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I've struggled for days reading about square roots of complex numbers and I get most of the problems but not this one. I really want to understand what is going on in this problem, hope someone can help!
1. The complex number (C) is [tex] C = 1/\sqrt{i*x} [/tex]. find the two roots of C. The solution for one of the roots is given as [tex] C = \sqrt{1/2x} - i\sqrt{1/2x} [/tex] can someone show me how to get to that solution?
2. Equations: the relationships between polar and rectangular coordinates for complex variables
3. I try taking the square root of i*x by using the relation:
[tex] \sqrt{a+ib} = \sqrt{r}\left(cos(\theta/2)+i*sin(\theta/2)) [/tex]
Then substitute with my variables (a+ib) = (0+ix) and because the real part is zero and the imaginary part is positive the angle is [tex] \pi/2 [/tex] so I get:
[tex]
\sqrt{0+ix} = \sqrt{\sqrt{0^2+x^2}}(cos(\pi/4)+i*sin(\pi/4))
[/tex]
=
[tex]
\sqrt{ix} = \sqrt{x}(cos(\pi/4)+i*sin(\pi/4)
[/tex]
Substitute it back into the first equation ([tex] C = 1/\sqrt{i*x} [/tex]) and get:
[tex] C = \frac{1}{\sqrt{x}(cos(\pi/4)+i*sin(\pi/4)} [/tex]
now I can multiply nominator and denominator with the conjugate of the denominator and come up with a solution:
[tex] C = \frac{\sqrt{x}}{2x} - i \frac{1}{2x} [/tex]
but this is not the solution that I have been given plus I was supposed to find two solutions so I'm clearly doing something wrong. But what?
1. The complex number (C) is [tex] C = 1/\sqrt{i*x} [/tex]. find the two roots of C. The solution for one of the roots is given as [tex] C = \sqrt{1/2x} - i\sqrt{1/2x} [/tex] can someone show me how to get to that solution?
2. Equations: the relationships between polar and rectangular coordinates for complex variables
3. I try taking the square root of i*x by using the relation:
[tex] \sqrt{a+ib} = \sqrt{r}\left(cos(\theta/2)+i*sin(\theta/2)) [/tex]
Then substitute with my variables (a+ib) = (0+ix) and because the real part is zero and the imaginary part is positive the angle is [tex] \pi/2 [/tex] so I get:
[tex]
\sqrt{0+ix} = \sqrt{\sqrt{0^2+x^2}}(cos(\pi/4)+i*sin(\pi/4))
[/tex]
=
[tex]
\sqrt{ix} = \sqrt{x}(cos(\pi/4)+i*sin(\pi/4)
[/tex]
Substitute it back into the first equation ([tex] C = 1/\sqrt{i*x} [/tex]) and get:
[tex] C = \frac{1}{\sqrt{x}(cos(\pi/4)+i*sin(\pi/4)} [/tex]
now I can multiply nominator and denominator with the conjugate of the denominator and come up with a solution:
[tex] C = \frac{\sqrt{x}}{2x} - i \frac{1}{2x} [/tex]
but this is not the solution that I have been given plus I was supposed to find two solutions so I'm clearly doing something wrong. But what?