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In a lot of chips I've worked with there is an internal threshold (sometimes external resistor) that if the voltage is above "X" it is on, if below it is off.
Serial ports work like this with either a 5vdc reference or a 2.5vdc reference and then is split down the middle. there is an effect called field magnetism (bleed)where other external voltages can influence (and will) the wire voltage so you never get a clean 0vdc. So chip manufactures tend to build this into the chips.
a signal could look like: 4.5, 1.2, 3.8, 0.6, (on,off,on,off) and still be perfectly valid, that's where oscilloscopes come in handy for troubleshooting.
The reality is that all circuits are really analog, they don't instantly change from one state to another.
So, the transition is never at one extreme or the other. If the chip claims to work at say 0 and +5 there is some point before +5 that the circuit decides the signal is a 1 and some point before 0 that it decides that the signal is a 0 (replace 0 and +5 with your favorite voltages for representing 1 and 0.)
In diagrams they like to draw straight lines and sharp corners but the reality that none of the lines are straight and none of the corners or sharp. In fact, there is usually some amount of overshoot at the corners and a certain amount of settle time. They have a couple of pictures here of what overshoot looks like.
So, if they used the 0 or the +5 for the actual 1 and 0 indicator it would get very tricky because the signal (at least at the corners) tends to bound around the min and the max voltage...
Not sure what you mean by currency. You generally have voltage, current, and resistance.
As for 0 and 1, generally this is a measure of voltage only. And most of the time 0 is near ground potential and 1 is near Vcc (the voltage common to the circuit). So if your wondering if there might be some amount of residual voltage - yes. In fact it with a typical circuit, most of the time the pins will not be exactly or Vcc. They will be slightly higher than 0 or slightly lower then Vcc. And occasionally, you can actually et something that is very close to half way between, but such a situation is generally a result of some problem in the circuit.
Having said all this, in most cases in working with a digital circuit it helps to set the scale and simply lance at the value. High and low will be very obvious.
I saw a lot of good answers here, but wanted to point out that there are some very common devices in personal computer architecture that use multiple voltage levels to represent more data - MLC solid state devices, one example of which is the Triple Level Cell SSD from Samsung. These use 8 voltage levels to represent 8 bits of information per cell (as far as I remember). The details have gotten a little hazy, but this^] article explains it pretty well. In other words, 0 is only represented by the accepted range of the voltage present, as any other value would be.