What does state mean in math?

Answer 1

The 'State' of something is its situation. An on/off switch has two states: on, and off. At any time the switch is in only one of the possible states: it is either on, or off.

You set the state, and it stays that way. "state" and "stay" come from the same root word, as does "stand".

The month on your calendar has twelve states: January, February, .... At any time it's in only one of those states. As I write this, it's in March. Your TV channel has states: whichever channel it's set to. It probably also has a state for what the source of the TV signal is: cable, satellite, DVD, or whatever.

You can say that your TV system has four properties that are all states:

* which cable channel it's set to (say there's 200 cable channels = 200 states)

* which satellite channel it's set to (say there's 400 satellite channels = 400 states)

* which DVD is in the DVD player (say you've got 150 DVDs = 150 states, or it could be empty = 151 states)

* which source the TV is showing: cable, satellite or DVD, three states. Oh wait - it could be off too, four states.

Another way of looking at your TV system is that it's got just one property with 752 states.

Those are discrete states, the on/off switch can't be somewhere between on or off, it's either one or the other. A dimmer switch, or a volume control, has continuous states: in between two different settings, you can always find a setting in between. Therefore, you have an infinite number of states - a 'continuum'.

In practice, modern televisions and ipods usually have many discrete settings for volume - maybe 5 or 20 or 50, because nobody needs a volume setting between 37 or 38, they sound almost the same anyway. Similarly, a volume control knob is kind of jumpy on a microscopic scale - if you tried to turn it from 4.137 to 4.138, you could easily overshoot to 4.142 . So in practice, you can think of most continuums as many discrete states.

The location of something requires three properties - the X, Y and Z coordinates. Each is a continuum - there's an infinite number of locations in your house, that's why you can't find your car keys. But really, if you moved your car keys by just a centimeter, you'd consider that the 'same place' - that is, the same state.

If you're designing something with discrete states, like a gear shift on a car, or an interactive web page, you might want to make a 'state diagram' to help you see what you're doing. It's got one box for each state, and arrows between that describe how the system can move between states.