will have a rounded-off version of it, and that's good enough. In actual math work with it, though, they treat pi sort of like you would a variable. Pi divided by 2 is just dealt with as pi/2, and 3 added to pi is just dealt with as pi + 3. If at the end of everything they need some decimal version of it, that's when they do a rounded off calculation, but in a lot of cases, that's not important.

∞ isn't a number (except in some obscure types of math), so an operation like ∞ + 3 is meaningless and isn't done. It'll come up in things like limits, where the idea is that a mathematician wants to know what happens to an equation when a variable gets bigger and bigger, but they're not actually doing any operations on ∞ itself.