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Thread: imaginary numbers

  1. #1
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    imaginary numbers

    When one converts between trigonometric and exponential equations one uses imaginary numbers and the other does not. This is the example I am using



    is the same as



    If something can be written without imaginary numbers, why would we use imaginary numbers. For me I can get my head around trigonometry, but not imaginary numbers.
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    Because it is often the case, especially when differentiating or integrating complex expressions, that the exponential/imaginary number representation is considerably easier to use. You can also extend complex numbers into quaternions which are useful in modern physics.

    You could turn it around and ask that if you can represent trigonometric results using imaginary numbers why would you ever use trig? It is down to what is easiest to use for the task and personal preference.

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    Quote Originally Posted by Shaula View Post
    Because it is often the case, especially when differentiating or integrating complex expressions, that the exponential/imaginary number representation is considerably easier to use. You can also extend complex numbers into quaternions which are useful in modern physics.

    You could turn it around and ask that if you can represent trigonometric results using imaginary numbers why would you ever use trig? It is down to what is easiest to use for the task and personal preference.
    That is what I figured. Sometimes I just wonder if all the mathematical tricks get people confused about what is real.
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    Quote Originally Posted by Copernicus View Post
    That is what I figured. Sometimes I just wonder if all the mathematical tricks get people confused about what is real.
    Probably has less impact than simplified, popular science analogies that confuse people about what a scientific model actually says.

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    Quote Originally Posted by Shaula View Post
    Probably has less impact than simplified, popular science analogies that confuse people about what a scientific model actually says.
    That is funny, but true! So many people think we will have warp speed and transport people through space eventually, just like star trek. https://www.reddit.com/r/TaylorSwift...string_theory/

    Really I wonder if the math, symbols, mathematical tricks subscripts, superscripts, Greek letters, jargon, and acronyms seem to make physics very difficult and I just wonder how many people in physics have the mental energy to slog through all of it and instead just talk about it.
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    Quote Originally Posted by Copernicus View Post
    That is funny, but true! So many people think we will have warp speed and transport people through space eventually, just like star trek. https://www.reddit.com/r/TaylorSwift...string_theory/

    Really I wonder if the math, symbols, mathematical tricks subscripts, superscripts, Greek letters, jargon, and acronyms seem to make physics very difficult and I just wonder how many people in physics have the mental energy to slog through all of it and instead just talk about it.
    personally I feel it is important to teach the practical side of physics before the maths so as to create enthusiasm, there are so many great classroom demonstrations and experiments that need only simple maths. Actually maths is the same, you don't expect to swallow it all at once. I do think you have pointed to a very important problem in the public understanding of science. It has become so hard to grasp the meaning that physics is in danger of looking just like inaccessible magic to a vast majority. That is very dangerous for our society.
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    Quote Originally Posted by Copernicus View Post
    Really I wonder if the math, symbols, mathematical tricks subscripts, superscripts, Greek letters, jargon, and acronyms seem to make physics very difficult and I just wonder how many people in physics have the mental energy to slog through all of it and instead just talk about it.
    Of course people talk about it - but when they are writing a research paper or planning an experiment they stop talking about it and go back to the maths. Because any other approach is fraught with danger and scientists in general know this. I think just about every researcher had had the experience, often early on, of talking themselves into something that seems so right but fails horribly when scrutinised.

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    Quote Originally Posted by profloater View Post
    I do think you have pointed to a very important problem in the public understanding of science. It has become so hard to grasp the meaning that physics is in danger of looking just like inaccessible magic to a vast majority. That is very dangerous for our society.
    What, are physicists not allowed to be professionals now? I don't always understand how medicines I am prescribed work. I don't understand how my car works (too many electronics). I don't understand how some cryptographic methods work. I could go on. But I trust the professionals in these fields to cure me, fix my car, secure my transactions etc. I know that if I spent the time and effort I could understand these things. Don't see why physics should be different!

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    Quote Originally Posted by Shaula View Post
    What, are physicists not allowed to be professionals now? I don't always understand how medicines I am prescribed work. I don't understand how my car works (too many electronics). I don't understand how some cryptographic methods work. I could go on. But I trust the professionals in these fields to cure me, fix my car, secure my transactions etc. I know that if I spent the time and effort I could understand these things. Don't see why physics should be different!
    Good point Shaula! But it seems it is too much work for physicists to understand each other. For example it appears that not many scientists are trying to understand Verlinde's work. http://backreaction.blogspot.com/201...t-gravity.html
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    Quote Originally Posted by Copernicus View Post
    Good point Shaula! But it seems it is too much work for physicists to understand each other. For example it appears that not many scientists are trying to understand Verlinde's work. http://backreaction.blogspot.com/201...t-gravity.html
    Physicists work to understand each other. That a reason why physics textbooks are written. The issue is rather with Verlinde's work which is specialized and so is of interest to a much smaller group of scientists. Look at the work the blog author did to better understand it:
    Verlinde’s paper is indeed not an easy read. I spent some time trying to make sense of it and originally didn’t get very far. The whole framework that he uses – dealing with an elastic medium and a strain-tensor and all that – isn’t only unfamiliar but also doesn’t fit together with general relativity.
    The basic tenet of general relativity is coordinate invariance, and it’s absolutely not clear how it’s respected in Verlinde’s framework. So, I tried to see whether there is a way to make Verlinde’s approach generally covariant. The answer is yes, it’s possible. And it actually works better than I expected. I’ve written up my findings in a paper which just appeared on the arxiv:

    A Covariant Version of Verlinde's Emergent Gravity
    S. Hossenfelder
    arXiv:1703.01415 [gr-qc]
    This is what usually happens with a new theory. The author(s) present it. A small group of scientists understand it. They write papers clarifying the theory. More scientists now understand it. They write papers clarifying and expanding the theory. Repeat.

