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Any real world use of imaginary numbers?

  1. Apr 22, 2013 #1
    Everybody says that it is used in engineering or somewhere but how can you use it.
    in real world it is impossible to take square of any number and get negative answer.
    how can it have any use when it does not even exist.
    and people talk about imaginary plane, what is it?
    Thanks for helping a curious young physicist learning maths just to do physics :tongue:
    Last edited by a moderator: Apr 23, 2013
  2. jcsd
  3. Apr 22, 2013 #2


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    The imaginary plane is a plane for which the [itex]``x''[/itex] axis is a real number, and the [itex]``y''[/itex] axis is an imaginary number. It may help to have a picture, so I'll "borrow" one from Wikipedia:
    Typically we write a complex number as [itex]z = a + jb[/itex], where [itex]a[/itex] is the real part (plotted on the horizontal axis) and [itex]b[/itex] is the imaginary part, plotted on the vertical axis. In this plot, [itex]r[/itex] is the magnitude of the complex number and [itex]\varphi[/itex] is the argument. Their definitions are straightforward from the geometry.

    One example from engineering is looking at the response of a dynamical system. A simple example is a second-order mass-spring damper, with an equation given by: [tex]\ddot{x} + 2\zeta\omega_n\dot{x} + \omega_n^2 = u(t)[/tex] where [itex]x[/itex] is displacement, [itex]\zeta[/itex] is the damping ratio, [itex]\omega_n[/itex] is the natural frequency, and [itex]u(t)[/itex] is an unspecified forcing function or control.

    If you take the Laplace transform of the system with zero initial conditions, you can get the transfer function [tex]\frac{X(s)}{U(s)} = \frac{1}{s^2 + 2\zeta\omega_ns + \omega_n^2}.[/tex] If the system is underdamped, i.e. [itex]\zeta < 1[/itex], the poles of the transfer function (which are also the eigenvalues of the system) are [itex]s \in \mathbb{C}[/itex]. For this second order system, we have [tex]s_{1,2} = -\zeta\omega_n \pm j\omega_n\sqrt{1-\zeta^2}[/tex] where [itex]j:=\sqrt{-1}[/itex]. We can plot the poles on the complex plane, which in control engineering is called the s-plane if the system is on a continuous domain, and have a graphical representation of the system response and stability.

    If a pole lies on the left half of the real axis, the system is dynamically stable. If the pole lies directly on the real axis (i.e. [itex]\Im(s)=0[/itex]) the system has no oscillatory part and is overdamped. If the pole lies somewhere off of the real axis, then the system is underdamped and the response takes the form of a damped sinusoid. If the pole is purely imaginary, then the system is neutrally stable and oscillatory, while any pole whose real part is positive is unstable. We call this a root locus plot, an example of which (again from Wikipedia) is:
    https://upload.wikimedia.org/wikipedia/commons/4/45/RL%26ZARL-%281_2%29-%281_3_5_1%29.png [Broken]
    In the plot, x's (x) signify poles and circles (o) signify roots. Using our transfer function notation [itex]\frac{X(s)}{U(s)}[/itex], roots are solutions to the equation [itex]X(s)=0[/itex] (i.e. set the numerator equal to zero and solve for [itex]s[/itex]) and poles are solutions to the equation [itex]U(s)=0[/itex] (i.e. set the denominator equal to zero and solve for [itex]s[/itex]).
    Last edited by a moderator: May 6, 2017
  4. Apr 22, 2013 #3


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    Alternating current analysis heavily depends on the use of imaginary numbers.

    Some hydrodynamic analysis and the solution of harmonic problems use imaginary numbers.

    In short, imaginary numbers provide additional tools for physicists and engineers to use in solving many real world problems.
  5. Apr 22, 2013 #4


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    Good question.

