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I am currently doing this problem and although I got the answer right, I do not completely understand how the concept works. In this case, what does "relative to infinity" mean? I have a hard time wrapping my head around that concept. Also, if all the charges were of equal sign and magnitude, shouldn't the electric field at $A$ be $0$ since the electric field vectors cancel out, and thus the electric potential will also be $0$? If I follow that logic, then the electric potential at $A$ for the given problem would not be the sum of the electric potentials of the charges, but rather $V_{q_2}-V_{q_4}+V_{q_1}-V_{q_3}$ (or something along those lines)?

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Your several questions show you're missing a single and simple aspect of the voltage which is that it's only defined to within a constant.

In some ways, think of voltage like height (or potential energy) in a gravitational field. It's fundamentally a relative term, which is why people often talk about height relative to sea level, or an individual's height (head relative to feet).

In practical applications, this usually means that you think of voltage as defined between two points, and refer to is as "voltage difference" (usually with "ground" as a reference and things relative to that, but this isn't the only possible reference). In physics it's interesting that it's meaningful to talk about voltage as a thing that can be meaningfully defined at each point in space (that is, it's the same value regardless of the path taken to get to that location).

"relative to infinity":
If all of space only had your 4 charges in it, then this phrase is telling you that you should say that the arbitrary constant in the voltage is set so that it's zero very far away from the charges. Then the voltage is also linearly related to how much work it would take to move a charge from very far away to that location.

"if all the charges were of equal sign":
You are right about the electric field, that it would be zero at the center. But the voltage is not zero just because the electric field is zero. It's like saying if you were standing on the top of hill that was a nice smooth Gaussian bump, so the slope at the top of the hill was zero, then saying the voltage is zero is like saying you weren't standing on a bump because it was locally flat. You know you're on a hill because of the work it took to get you to the top of the hill, not because of how flat it is or isn't.

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In this context, it means "set the potential to zero at infinity". You can think of this as the boundary condition that you get when you integrate the field to get the potential, or you can think of this as setting where the "ground" of your electrical system is. If you're getting yet fancier, you can think of this as a simplified version of 'choosing the gauge for your electromagnetic system'

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