This is for teaching math. I'm wondering if someone knows some striking near-equalities between simple arithmetic expressions. I vaguely remember that such things exist (e.g., numbers that look alike out 10 digits, but that are really different), but I don't remember where I saw them or what they are. I think there are even some famous historical instances where it was debated whether certain expressions were equal or not.

One possibility I am particularly interested in is integer combinations of (integer) square roots that turn out to be very very close to zero (but that are nonzero). Does anyone know how to construct these? I am assuming they exist because I read somewhere (and forgotten again!) that there is currently no efficient algorithm---or maybe even no algorithm at all, can't remember---for determining the sign of such a sum.

Thank you!

PS: I'm also interested in sources (e.g., book of number puzzles or relevant number theory, etc).