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Name: Karl J.
Status: student	
Age:  N/A
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Date: 9/15/2004

As we all know, i is the square root of minus 1. But what is the square root of minus i?

Powers and roots of complex numbers yield some pretty weird looking results unless you read through the algebra of complex numbers found in any introductory text on the subject. A readable book on the topic that does not get too involved is Paul J. Nahin's book "An Imaginary Tale: The Story of i". It follows directly from the definition of the multiplication rules for complex numbers that (-i)^1/2 = (1/(2)^1/2) - i * (1/(2)^1/2). You can convince yourself that this is so by squaring the right hand side of the equation: = 1/2 -1*i -1/2 = -i. Perhaps even more mysterious (but it really isn't) is: (i)^i = exp(-pi/2) = 0.2078... which is a real number!! And of course there is that very remarkable relation: exp(2*pi*i) - 1 = 0 which incorporates all of the major features of numbers: 0, 1, =, minus, exp, pi, and i.

Vince Calder

Karl J.,

Once you get into complex numbers, any root can be expressed as a real number plus or minus an imaginary number. Take (a+bi) and square it. Set what you have as real equal to zero. Set what you have as imaginary equal to 1. You get (a^2-b^2)=0 and 2ab=1. First, notice that a=b is needed. You cannot use a=-b, because ab must be positive. Since a=b, 2ab requires that a=b=1/sqrt(2). We end up with (1+i)/sqrt(2).

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College

If you use exp(i*theta) = cos(theta) + i*sin(theta), then you can work it out. This relation is from DeMorgan in 1800's. So if theta = -i*pi/2, then the exp = -i. Taking sqrt of both sides, exp(-i pi/4) = sqrt(-i), but also = cos(-pi/4) + isin(-pi/4) = cos(-45) + isin(-45) = 1/sqrt(2) - i 1/sqrt(2).

There is a good book on all this, a "popular" math book, called An Imaginary Tale by Paul Nahin. It goes through the history of all this math, going back before people had all these great short-cuts to figure things out, back to the time when people wondered just what does sqrt(-1) mean.

Steve Ross


Sqrt(-i) = Sqrt((-1)*i) = Sqrt(-1)*Sqrt(i) = i*Sqrt(i) or i^(3/2)

Todd Clark, Office of Science
U.S. Department of Energy

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