Name: Karl J.
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.
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
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) +
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(-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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Update: June 2012