# square root of i

• Nov 21st 2013, 03:23 AM
Tweety
square root of i
Find the square roots of i

?

would the modulus = i ? and not sure how to work out its argument?

any help apprecited
• Nov 21st 2013, 03:36 AM
Plato
Re: square root of i
Quote:

Originally Posted by Tweety
Find the square roots of i ?

Well of course as with any complex number there are two square roots:
$\displaystyle \sqrt {|z|} \exp \left( {\frac{{\arg (z)i}}{2} + \pi ki} \right),~k=0,~1$
• Nov 21st 2013, 03:37 AM
SlipEternal
Re: square root of i
$\displaystyle e^{i\theta} = \cos \theta + i \sin \theta$

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 $\displaystyle [0,2\pi)$:

$\displaystyle e^{\pi i/4},e^{5\pi i/4}$
• Nov 21st 2013, 03:50 AM
Tweety
Re: square root of i
why plug in pi/2 ?
• Nov 21st 2013, 04:00 AM
Tweety
Re: square root of i
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

$\displaystyle \frac{1}{\sqrt{2}}(1+i)$

$\displaystyle -\frac{1}{\sqrt{2}}(1+i)$
• Nov 21st 2013, 07:02 AM
HallsofIvy
Re: square root of 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 $\pi/4$. 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 $\displaystyle \frac{\sqrt{2}}{2}(1- i)$.