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I^i means WHAT?

  1. Aug 10, 2010 #1
    i^i means WHAT?!?!?!?!?!

    My friend gave me the brain-teaser "i^i = what?", and with a little bit of coaching I finally discovered that

    i^i = e^(-pi/2)

    Which is cool, I suppose. But the more I think about it, the more I wonder:

    what the heck does it mean???

    For that matter, what does anything raised to an imaginary power mean?
    For instance:
    e^(ix) = cos(x)+i*sin(x)

    It's so counter-intuitive, is there some way to make sense of it by analogy with real numbers? Or will it just remain abstract until the end of time?
  2. jcsd
  3. Aug 10, 2010 #2
    Re: i^i means WHAT?!?!?!?!?!

    For e^(ix) one approach is to note that for real constants a, y=e^(ax) is the solution of y'(x) = ay(x), y(0)=1. So for complex number i, to evaluate e^(ix) we want complex-valued function y such that y'(x) = iy(x), y(0)=1. The solution is y(x) = cos(x) + i sin(x).
  4. Aug 10, 2010 #3


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    Re: i^i means WHAT?!?!?!?!?!

    e^(ix) = cos(x)+i*sin(x) may not be intuitive, but you can prove it by expanding both sides into power series and see they are the same.
  5. Aug 12, 2010 #4
    Re: i^i means WHAT?!?!?!?!?!

    Why are real numbers any more 'real' that complex numbers? They are just convenient representations of things that we observe around us.

    The function [tex]e^x[/tex] gives us an easy/convenient way to represent other quantities. I assume you're taking some form of coaching for competetive examinations from your post. At this level, there arent a lot of examples that I can think of. It's a very useful function when dealing with geometrical rotation of a quantity or geometrical figures in general (eg. a circle of unit radius or other regular polygons).

    You can also define any possible mathematical function in terms of [tex]e^x[/tex] with something called Fourier Analysis, which is one of the corner stones of signal analysis and digital electronics.

    In Electrical engineering, I doubt you can go through a single topic without encountering the exponential function in some form. Its also lends itself to very very convenient representations of AC waveforms and provides a graphical analogy to the same (Phaser Diagrams).

    There are countless examples of the use of the exponential function in 'real' life. Hope that helps.
  6. Aug 16, 2010 #5
    Re: i^i means WHAT?!?!?!?!?!

    It doesn't mean that, [tex]i^{i}[/tex] means an infinite number of different values.

    Think about square roots, or things raised to the power of 1/2. Then this means a set of two different numbers (except for 0). For irrational numbers, you can get an infinite number of (complex) values.

    For complex numbers, we can define a general power [tex]z^{v}[/tex] as:


    and log(z) can have an infinite number of values. This definition of exponentiation is basically just a rule which is consistent with all the older laws for use with real numbers.

    So in your case:

    [tex]i^{i}=e^{i.log(i)}=e^{i((-1/2+2n).\pi .i)}[/tex]

    for n any natural number. So [tex]e^{-\pi /2}[/tex] is an answer, but there are many other values also.
  7. Sep 7, 2010 #6
    Re: i^i means WHAT?!?!?!?!?!

    I like to think of imaginary numbers to be a convenient way to "extend the number-line" so to speak... If you look back in history to when we first conceived the "real" numbers, to the time when we formalized them into the set [tex]R[/tex] and then to the point where imaginary numbers came into play -- none of it is intuitive. It's only intuitive to us because we're comfortable with it.

    There's a nice documentary on this subject, I think it was put out by BBC... It's on YouTube somewhere. You'll find it if you search "Mathematics Documentary".
  8. Sep 10, 2010 #7
    Re: i^i means WHAT?!?!?!?!?!

    What does it mean for a number to be raised to a fractional power? if 2^2 = 2 x 2, what the heck does 2^(1/2) mean?

    Generalizations don't always make sense. The power of generalizing in math is that you throw away some of your assumptions (exponentiation has a literal interpretation in terms of multiplication) while maintaining handy invariants (x^(a+b) = x^a * x^b) in order to describe a larger (and often more abstract) class of mathematical objects (real numbers, as opposed to the integers).

    Exponentiation is interesting, too, in that it loses one interpretation (iterated multiplication) and gains a new one (converting rectangular form to polar form).
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