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Fast-Nop3202829dThe main reason is that it's easy to use transistors as switches. What they don't have is a useful plateau between those states (and also the power losses are not nice there), that's why base 3 math isn't really useful.
RememberMe1478129dThe mathematics behind computation don't rely on binary logic at all, it just happens to be an implementation detail, so you can switch it out for any sufficiently powerful logic system, including n-ary, quantum binary (or n-ary), and other more exotic types like probabilistic models. There are many other possible types of computers than the regular one.
It really is a function of implementation technology. It's very easy to make 2-state devices via silicon and guarantee that they'll always work because your noise immunity has to cover two states only, but at the densities we're seeing now, even this is very, very hard to do (how do you distinguish between 0 state and 1 state in a device that's a bunch atoms wide?). With ternary devices ensuring that 0, 1, and 2 are clearly distinguishable at single digit nanometer scale and 1+ GHz is very difficult, so the gains from process technology outweigh the gains from ternary efficiency. You might argue that we don't need to do that because ternary logic is more efficient and that's absolutely correct, but I haven't heard of any implementation that manages to do that trade-off successfully. Who knows, with new n-stable devices coming and optical switching and all that good stuf, we might see n-ary logic (why stop at three after all).
Demolishun1049229dI did see some information about potentially using ternary for communication. This is due to how optical data is sent down a fiber. But that is pretty specific to a pipe that just packs more bits.