    I am an electrical engineer and hardly a day goes by where I do not extensively rely on complex numbers. Many of the applications use Euler's equation
    e^{i \omega t} = \cos(\omega t) + i \sin(\omega t),
    and Fourier series/transforms. The simplest examples are in linear circuit analysis, which are described by constant coefficient ordinary differential equations. If you have a sinusoidal voltage source driving the circuit, then the solution will be of the form [itex]A \cos(\omega t + \phi)[/itex]. The algebra to find [itex]\phi[/itex] in particular can be awful. However, if you represent your source as a complex exponential, you assume your solution is of the form [itex] B e^{i\omega t}[/itex] where [itex]B=|B|e^{i\phi}[/itex] is now a complex number that only requires straightforward algebra to find. At the end of the day, you take the real part of your complex solutions, which is how you relate the solution to the real world. The quantity [itex]\phi[/itex] is called the phase, and dealing with phase is WAY easier using complex numbers. Dealing with phase is also why complex numbers are often used in signal processing and designing/analyzing communications systems. Indeed, the digital data inside of a modern radar or communication system (after the analog to digital converter) is typically represented with complex numbers precisely to enable the easy exploitation of phase.

    For lots of examples, you can look at EECS courses at MIT:
    Many of the courses certainly use complex numbers all over the place. As examples, check out 6.002 (circuits), 6.003 (signals and systems), 6.011 (signal processing/comms). Note EEs usually use [itex]j=\sqrt{-1}[/itex], instead if [itex]i[/itex].

    Hope that helps.

    Last edited by a moderator: Apr 23, 2013
  6. Apr 22, 2013 #5
    Let's follow the idea that the numbers don't exist. Perhaps that is true. You can view it as being the algebraic completion of the real numbers. You can create methods which work very generally. Often, these methods give you a very general form of the solution, and a couple steps later, you arrive at the real world solution. It would have been *much* harder to derive the solution to your real world problem without referring to the complex numbers. They are a tool. If you don't use it, it will be much harder to come up with the solution. They do not have to appear in the final answer to a real world problem, but the method of finding that answer will go much better if you take advantage of the more robust math offered by considering the complex number system.

    I think of it as a switchblade, or a letterman's tool. You're working on the real line, arrive at a difficult problem, and flip out the imaginary axis to help solve the problem.
  7. Apr 23, 2013 #6
    Complex numbers are much more convenient than "real" numbers for circles and periodic phenomena like waves. Waves are very common in physics, so complex numbers are too.

    Waves are so basic that it is possible to convince yourself that complex numbers are the true mathematics. Natural numbers are for people who count on their fingers.:!!)

    Forget about all that "square root of negative one" stuff. It's historical and mostly serves to confuse the situation. The operations of addition and subtraction are pretty useless too. The geometric view is more useful.
    Last edited: Apr 23, 2013
  8. Apr 23, 2013 #7
    It is not possible based on real multiplication.

    But consider ℂ simply an algebra over a field based on the (ℝ²,+) vector space over ℝ.
    (0,1) is the number, for which, by the multiplication rule of the field (ℂ,+,⋅)
    (0,1) ⋅ (0,1) = (-1, 0)
    holds, and one usually defines
    i := (0,1).

    There's no voodoo of any kind.

    It exists and there's much use of it:

    Consider e.g. a π/2 rotation:
    [tex]\left( \begin{array}{cc}0 &-1\\1 &0\end{array}\right) \left( \begin{array}{cc}x\\y \end{array}\right) = \left( \begin{array}{cc}-y\\x \end{array}\right)[/tex]
    Matrices of the type [tex]\left( \begin{array}{cc}a &-b\\b &a\end{array}\right)[/tex] with [tex]a,b \in ℝ[/tex] with usual addition and multiplication form a field which is ≅ ℂ under this map.
    [tex]\left( \begin{array}{cc}a &-b\\b &a\end{array}\right) \rightarrow a + ib[/tex]
    So one could also define
    [tex]i := \left( \begin{array}{cc}0 &-1\\1 &0\end{array}\right)[/tex]

    And that is for sure a most useful matrix.
  9. Apr 23, 2013 #8


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    In addition to the engineering examples, they're used all the time in quantum mechanics.

    It's impossible in the set of real numbers. The real world has nothing to do with it.

    You might as well ask that about the real numbers. They are no more and no less real than the complex numbers. The axioms of set theory allow us to construct a set whose members have the properties of real numbers, and a set whose members have the properties of complex numbers. It is only in this sense that either of these types of numbers can be said to "exist".

