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With a push button on an Arduino (or Raspberry Pi), instead of wiring a high voltage (5v) through a button and to a pin to read the input, you often want your input pin set to HIGH (high impedance) and wired through a switch to ground to avoid interference like EM waves. My question is what exactly does setting the input pin to HIGH do (besides making it high impedance)? Does it make the circuit less sensitive to EM interference? I like to understand more of the physics behind circuit design.

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In Arduino, just setting the pin as input already makes it high impedance.
Writing HIGH on an input pin actually enables its internal pull-up resistor.
This way, the voltage at the pin is always defined (instead of floating when the switch is open).
Just search for "pull-up resistor" if you need more information.
Note that this code:

pinMode(n, INPUT); // set as input, high-impedance
digitalWrite(n, HIGH); // activate the internal pull-up resistor

is equivalent to this (in recent Arduino IDEs):

pinMode(n, INPUT_PULLUP)

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