Decibels | James Howard Decibels | James Howard

### James HowardA Mathematician, a Different Kind of Mathematician, and a Statistician # Decibels

According to Wikipedia, “[t]he decibel (dB) is a logarithmic unit used to express the ratio between two values of a physical quantity, often power or intensity.” We usually hear it applied to sound. It is also applied to radio communications. For instance, your phone usually has a setting that gives the signal strength in decibels.

In practice, this means there needs to be two measured values in order to have a ratio. The numerator of the ratio is the measured value. The denominator is a reference value, a standard number we already know. For measuring sound this $p_0$, or the pressure at 20 micropascals in air. So when measuring sound,

$L_p = 20\log_{10} \frac{p}{p_0}\,\text{dB.}$

But this definition implies I could apply it to any other physical quantities provided the base units were the same. So, being ridiculous, and using Earth as a reference value, we know the mass of Jupiter is,

$M_j = \log_{10} \frac{M_J}{M_\oplus} = \log_{10} \frac{1.898\times 10^{27}\,\text{kg}}{5.972\times 10^{24}\,\text{kg}} \approx 2.5\,\text{dB.}$

Neat.

Image by Kevin Gill / Flickr.