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Depends on the language and context, but mathematics treats division by zero as resulting in 'undefined' (essentially Not A Number) rather than infinity - again, depending on the field and context, but treating it as infinity can produce paradoxes.
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Yes. I thought as much, but I don't have the hardcore maths background to discuss such matters.
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No, for two reasons. One, the button won't be lowered again unless it's a sticky piston.
Two, buttons only produce a second or so of power when pressed. You'd need a lever to toggle between perpetual on/off states.
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Very well, I stand corrected. Again.