1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I have a question

  1. Jun 19, 2010 #1

    Is the following mathematical argument algebraically valid:

    sqrt(-1) x sqrt(-1) = [sqrt(-1)]2 = -1

    I know sqrt(-1) is a complex number, I just want to know if the argument above is valid.

  2. jcsd
  3. Jun 19, 2010 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Yes, it's valid.

    There is one subtlety: there are two numbers whose square is -1, namely [itex]i[/itex] and [itex]-i[/itex].

    Your argument works fine if you choose sqrt(-1) = [itex]i[/itex]:

    [tex]\sqrt{-1} \times \sqrt{-1} = i \times i = i^2 = -1[/tex]

    and equally well if you choose sqrt(-1) = [itex]-i[/itex]:

    [tex]\sqrt{-1} \times \sqrt{-1} = (-i) \times (-i) = (-i)^2 = -1[/tex]

    But what happens if I choose sqrt(-1) = [itex]i[/itex] for one of the square roots, and sqrt(-1) = [itex]-i[/itex] for the other one? Then I get

    [tex]\sqrt{-1} \times \sqrt{-1} = (i) \times (-i) = -(i^2) = 1[/tex]

    Uh oh!
  4. Jun 19, 2010 #3
    LOL @ "uh oh!"

    Thanks. I actually just searched through the threads and see that this question has been asked several times in different ways. But it make sense now. Thanks again.
  5. Jun 19, 2010 #4
    That so I was told, is the reason for i. Then the matter becomes [tex]\sqrt(-1)x\sqrt(-1) = i^2 = -1.[/tex]
    Last edited: Jun 19, 2010
  6. Jun 20, 2010 #5
    The point is that you have to define sqrt() to be a function, i.e. return only one value. I think this is called principal value. Then your calculations works. You just have to remember that [itex]sqrt(x^2)\neq x[/itex] in general.

    But I'm not completely sure about this Riemann surface business and what it's for :)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook