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I understand why voltmeters are connected in parallel and ammeters are connected in series, but why is it that to measure voltage, you must have high resistance, and to measure current, you must have low resistance? Perhaps this is not within the scope of the question, but how exactly do either devices measure what they measure and what do their resistances have to do with it?

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Theoretically, these requirements arise from the way you connect the measurement devices to the rest of the circuit.

A voltmeter is connected in parallel, as you said. Say that you are trying to measure the voltage drop across a resistor $R$ through which passes a current $i$. If the internal resistance of the voltmeter is comparable to $R$, then the current $i$ will be divided through both branches, and the voltage value you read will be different from when the resistor is connected alone. If you make the internal resistance very large, then the current flowing through the voltmeter will be negligible, and the voltage drop across the resistor will not change.

The same applies to ammeter; since you're connecting them in series with the rest of the circuit, you need the device's internal resistance to be negligible so that it doesn't affect the current flowing through the branch. If you connect a voltage source $V$ to a resistor $R$ and want to measure the current flowing through the resistor, if the ammeter's internal resistance is not negligible, the total resistance the voltage source "sees" will be different, so by Ohm's Law the current will also be different.

As for the actual electronics of these measurement devices, I'm afraid I can't help you, as I don't know them in detail myself.

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    $\begingroup$ A simple diagram would really help :) $\endgroup$ – DanielSank Sep 30 '15 at 21:52
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Hope this helps!

The current doesnt pass through the voltmeter so the the resistance of load R is not affected.

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