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B Complex Numbers in a Simple Example that I am Very Confused

  1. Jan 23, 2017 #1
    There a simple math example that I am confused ##(\sqrt {-4})^2##
    Theres two ways to think
    1-##\sqrt {-4}=2i## so ##(2i)^2=4i^2## which its ##-4##
    2-##\sqrt {-4}##.##\sqrt {-4}##=##\sqrt {-4.-4}=\sqrt{16} =4##

    I think second one is wrong but I couldnt prove how, but I think its cause ##\sqrt {-4}## is not "reel" number so we cannot take them into one square root
     
    Last edited by a moderator: Jan 23, 2017
  2. jcsd
  3. Jan 23, 2017 #2

    fresh_42

    Staff: Mentor

  4. Jan 23, 2017 #3
    Last edited: Jan 23, 2017
  5. Jan 23, 2017 #4

    Mark44

    Staff: Mentor

    It's "real" number, not "reel" number.

    Formula 2 is incorrect. ##\sqrt a \sqrt b = \sqrt{ab}## only if both a and b are nonnegative.
     
  6. Jan 23, 2017 #5
    Thx a lot.My typo
     
  7. Jan 24, 2017 #6

    Svein

    User Avatar
    Science Advisor

    Well - in the complex domain things are not the same as they are in the real domain. First, you have [itex]-4=4e^{\pi i} [/itex] so you might think that [itex]\sqrt{-4}=2e^{\frac{\pi}{2} i} = 2i[/itex]. But you also have [itex] -4=4e^{3\pi i}[/itex] and therefore [itex] \sqrt{-4}=2e^{\frac{3\pi}{2} i}=-2i [/itex]. Squaring either of the roots brings you back to [itex] -4[/itex].
     
    Last edited: Jan 24, 2017
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