    It's the "complex plane" (sometimes called "Argand plane"). There are two lines in it called "the real axis" and "the imaginary axis".

    Complex numbers can be defined as ordered pairs of real numbers, for which we have defined the following addition and multiplication operations:


    Ordered pairs are defined so that (a,b)=(c,d) if and only if a=c and b=d. The real axis is the set of all pairs whose second component is 0. The imaginary axis is the set of all ordered pairs whose first component is 0.

    Since the ordered pairs that belong to the real axis have exactly the same properties as the real numbers we started with, we can now think of them as real numbers and simplify the notation from (x,0) to just x. We also introduce the notation i=(0,1). Note that for all real numbers a and b, we have
    a+ib&=(a,0)+(0,1)(b,0)=(a,0)+(0\cdot b-1\cdot 0,0\cdot 0+1\cdot b)\\
    \end{align} and
    $$i^2=(0,1)(0,1)=(0\cdot 0-1\cdot 1,0\cdot 1+1\cdot 0)=(-1,0)=-1.$$ The set of real numbers are often represented geometrically as a line, so it makes sense to represent the set of "complex numbers" (=ordered pairs of real numbers) as a plane.
  10. Apr 23, 2013 #9
    You can derive almost every trigonometric identity using "Euler's Formula"

    $$e^{i\phi} = \cos \phi + i \sin \phi$$​

    I am sure you can easily find applications to Trigonometry.
  11. Apr 23, 2013 #10
    And we would make a new field ##\cong \mathbb{R^{3}}## if we could find a way to multiply those vectors.
    Last edited by a moderator: Apr 23, 2013
  12. Apr 23, 2013 #11
    The simplest way that I think about it is to recognize that the real numbers describe things that go on in the real world for certain kinds of things, eg, linear translation and things with numeracy (ordering). In that sense, all numbers are imaginary, they just have relations that correspond to part of the physical world.

    The complex numbers, as in the previous answers, are better at describing periodic motion and rotation. You can call sqrt (-1) anything you like but what is important is that it is the symbol which, when used with multiplication will rotate your object 90 degrees. If you multiply twice you will wind up on the other side of the "mirror" (that's why you use the mathematical connection with sqrt (-1)). Bottom line: any number system has internal consistencies and operations which are useful or not depending on the correspondence with the physical system.

    Does that help?
  13. Apr 23, 2013 #12
    Last edited by a moderator: Apr 23, 2013
  14. Apr 23, 2013 #13
    Also historically, in BCE in Egypt, a guy was calculating the volume of a pyramidic frustum, and needed imaginary numbers to solve the equations. That is the first known usage of imaginary numbers.
  15. Apr 23, 2013 #14
    Thank you everybody!
  16. Apr 24, 2013 #15
    Was it Moses? :tongue:
  17. Apr 24, 2013 #16
  18. Apr 24, 2013 #17
    Nope, Heron of Alexandria :)
  19. Apr 26, 2013 #18
    Let me put it this way: you try evaluating ##\int t^2 e^{t/2} \cos(\frac{\sqrt{3}}{2}t) \ dx##, an expression that may easily show up in the solving of differential equations in physics (notably ##L[y]=\frac{d^2y}{dt^2}+\frac{dy}{dt}+y=t^2## for a linear operator L), without complex numbers.

    ...Yeah, I thought so. :tongue:
  20. Apr 27, 2013 #19
    Solving integration problems in the complex plane can be easier than doing the integration in the reals.

    Similar to the complex numbers, there is another algebraic structure called the quaternions. This can also be used in physics. It is most often used for computer graphics because quaternions model rotations with movement very well.
  21. Dec 15, 2013 #20
    Think of it the other way around. You specify points in the plane with two numbers. Now set up a rule so that multiplication by a number, call it i, will have the effect of rotating the point in the plane by 90o. Too make it all consistent, see how i lines up with the real numbers. Turns out i^2 = -1 so useful for geometry but not for arithmetic. All numbers imaginary. It just depends what physical process they correspond to.
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