E.g. decibels (dB) is a ratio of identical quantities $q_1$ and $q_2$. So the ratio's unit will be:

$[\frac{q_1}{q_2}] = \frac{[q_1]}{[q_1]} = \frac{1 [whatever]}{1 [whatever]} = 1 = 0dB$

So the units are gone.

If you express the ratio in logrithmic terms, you need to be notified about it. That's what the non-SI unit dB is doing: *!nota bene, I'm a logarithmic number!* 

This comes from telecommunications. *dB* is the newer term, indicating $lg$ has been used. Before, at the times of first telephones, Neper was used (*Np*) to express ratios by $ln$.

You are free to use ratios on an anything. E.g. you could say:
- my shoes are 10 % larger than yours
- which is a 1.1 ratio
- which is $+.83$ dB  (I leave it up to you to discover the hidden square and 10)


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An important point: In physics the units denote both the phenomenon and its measurement, like:

- force F = m * a <=> 1 [kg m s^-2]
- mass m <=> 1 [kg]
- acceleration a <=> 1 [m s^-2]

This is also a unique error check when doing more complicated calculations, you won't find anywhere else. Whatever you calculated, units must still equal on both sides. If they don't: error. If they do: perhaps it's correct.