But it is worth pointing out that it is actually possible to maintain "perfect" accuracy when doing arithmetic even when sqrts (or other irrational numbers) are involved.
The idea is to store numbers as an AST consisting of an expression. Then you can perform any arithmetic on them by growing the AST. After each operation you simplify as much as you can. Once you have your final result you can evaluate the AST up to any precision that you need (and sometimes your final AST will already be a rational number because sqrts cancelled out)
I believe you're basically referring to the computable reals; for the curious, here's the wikipedia page (kind of abstract): https://en.wikipedia.org/wiki/Computable_number and here's the paper where they implement them programmatically, including for the android calculator application (I believe this is how the android calculator works now): https://dl.acm.org/doi/abs/10.1145/3385412.3386037
But it is worth pointing out that it is actually possible to maintain "perfect" accuracy when doing arithmetic even when sqrts (or other irrational numbers) are involved.
The idea is to store numbers as an AST consisting of an expression. Then you can perform any arithmetic on them by growing the AST. After each operation you simplify as much as you can. Once you have your final result you can evaluate the AST up to any precision that you need (and sometimes your final AST will already be a rational number because sqrts cancelled out)