1. ## Complex number

Find the square roots of the following complex numbers:
a)- i
b) x+ i(x^4+x^2+1)^1/2

Thanks

2. Originally Posted by matsci0000
Find the square roots of the following complex numbers:
a) i
well $\displaystyle i = \sqrt{-1}$ so therefore

$\displaystyle \sqrt{i} = \sqrt{\sqrt{-1}} = \sqrt[4]{-1}$

3. a) Let $\displaystyle z=-i$. We have to find the complex numbers $\displaystyle w=a+bi$ such as $\displaystyle z=w^2$

$\displaystyle -i=a^2-b^2+2abi$ and $\displaystyle |z|=|w|^2$. Then

$\displaystyle \left\{\begin{array}{ll}a^2-b^2=0\\2ab=-1\\a^2+b^2=1\end{array}\right.$

From the first and the tird equation we get (using the sign for a and b from the second equation)

$\displaystyle a=\pm\frac{\sqrt{2}}{2}, \ b=\mp\frac{\sqrt{2}}{2}$

Then, $\displaystyle w_1=\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i, \ w_2=-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i$

b) Using the same method we have

$\displaystyle \left\{\begin{array}{ll}a^2-b^2=x\\2ab=\sqrt{x^4+x^2+1}\\a^2+b^2=x^2+1\end{arr ay}\right.$

We get $\displaystyle a=\pm\sqrt{\frac{x^2+x+1}{2}}, \ b=\pm\sqrt{\frac{x^2-x+1}{2}}$

Then, $\displaystyle w_1=\sqrt{\frac{x^2+x+1}{2}}+i\sqrt{\frac{x^2-x+1}{2}}, \ w_2=-\sqrt{\frac{x^2+x+1}{2}}-i\sqrt{\frac{x^2-x+1}{2}}$