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Imagine you need to boost a weak sensor signal by a factor of ten, but your downstream circuit expects the signal to be completely upside-down. An inverting amplifier does both jobs simultaneously with just two resistors and an operational amplifier.
How do two passive resistors dictate the exact amplification factor of an active circuit?

The Inverting Configuration

In an inverting amplifier, the non-inverting terminal (++) connects directly to ground (0 V0\text{ V}). The input voltage VinV_{\text{in}} feeds through an input resistor RinR_{\text{in}} into the inverting terminal (−-), while a feedback resistor RfR_f loops back from the output VoutV_{\text{out}} to that same inverting input.
Why does the inverting input behave like a ground connection even though it has no wire attached to 0 V0\text{ V}?

The Virtual Earth and KCL

Because an ideal op-amp has near-infinite open-loop gain, negative feedback forces the voltage difference between the two input pins to zero. Since the non-inverting pin sits at 0 V0\text{ V}, the inverting pin is held at 0 V0\text{ V} as well — creating a virtual earth (a point held at ground potential without a direct physical ground connection).
In 1845, German physicist Gustav Kirchhoff formulated Kirchhoff's Current Law (KCL) to solve complex circuit loops: the sum of all electrical currents entering a junction must equal the sum of all currents leaving it.
An ideal op-amp has infinite input impedance (resistance to incoming electrical current), meaning no current can flow into the inverting input pin itself (I−=0 AI_{-} = 0\text{ A}). Therefore, all current arriving through RinR_{\text{in}} must continue entirely through RfR_f.
Now, how do we translate this simple current equality into a direct formula relating VoutV_{\text{out}} to VinV_{\text{in}}?

Deriving the Voltage Gain Formula