Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Could anybody explain to me why after branch cuts some times $\sqrt {-1}$ is taken to be $i$ whereas at others, it is taken to be $-i$? Reference notes would also be appreciated. Thank you.

share|improve this question
add comment

2 Answers 2

up vote 0 down vote accepted

Look at the square root function. Start at $z=-1$, and choose the value there to be $i$, then move one revolution around the origin.

If you assume continuity all the way around you would have $\sqrt{-1} = -i$, since the whole revolution in the argument variable corresponds to half a revolution of the function value. So in order for the square root to be a function (i.e. not give two different values for the same argument), it needs to be discontinious somewhere, and that somewhere is the branch cut, usually taken as the negative x-axis.

As to which value to choose, it's all the same. You try to chose a branch cut in each case so it doesn't interfere too much with whatever you are doing, and you choose one principal value. $-i$ is generally just as good as $i$, but in some cases one of them might yield prettier expressions.

share|improve this answer
    
Thank you for the answer, Arthur. –  Lev Oct 10 '12 at 18:46
add comment

It all depends on what you're doing, but note that $i$ and $-i$ are two expressions for the same thing ; a priori they are not algebraically distinguishable. So writing $\sqrt{-1}$ itself is ambiguous, whether you're talking about branch cuts or not ; the expression "$\sqrt{-1}$" refers to a root of $x^2 + 1$, because when we use the symbol $\sqrt{-}$ in a context where it is well-defined, it is when we work with positive arguments and assume that the extracted root is positive. With negative arguments we have no such luck.

In other words, $\sqrt{-1}$ refers to a root of $x^2+1$, and if $i$ is such a root, then $-i$ is the other one. But if we call the roots $a$ and $b$ instead, all we know is $b = -a$, but nothing prevents us from reversing the roles.

So don't worry about it all that much.

Hope that helps,

share|improve this answer
    
Thank you for the answer, Patrick. –  Lev Oct 10 '12 at 19:02
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.