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Radical i
Name: Karl J.
Status: student
Age: N/A
Location: N/A
Country: N/A
Date: 9/15/2004
Question:
As we all know, i is the square root of minus 1. But
what is the square root of minus i?
Replies:
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
Karl,
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
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