What does a bitwise shift (left or right) do and what is it used for?

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I've seen the operators >> and << in various code that I've looked at (none of which I actually understood), but I'm just wondering what they actually do and what some practical uses of them are.

If the shifts are like x * 2 and x / 2, what is the real difference from actually using the * and / operators? Is there a performance difference?

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