Originally posted by mazza_728
this is how i was taught to do it but i really dont understand it.. can u explain all steps in laymens terms by any chance.. explain it like ur explaining to a three year old because that is how my brain is functioning at the moment .
Heheh ok.
z^2=i
Firstly, remember that z can be expressed in mod-arg form as |z|(cos@+isin@), and by de moivre's theorem:
z^2=|z|^2cis2@
so now we have:
|z|^2cis2@=i
also, i can be expressed as cos(pi/2)+isin(pi/2)
so:
|z|^2cis2@=cis(pi/2)
Now if 2 complex numbers are equal, their modulii are equal, and so are their arguments, so:
|z|^2=1 (equating modulii)
|z|=1
and equating arguments:
cis2@=cis(pi/2)
taking the inverse cis:
2@=pi/2 +2k*pi (because an angle outside of the principle range is has an equivalent, which is +2k*pi where k is an integer)
@=pi/4 + k*pi
Now in the roots, the only difference will be the value of @, which varies with k.
so firstly let k=0, then z=cis(pi/4).
secondly let k=-1, z= cis(-3/4)
No matter which integer you let k equal, it will always be one of these two roots. (for example, cis(5pi/4)=cis(pi/4))
i really like this method but when i try and solve say:
z4+1=0
how does it work??
If you wanted to use that method for this, you would need to do:
z^4=-1
(a+ib)^4=-1
then expand out (a+ib)^4, and equate reals and imaginaries, but you do
not want to do this with ^4, it will get too messy.
