Find the square roots of i
?
would the modulus = i ? and not sure how to work out its argument?
any help apprecited
$\displaystyle e^{i\theta} = \cos \theta + i \sin \theta$
Plug in $\displaystyle \dfrac{\pi}{2} + 2\pi n$:
$\displaystyle e^{i \left(\pi/2 + 2\pi n\right)} = i$
Take the square root of both sides:
$\displaystyle e^{i\left(\pi/4 + \pi n\right)} = \sqrt{i}$
There are two solutions in the range $\displaystyle [0,2\pi)$:
$\displaystyle e^{\pi i/4},e^{5\pi i/4}$
why cant I just work it out like this, the above looks quite complicated,;.....
let z = 0 + 1*i
than the modulus of z = $\displaystyle \sqrt{0^{2} + 1^{2}} = 1 $
arg(z) = $\displaystyle tan \alpa = 1 = \frac{\pi}{4}} $
but not sure how to go from here, the correct answers are
$\displaystyle \frac{1}{\sqrt{2}}(1+i) $
$\displaystyle -\frac{1}{\sqrt{2}}(1+i) $
First, what made you think, originally, that there were four square roots of i?
Second, while the modulus of z= i is 1, its argument is NOT [itex]\pi/4[/itex]. I don't know where you got that "1" in "$\displaystyle arctan(1)$". Surely you don't think argument is the arctangent of the modulus? The argument of "a+ bi" is the arctangent of b/a, if a is not 0. Of course, if a= 0 then a+ bi= bi which has modulus $\displaystyle \frac{\pi}{2}$ if b> 0, $\displaystyle -\frac{\pi}{2}$ if b< 0. (0 has no "argument").
So if z= i, $\displaystyle arg(z)= \frac{\pi}{2}$. HALF of that is $\displaystyle \frac{\pi}{4}$ and half of $\displaystyle \frac{\pi}{2}+ 2\pi= \frac{5\pi}{2}$ is $\displaystyle \frac{5\pi}{4}$. Those are the arguments of the two square roots of i which, of course, have modulus $\displaystyle 1^{1/2}= 1$, so the two square roots are $\displaystyle \frac{\sqrt{2}}{2}(1+ i)$ and $\displaystyle \frac{\sqrt{2}}{2}(1- i)$.