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    Quote Originally Posted by Copernicus View Post
    If something can be written without imaginary numbers, why would we use imaginary numbers. For me I can get my head around trigonometry, but not imaginary numbers.
    It took me a long time to realize that what an be done with imaginary numbers can also be done without them. And that the reason we do use imaginary numbers is pretty much because it makes the math much easier.

    I came up with an analogy, to help me understand the rationale.


    Consider drawing the reflection in a flat mirror of a complicated object. You have to
    1. measure the angle of every important incoming ray (eg. vertices),
    2. reflect that ray 180 degrees,
    3. draw it back out with the previously measured angle to its final position.
    This is the "real world" way reflections work.

    But a far simpler way to render the reflection of an object is simply to
    1. draw straight lines right through the mirror to its virtual reflection.
    By its nature, every straight line that intersects the mirror will be at the correct angle in the virtual image behind the mirror. No matter how many points you decide to trace - you literally don't have to measure a single angle!

    The "world" behind a mirror is virtual, but it serves a very useful purpose. Much like imaginary numbers.
    Last edited by DaveC426913; 2017-Nov-28 at 11:59 PM.

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    A friend of mine asked me if imaginary numbers might be God-like. I told him no, they are an invention of our intellect that adds to our set of tools for doing useful calculations. The complex number system with its "real" and "imaginary" components is very handy for making valid calculations of the behavior of very real alternating current in electrical circuits that include capacitors and inductors. I shared my opinion that real and imaginary were unfortunate choices of words, with neither one being any more "real" than the other.

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    You can represent complex numbers as a pair of numbers if you like. Instead of , we can just have . Then we can define arithmetic operations like



    and

    .

    But this is really just semantics; I'm esssentially writing down the rules for complex arithmetic, without calling them complex numbers.

    I would draw an analogy with vectors and matrices. You don't have to write down a matrix equation; you can write down a system of equations for each individual element of the result. But if you understand how matrices work, and are comfortable with them, then writing an equation in matrix form can be simpler and more intuitive than writing a complicated system of equations without matrices. Complex numbers are no different; we can rewrite a lot of mathematics to avoid them, but if we are OK with using them, and do it enough that we are comfortable with them, mathematics can be a lot more clean, elegant, and intuitive with them rather than without.

    Take an example of a differential equation,



    where is a function of . If , the solution is



    where and are arbitrary constants. (Typically for some real-world physics problem, there would be additional constraints that would allow us to identify and .)

    But what if ? We can write the solution as

    .

    So this method completely avoids the use of complex numbers, and provides one solution for , and another solution for .

    But, you could also just use the first solution in both cases. It is



    This solution is perfectly valid for ; then the square root operations produce imaginary arguments, and the exponential functions of the imaginary exponents become trigonometric functions.

    So there are two ways to solve the problem. One avoids the use of imaginary numbers completely, and has two different forms for and ; the other uses imaginary numbers for , but has exactly the same form in both cases.

    Which is better? It's a matter of preference.

    Other examples - I don't know how you feel about power series. But if you are familiar with them, sometimes they converge, and sometimes they don't. Determining when they converge and when they don't can be done in a very simple, elegant way if you consider the extension of the function to complex arguments. Functions that seem very well behaved for real-valued arguments can still have power series that don't converge - the reason is that the function is badly behaved for some complex-valued arguments.

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    Quote Originally Posted by Hornblower View Post
    A friend of mine asked me if imaginary numbers might be God-like. I told him no, they are an invention of our intellect that adds to our set of tools for doing useful calculations. The complex number system with its "real" and "imaginary" components is very handy for making valid calculations of the behavior of very real alternating current in electrical circuits that include capacitors and inductors. I shared my opinion that real and imaginary were unfortunate choices of words, with neither one being any more "real" than the other.
    I've had this argument with people. Most of them disagree with me, and feel that "three" is more "real" than "three times i". The reason is that they can easily visualise three rocks, or even go out and find three rocks. The distinction between "three" and "three rocks" is somewhat lost on them.

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    Thanks all!
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    Quote Originally Posted by Copernicus View Post
    Good point Shaula! But it seems it is too much work for physicists to understand each other. For example it appears that not many scientists are trying to understand Verlinde's work. http://backreaction.blogspot.com/201...t-gravity.html
    Disagree with that completely. At least three experimental tests of it have been tried, there are a hundred or more citations for the paper, it was presented at Strings 11, there are at least three derivative theories of it already, it has been used in LQG work... One anecdote doesn't seem to counter that. Especially when the point of the anecdote appears to be to discredit objections to the theory as being by people who 'just do not get it'

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    Quote Originally Posted by Shaula View Post
    Disagree with that completely. At least three experimental tests of it have been tried, there are a hundred or more citations for the paper, it was presented at Strings 11, there are at least three derivative theories of it already, it has been used in LQG work... One anecdote doesn't seem to counter that. Especially when the point of the anecdote appears to be to discredit objections to the theory as being by people who 'just do not get it'
    Thanks for clarifying this Shaula. I don't really have an opinion on Verlinde's work, except that I think it must represent some part of reality, based off of his credentials.
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