On Thu, Oct 2, 2008 at 9:31 AM, jdmuys <[EMAIL PROTECTED]> wrote:
>
> Hi,
>
> I am a total newcomer, and here is very simple high-school level
> question for which I could not find an answer in several hours of
> searching:
>
> How can I use Sage to simplify ratios involving complex numbers?
>
> By simplify, I mean, to put into the canonical form a+b*i.
>
> For a very simple example: simplifying x=1/(1+i) would yield (1/2 - i/
> 2)
>
> Note: this is simple to do by hand: multiply both numerator and
> denominator by the conjugate of the denominator. For my example, this
> leads to:
>
> x= (1-i)/[(1+i)(1-i)]
> x = (1-i)/[1^2-i^2]
> x = (1-i)/[1+1]
> x = (1-i)/2
> x = 1/2 -i/2
>
> I tried quite a number of things, none of which worked.
>
> Thanks, and sorry if my question is easy (well actually, I hope it's
> easy ;-)
>
You could get the real and imaginary parts, as follows:
sage: a = (1-I)/(1 + I)
sage: a.real() + I*a.imag()
-1*I
If you're coefficients are all rational numbers, you could
alternatively define I to be the generator for the "ring" QQ[sqrt(-1)],
as follows, and all such expressions will automatically
be simplified the moment you type them in:
sage: I = QQ[sqrt(-1)].gen()
sage: 1/1 + I
I + 1
sage: 1/(1 + I)
-1/2*I + 1/2
sage: (1-I)/(1 + I)
-I
Note that expressions like sqrt(2)*I will no longer work
with this new version of I. To get back the old I, you
can do
sage: reset('I')
William Stein
Associate Professor of Mathematics
University of Washington
http://wstein.org
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