and you just mixed all the concepts up
it's 100% pointless to care about how the calculation is working at the machine code level
like, will you ever care what's going on with the neurons inside someone's brain when he's doing some math?
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I think you're asking user level calculation. There are high-end programs that can do such stuffs. Heck, even calculators can do that.
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that's my point, it should stop being just high end programs, it should be universal thru all programs
obviously "Pi" is a string (some special symbol) at the machine code level, so it's still precise and finite, and how much is a string is pointless, that's something the higher level program will take care of