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So I know how to solve $x^2= 1-i$. We would get two roots: $2^{1/4}e^{-i\pi/8}$ and $2^{1/4}e^{7i\pi/8}$.

The question I am actually asked is what is $\sqrt{1-i}$.

I assume both roots would be the answer since there is no "positive" root concept when imaginary numbers are involved?

jjagmath
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HFM
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  • Yes thanks - there is a quote from that link which I think answers it "The square root is not a well defined function on complex numbers. Because of the fundamental theorem of algebra, you will always have two different square roots for a given number." – HFM May 23 '23 at 01:49
  • While there's about a trillion more-or-less duplicates to this question, yours is one of the better crafted ones. You are correct: if someone is asking you for $\sqrt{1-i}$ the proper answer is "It does not have a conventional and well-defined value, not like the square root of a positive number, so here, you get two of them". – Lee Mosher May 23 '23 at 01:50
  • @HFM That's great! If you feel satisfied with the other answers and comments, could you mark your question as a duplicate so it will redirect all future visitors? – bobeyt6 May 23 '23 at 01:52

1 Answers1

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Maybe.

While we can't say "pick the positive value" when we're dealing with complex numbers, there is an analogous procedure, called "picking a branch", that we can use to make the square root a proper function with just a single value. And "the principal branch" for the complex square root is what more-or-less corresponds to picking the positive number for the positive reals.

So if you've discussed "the principal branch of the square root" in your class or text, use that. Otherwise, yeah, I'd report both values.

JonathanZ
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