It's well-established that 0 decibans means 1:1 odds or 50% confidence; that 10 decibans means 10:1 odds; that -10 decibans means 1:10 odds; and that fractional numbers of decibans have similar meaning.
Does it make sense to talk about "i decibans", or "10 + 20i decibans"? If so, what does that actually mean?
I'm currently roughing out what may eventually become a formal specification for a protocol. It includes a numerical field for a level of confidence, measured in decibans. I'd like to know if I should simply define the spec as only allowing real numbers, or if there could be some purpose in allowing for complex numbers, as well.
x decibans means 10^(x/10):1 odds. Using that, a+b i decibans means 10^((a+b i)/10):1 = 10^(a/10)10^(b i/10):1 = 10^(a/10)e^(ln(10)b i/10):1 = 10^(a/10)e^(2 pi b ln(10)/(20 pi) i):1 = 10^(a/10)(cos(2 pi ln(10)/(20 pi))+i sin(2 pi ln(10)/(20 pi))):1. It's just a complex number, unless the imaginary part is divisible by ln(10)/(20*pi), in which case it's the same as just using the real part. A non-real deciban is no more meaningful than a non-real probability.