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If $i=\sqrt{-1}$, is $\large\sqrt{i}$ imaginary?

Is it used or considered often in mathematics? How is it notated?

  • 10
    The square root of i is (1 + i)/sqrt(2). [Try it out my multiplying it by itself.] It has no special notation beyond other complex numbers; in my discipline, at least, it comes up about half as often as the square root of 2 does --- that is, it isn't rare, but it arises only because of our prejudice for things which can be expressed using small integers.2010-08-25
  • 0
    Do you know the exponential notation for complex numbers, i.e. $e^{ix}=\cos x+i\sin x$?In my unser bellow I assumed you would.2010-08-26
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    Do you want an explanation just using algebraic properties of complex numbers?2010-08-27
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    This is a fairly common and natural question, you should lookup the similar threads http://math.stackexchange.com/questions/2914/how-to-combine-complex-powers http://math.stackexchange.com/questions/3210/simple-complex-number-problem-1-1 http://math.stackexchange.com/questions/1211/non-integer-powers-of-negative-numbers etc.2010-08-27
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    Just adding, complex numbers exhibit a property known as closure; any operations on them involving +, −, ×, ÷, $\sqrt{\cdots}$, $\sqrt[n]{\cdots}$, etc. will always produce an answer that is also complex.2016-12-11

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