Do we use imaginary numbers just in the intermediary steps of a predictive theory? For example, in QM, in order to make predictions in the real world, you square the wave function. The wave function might have have all the information, but in order to predict something you must operate on it to remove the imaginary numbers. Is it like this in other areas, like EE, or can you have a predictive answer in terms of imaginary numbers?
In signal processing you certainly can have meaningful imaginary numbers. Basically, all you need is to have a signal where the magnitude and the phase both encode meaningful information.
Complex numbers are very useful as a means to an end. But even if they weren't so useful, we'd still study them before they're so fascinating in their own right.
You cant measure an imaginary number in nature...You can measure the coefficient (real number) of [itex]i[/itex] of the imaginary part of a complex number, but not actually measure an imaginary number. Laws of physics may take a more convenient form in terms of complex numbers, but I think it can be proved that all algebraic manipulations can be done without recourse to complex numbers as well as contour integration.
Now we are getting into the nature of reality and numbers...interesting mathematical and philosophy issue? Not qualified to say much. If others know more about this I'd like to hear about it.
Recently looking back at some fluid mechanics notes and found that problems that can be formulated as a complex o.d. diff equation were equivalent to a pair of real coupled partial diff equations. So they are not just convenient for QM calculations...
Well, you were the one who said "You cant measure an imaginary number in nature...You can measure the coefficient (real number) of i of the imaginary part of a complex number, but not actually measure an imaginary number." What did you mean by that? I see no reason to think that "complex numbers are just a means to an end" is any more true (or false) than the same statement about real numbers.
Fluid mechanics: It isn't just 2D flow. There are problems that are more easily solved by a complex o.d. diff equation, namely problems involving sinusoidal oscillations. QM involves sinusoidal oscillations - hmm.
Well you could call it measuring a complex number but really you are measuring the real numbers that can be extracted from the complex number, for example [itex]a[/itex] and [itex]b[/itex] from [itex]z = a + ib[/itex] or with [itex]z = R e^{i \theta}[/itex] we can extract [itex]R[/itex] and [itex]\theta[/itex]. Formulation of problem to solution can be done without recourse to complex numbers though.
Except you can't measure (observe) real numbers either because we don't have infinite precision. We model with the real numbers. Heck you can still do the most important aspects of calculus without the reals, this is called computable analysis.
It brings up the Gödel's incompleteness theorem by which we as humans can understand things that would take and infinite number of computations.
How do you prove that axb = bxa in a computable manner? It is obvious that this is right, but how do you do this via a computable manner for all a and b? Possibly I could be misunderstanding.
If you can measure a real number, you can measure a complex one. You can measure phase and amplitude of some physical quantity.
And you cannot measure a physical quantity to be [itex]\sqrt{2}[/itex] (or even 2). That was my (and others) point.
The first paragraph or so of http://math.buffalostate.edu/~MED600/signednumbrs/jacobstm.pdf quotes an amusing and relevant anecdote by Isaac Asimov.
What if a certain quantum observable takes discrete values? Assuming QM and its predictions are right. If you measure it to be in a definite discrete quantum state and then that eigenvalue is determined exactly. EDIT: fundamental constants may not be determined exactly.
This is deviating away from the OP question. There will be a proof that what can be expressed via the real numbers associated with complex numbers, and the intermediate steps in a calculation involving complex number's rules can all be done without recourse to complex numbers. This is the case in physics where nothing can be measured as [itex]i[/itex] or proportional to [itex]i[/itex], however maths people want solutions to [itex]\sqrt{-1}[/itex].