# Are imaginary and complex numbers of decibans meaningful?

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.

## Comments (15)

BestIf you needed to worry about quantum probability amplitudes, you'd already know about it. Just use real numbers.

After these first two responses, I'm tempted to write something like this: "While complex numbers are theoretically possible here, unless quantum superpositions and eigenstates are important to what is being asserted, the imaginary component will be so small as to be negligible, and should be discarded."

*4 points [-]So far, I haven't seen any reason to think that complex numbers are, in fact, theoretically possible there. Quantum mechanics does involve certain things called "probability amplitudes", which can be non-real, but probability amplitudes aren't probabilities.

*3 points [-]It's not that the imaginary parts are small - they're defined on a range 2π wide (if you're working in natural logs, and some weird width otherwise). It's that they're hard to interpret, and if you know (and need to know) enough to be working with them there's little point to taking the log.

No, in accordance with whatchamacallit's law.

If you end up with complex probabilities, you won't be able to plug them into an expected utility formula to get a preference ordering. This has always been the knockdown argument for quantitatively scaled real-number subjective probabilities in my book. Even if underlying physics turns out to use complex-numbered reality fluid, I don't see how I can make choices if my degree of anticipation for something happening to me is not a real number - I don't know of any complex analogue of the von Neumann-Morgenstern theorem which yields actual decision outputs.

To put it simply, complex numbers lack an order compatible with the algebraic structure: just that makes them unsuitable for decision theoretic criteria.

After the first dozen responses, I'm currently thinking of writing something along the lines: "While the unusual math of noncommutative probabilities allows for complex probabilities, which have applications in quantum superpositions and eigenstates, there is little likelihood of any practical application involving (the protocol). A (protocol) statement may be written with a complex number for its confidence, but a (protocol) reader or interpreter need only concern itself with the real portion of that number."

Either that, or just stating 'real numbers only'.

(PS: I've never written anything which has even a chance at being an Internet Draft, let alone an RFC; but if the tag: URI made it in, nym: just might pass muster, too - and I would welcome any and all advice.)

I don't think you need algebraic closure to solve any probabalistic problems.

*3 points [-]A little while back, Behavioral and Brain Sciences showcased a paper arguing that quantum probabilities might be useful for cognitive modeling. I still haven't read more than a bit of the beginning, but it being in BBS means that it can't be entirely bad.

Do note that they explicitly do

notsuggest that quantum mechanics would have anything to do with the way the brain functions:Rather, they take the general mathematical framework of quantum probability and apply it to cognitive phenomena:

"Quantum probability" in this case should really be called "noncommutative probability": that emphasizes that there's no direct connection to quantum mechanics as a physical theory. The basic mathematical feature is that random variables don't commute.

Live and learn.

*2 points [-]Taking the log moves nonzero complex numbers to complex numbers, so it makes exactly as much sense as expecting to observe complex odds.

Odds are the ratios of outcomes. How might you get complex outcomes like that?

Well, if the outcomes are quantum states, you could get something like 0.5 |a> + 0.5(1+i) |b> - 0.5|c> in which case the odds of each outcome are 1:2:1 if you make them decohere right then. But let's suppose you don't decohere them and work with the amplitudes.

What would taking the logarithm do to these things?

If your states are energy eigenstates, taking the log has the interesting effect of extracting the time dependence, which would manifest as an oscillation between various cases of interference between the states.

If the states are angular momentum eigenstates, taking the log would extract the angle dependence of the outcome. What this means depends more on your context than the first case. Also, you're not going to be able to predict the time dependence from this (the combined state might already be an energy eigenstate, which would help).

And so on, with complementary pairs of observables. If they're eigenstates of one observable operator in a pair of complementary observables, this will encode some information about the dependence on the complementary operator.

If your choice of states don't imply (as far as you can tell) a complementary operator, you're out of luck.

In short:

The real part of your deciban figure would be the log of the amplitude. That acts pretty much like you're used to it acting for normal real quantities. Just throw in a correction factor of 2 to use the squared amplitude instead of the amplitude.

To make any use of that imaginary part, you'd need to know a lot of details about the exact definitions of your states and do a lot of math. If you don't, then you just throw out the imaginary part of that log and use the real part.

*1 point [-]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.

You can think of complex number in some sense as two dimensional numbers. If you think about allowing two dimensions than why not three? What argument do you have to prefer two over three?

On the other hand you do have arguments for keeping it one dimensional.

Beyond the complex numbers are a whole host of fun number systems, the 'hypercomplex' numbers, of which the most popular seem to be the quaternions (with four dimensions) and octonions (with eight).

It seemed a good idea to ask about just ordinary, two-dimensional complex numbers, and find out if there was any use for them in this context, before inquiring about anything more complicated. :)