Please help me find the square roots of a complex number

[val1]

Determine the square roots of the given complex numbers using
i) algebraic method
ii) De Moivre’s Theorum

and show the results on an Argand diagram.

\(\displaystyle \L
1 + i\)

I appreciate your help in finding the solution.
:?

 

[Unco]

ii) De Moivre’s Theorum

Let \(\displaystyle \L z \, = \, 1 \, + \, i\)

Although this is relatively ok, it's always a good idea to sketch an Argand diagram to ensure we calculate the correct argument.

             /|\ Im
              |
              |
              |
            1 + - - - - *
              |       * :
              |     *   :
              |   *\    :
              | *   |   :
      --------+---------+----->
              |         1     Re
              |
              |

\(\displaystyle \L |z| \, = \sqrt{1^2 \, + \, 1^2} \, = \, \sqrt{2} \, = \, 2^{\frac{1}{2}}\)

\(\displaystyle \L Arg(z) \, = \, \tan^{-1}{\left(\frac{ \, 1 \, }{ \, 1 \, }\right)} \, = \, \frac{\pi}{4}\)

So \(\displaystyle \L z \, = \, 2^{\frac{1}{2}} \left(\cos{\left(\frac{\pi}{4}\right)} \, + \, i\sin{\left(\frac{\pi}{4}\right)}\right)\)

Now apply De Moivre's Theorem to find \(\displaystyle \L z^{\frac{ \, 1 \, }{ \, 2 \, }}\) .

 

[val1]

Thanks for your reply.

I need some further information as I'm not sure I understand how this has been worked out.

The first statement is:

Let z = 1 + i

I'm not sure if I'm reading the Argand diagram right but it appears to show 1 on both the Real axis and on the Imaginary one - should the value on the Imaginary axis be i rather than 1?

Also, are the statements showing the modulus of z and the Argument of z the 'algebraic method'?

:?:

Thank you

 

[TchrQbic]

Thanks for your reply.

I need some further information as I'm not sure I understand how this has been worked out.

The first statement is:

Let z = 1 + i

I'm not sure if I'm reading the Argand diagram right but it appears to show 1 on both the Real axis and on the Imaginary one - should the value on the Imaginary axis be i rather than 1?



Also, are the statements showing the modulus of z and the Argument of z the 'algebraic method'?

:?:

Thank you

The Imaginary axis is in i units -- that is, here the graph is shown correctly for 1 + 1i . The method shown is the algebraic method.

 

[val1]

Thanks very much for clarifying that.