A resistor with a capacitor, or a resistor with a coil, makes a filter: some frequencies get through and some do not. The handover is not a wall, it is a slope, and we mark it at one agreed place — the corner, or cutoff, where the reactance has come to equal the resistance.

$f_{c} = \dfrac{1}{2\pi R C}$ — f-c equals one over two pi R C, with $f_{c}$ the corner frequency in hertz, $R$ in ohms and $C$ in farads. Its inductive twin is $f_{c} = \dfrac{R}{2\pi L}$, with $L$ in henries — note that R has moved to the TOP, because a coil's reactance rises with frequency while a capacitor's falls. Check the units and the arrangement declares itself: ohms times farads is seconds, ohms over henries is per second, and either way the reciprocal comes out in hertz.

At that corner the output has fallen to $1/\sqrt{2}$ — about 0.707 — of the input, which means the POWER has fallen to exactly half. That is why the corner is also called the half-power point, and why it is written −3 dB.

Which brings us to the decibel: a ratio, expressed logarithmically, carrying no unit whatever. $G_{dB} = 10\log_{10}\!\left(\dfrac{P_{2}}{P_{1}}\right)$ for powers — G equals ten log P-two over P-one — where $P_{1}$ is the input or reference power and $P_{2}$ the output, both in watts, and the subscript convention is worth stating outright: 1 is the reference going in, 2 is what comes out. For voltages the constant is twenty, not ten: $G_{dB} = 20\log_{10}\!\left(\dfrac{V_{2}}{V_{1}}\right)$, with $V_{1}$ and $V_{2}$ in volts. The 20 is not a different definition — power goes as the square of voltage, and squaring inside a logarithm doubles the constant outside it. Using 10 on a voltage ratio (or 20 on a power ratio) is the marquee error of this lesson, and it is wrong by exactly a factor of two every single time.