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April 20th, 2015, 03:39 AM   #1
Tau
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What's the deal about complex numbers?

I read in Penrose's book The Road to Reality about the usefulness of complex numbers and in a nutshell I got that by using complex numbers and the complex plane, we get a deeper understanding of functions. The problem is that again there I read about the miraculous properties of complex numbers, but he didn't comment on why they are so useful, the deeper reason.

Is it because they quantify "errors"? So instead of simply discarding a solution that has an imaginary number, as was done by mathematicians prior to their establishment, we keep it as a sort of quantifying "how much wrong" something is in place of just labeling it wrong and discarding it.

So what's the mechanism that makes them work like they do and be used from electrical engineering to quantum mechanics and beyond? Am I completely wrong about the "quantifying errors" idea?

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April 20th, 2015, 01:07 PM   #2
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Google "imaginary numbers history".
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April 21st, 2015, 06:17 AM   #3
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I did that and one thing I found interesting is this quote by Augustus De Morgan:

"The imaginary expression √-a and the negative expression -b have this resemblance, that either of them occurring as the solution of a problem indicates some inconsistency or absurdity. As far as real meaning is concerned, both are equally imaginary, since 0 - a is as inconceivable as √-a."

The important part is "either of them occurring as the solution of a problem indicates some inconsistency or absurdity". This seems to cement my idea of "quantifying errors". Rather than labelling an expression that leads to a complex solution wrong and leaving it at that, we keep the complex solution that actually quantifies "how much" wrong it is.
Similarly if I owe to a bank we can just say that I owe or we can have a useful amount for example $-100. In the first case I have zero dollars and I owe (vague and not useful), or if we use negative numbers we can have the correct solution that I have zero dollars and I owe a specific amount, $-100. Similarly in a problem that involves complex numbers.

I'm continuing reading about their application but the heart of it seems to me to be that.

Last edited by Tau; April 21st, 2015 at 06:19 AM.
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April 21st, 2015, 07:05 AM   #4
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In physics, complex numbers often pop up to describe parameters that change upon interactions between things or combinations with other related things. Examples of these are nucleon-nucleon interactions in nuclear reactions or complex impedance in RL and RC circuits.

Complex numbers are also useful for simple harmonic motion when looking at displacement, velocity and acceleration vectors. It's cool
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April 21st, 2015, 07:36 AM   #5
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Where they become indispensable is when you have, for example, a singularity in the function you are trying to analyse. In real analysis, singularities are more or less the end of the road. The function blows up and there is nothing you can do about it other than say how quickly it does so.

Another example is where a function cannot be evaluated because we can't get a closed form for an integral.

These problems occur because we have only one "path" on which to seek a solution - we can only travel up and down the real number line. The complex plane provides us with another dimension - we can sometimes get results by travelling around the troublesome parts of the real number line.
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April 21st, 2015, 07:47 AM   #6
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Also the complex numbers give us more information about functions. You can have a function which is infinitely differentiable on the real numbers but still seems badly behaved for some reason. When you generalize to the complex numbers you can distinguish the analytic functions from the ones only differentiable on the reals. The complex singularities can still affect the function on the real line, even if it's hard to see from that angle.
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April 21st, 2015, 09:11 AM   #7
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All these I have read before but there must a be a good reason beyond "it happens that way" or "they pop up". Isn't this reason the "quantifying errors" argument?
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April 21st, 2015, 09:28 AM   #8
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I don't see them as quantifying errors. They provide functionality not available in the reals. Note that the answers are not necessarily complex, but you need the complex space to get to them.
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April 21st, 2015, 09:57 AM   #9
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Do you agree with De Morgan's observation that they arise only whenever we deal with an error ("inconsistency or absurdity")?
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April 21st, 2015, 10:18 AM   #10
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Most certainly not when they appear in the calculation rather than the answer! In the case of negative numbers, there is usually an obvious meaning to them even if we can justify not using them. For physical applications complex results can be harder to interpret, but they sometimes simply imply a lack of real-world solutions, where the mathematics requires solutions to the equations.
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