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Math Help - complex no. again

  1. #1
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    complex no. again

    p306
    in my textbook it writes:
    given: z^4 =1+i
    z = [\sqrt 2 (cos \frac \pi 4 + i sin\frac \pi 4  )]^ {\frac 1 4}
    where does the \pm go?
    i don't understand why it shouldn't be z = \pm [\sqrt 2 (cos \frac \pi 4 + i sin\frac \pi  4)]^ {\frac 1 4}
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  2. #2
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    Quote Originally Posted by afeasfaerw23231233 View Post
    p306
    in my textbook it writes:
    given: z^4 =1+i
    z = [\sqrt 2 (cos \frac \pi 4 + i sin\frac \pi 4  )]^ {\frac 1 4}
    where does the \pm go?
    i don't understand why it shouldn't be z = \pm [\sqrt 2 (cos \frac \pi 4 + i sin\frac \pi  4)]^ {\frac 1 4}
    Is there a constraint?
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  3. #3
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    Quote Originally Posted by afeasfaerw23231233 View Post
    p306
    in my textbook it writes:
    given: z^4 =1+i
    z = [\sqrt 2 (cos \frac \pi 4 + i sin\frac \pi 4  )]^ {\frac 1 4}
    where does the \pm go?
    i don't understand why it shouldn't be z = \pm [\sqrt 2 (cos \frac \pi 4 + i sin\frac \pi  4)]^ {\frac 1 4}
    A little note. If you have x^2 = y^2 the solution are given by x = \pm y. The general solution x^4 = y^4 is not just x = \pm y you also have x = \pm i y as a solution as well.

    Bobak
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  4. #4
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    no. it doesn't have a constraint. it's an example in my textbook:
    e.g.3
    Find the fourth roots of 1+i
    solution:
    let z^4 =1+i
     z = [\sqrt 2 (cos \frac \pi 4 + i sin\frac \pi 4  )]^ {\frac 1 4}
    yakyakyak.... ,etc
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  5. #5
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    Quote Originally Posted by afeasfaerw23231233 View Post
    p306
    in my textbook it writes:
    given: z^4 =1+i
    z = [\sqrt 2 (cos \frac \pi 4 + i sin\frac \pi 4 )]^ {\frac 1 4}
    where does the \pm go?
    i don't understand why it shouldn't be z = \pm [\sqrt 2 (cos \frac \pi 4 + i sin\frac \pi 4)]^ {\frac 1 4}
    The book is wrong.

    It should say:

    z = \left( \sqrt 2 \left[ \cos \left( \frac \pi 4 + 2 n \pi \right) + i \sin \left( \frac \pi 4 + 2 n \pi \right) \right] \right)^{\frac 1 4}

    where n is an integer.

    When n = 0 you get one of the real fourth roots. This is the one given in the book.

    There is obviously another real fourth root, as you rightly point out. It's given by n = 2.

    Then there are two other fourth roots, both non-real, given by n = -1 and n = 1. These are the ones that bobak makes mention of.

    Four fourth roots in total, as expected.

    Other values of n just give you the same four roots again.
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