Op Amps: how do they know whether the supply is split?
Your first equation is not correct. It should be identical to the second equation, assuming the bias point of the output is really halfway between the supplies. (In practice, there's a small DC offset.) An op amp is a differential amplifier with high gain. It doesn't "know" what kind of supplies it's hooked up to, and in fact there is no way to know. The choice of ground is arbitrary. If you have +/-12V supplies with an input referenced to ground, it's equally correct to say that you have a +24V/0V supply with an input referenced to +12V.

Adding negative feedback is what really references the output ground to the signal ground. For example, consider a voltage follower:







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The op amp will change its output to make the inverting input voltage equal to the non-inverting input voltage (V1). Since V1 is referenced to ground, the output voltage will be as well. This is independent of the supply voltages

That's usually okay with split supplies, but with a single-supply the output can't go below zero volts. If you have a ground-referenced AC signal, that's problematic. For example, here's an inverting amplifier:







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When V1 is negative, the amplifier works as expected. But when V1 is positive, the output runs into the 0V rail. The solution is to add a negative offset to the signal. Because the op amp is a differential amplifier, that's the same as adding a positive offset to the inverting input. (Remember, the choice of which node to call ground is arbitrary!)







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This offset is called a virtual ground, and is usually half of your single supply voltage.