On 02/15/2012 12:34 AM, William Stein wrote:
On Tue, Feb 14, 2012 at 10:15 PM, Dr. David Kirkby
<david.kir...@onetel.net> wrote:
On 02/15/12 05:58 AM, William Stein wrote:
Hi,
A student in my class (Andrey Sarantsev) just pointed out to me that
in Sage-4.8 and Sage-5.0, we have
sage: I^(0.5)
None
What? That's not good.
I'm not just putting this on trac, because I don't even know how to
search for whether this is there already. This seems like it should
be a blocker bug.
-- William
Put me out of my misery. What is sqrt(i), and why? I've only ever raised i
(or j as us engineers use) to integer powers.
For starters, sqrt(i) is a number whose square is i, and engineers
know what "squaring" is.
If you write a complex number z in polar coordinates as z =
rho*exp(I*theta) and alpha is real number, a common choice for the
meaning of z^alpha is
z^alpha = rho^(alpha) * exp(I*theta*alpha)
We have I = exp(I*pi/2), so I^(1/2) = exp(I*pi/4). In terms of the
standard picture, it's a point on the unit circle at a 45 degree angle
from the x-axis.
sage: CC(I^(1/2))
0.707106781186548 + 0.707106781186547*I
I have a concern over I^(0.5). It is plus or minus (1+I)/sqrt(2). The
difficulty I have is with the "plus or minus". It is possible to define
the positive square root of a real number so that
sqrt(ab)=sqrt(a)*sqrt(b). But you cannot define the "positive" square
root of a complex number and still retain that identity.
--
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