1. ## [SOLVED] Is this answer and steps correct?

If not, please do correct me.

The question is,

The complex number z is given by z = 1 - i. Express $z^2 - \frac{1}{z}$ in the form p+qi where p and q are real and find the modulus and argument of $z^2 - \frac{1}{z}$.

I don't know, I just feel that something is not right here even though I've checked many times. Here's my answer:

$-1/2 - (5/2)i$

Ok. I've figured out how to use LaTex.

The complex number z is given by $z = 1 - i$. Express $z^2 - \frac{1}{z}$ in the form p+qi where p and q are real and find the modulus and argument of $z^2 - \frac{1}{z}$

Here is how I expressed it in the form p + qi:

= $z^2 - \frac{1}{z}$

= $(1 - i)^2 - \frac{1}{1 - i}$

= $(1 - 2i - 1) - \frac{1}{1 - i}$

= $\frac{(1 - 2i - 1)(1 - i) - 1}{1 - i}$

= $\frac{(-2i - 2 - 1)}{1 - i}$

= $\frac{(-2i - 2 - 1)}{1 - i}\;\text{x}\,\frac{1 + i}{1 + i}$

= $\frac{(-3 - 2i)(1 + i)}{2}$

= $\frac{-3 - 5i + 2}{2}$

= $\frac{-1 - 5i}{2}$

= $\text{-}\,\frac{1}{2}\;\text{-}\,\frac{5}{2}\;\text{i}$

=Correct or Wrong? Please help to evaluate! Thanks. If this is correct then that means my modulus and argument are also correct.

2. try again

3. Originally Posted by mark1950
If not, please do correct me.
The question is,
The complex number z is given by z = 1 - i. Express $z^2 - 1/z$ in the form p+qi where p and q are real and find the modulus and argument of $z^2 - 1/z$.
First, notice that your exponents did not show up in your OP.
I added the correct LaTeX codes.
Now it is not clear what you are asking.
Is it $z^2 - \frac{1}
{z}\;\text{ or } \,\frac{{z^2 - 1}}
{z}$

4. Ok. I've figured out how to use LaTex.

The complex number z is given by $z = 1 - i$. Express $z^2 - \frac{1}{z}$ in the form p+qi where p and q are real and find the modulus and argument of $z^2 - \frac{1}{z}$

Here is how I expressed it in the form p + qi:

= $z^2 - \frac{1}{z}$

= $(1 - i)^2 - \frac{1}{1 - i}$

= $(1 - 2i - 1) - \frac{1}{1 - i}$

= $\frac{(1 - 2i - 1)(1 - i) - 1}{1 - i}$

= $\frac{(-2i - 2 - 1)}{1 - i}$

= $\frac{(-2i - 2 - 1)}{1 - i}\;\text{x}\,\frac{1 + i}{1 + i}$

= $\frac{(-3 - 2i)(1 + i)}{2}$

= $\frac{-3 - 5i + 2}{2}$

= $\frac{-1 - 5i}{2}$

= $\text{-}\,\frac{1}{2}\;\text{-}\,\frac{5}{2}\;\text{i}$

=Correct or Wrong? Please help to evaluate! Thanks. If this is correct then that means my modulus and argument are also correct.

5. Originally Posted by mark1950
Ok. I've figured out how to use LaTex.

The complex number z is given by $z = 1 - i$. Express $z^2 - \frac{1}{z}$ in the form p+qi where p and q are real and find the modulus and argument of $z^2 - \frac{1}{z}$

Here is how I expressed it in the form p + qi:

= $z^2 - \frac{1}{z}$

= $(1 - i)^2 - \frac{1}{1 - i}$

= $(1 - 2i - 1) - \frac{1}{1 - i}$
At this point I would combine like terms to simplify things: (1 - 2i - 1) = -2i:

= $-2i - \frac{1}{1 - i}$

Now combine:

= $\frac{-2i(1 - i) - 1}{1 - i}$

= $\frac{-2i - 2 - 1}{1 - i}$

= $\frac{-2i - 3}{1 - i}$

Now multiply the numerator & denominator by the denominator's conjugate:

= $\frac{-2i - 3}{1 - i}$

= $\frac{-2i - 3}{1 - i} \times \frac{1 + i}{1 + i}$

= $\frac{-2i + 2 - 3 -3i}{1 + 1}$

= $\frac{-1 -5i}{2}$

= $-\frac{1}{2} -\frac{5}{2}i$

We get the same answer nevertheless.

The modulus of this complex number would be

$|z| = \sqrt{\left( -\frac{1}{2} \right)^2 + \left( -\frac{5}{2} \right)^2} = \sqrt{\frac{1}{4} + \frac{25}{4}} = \frac{\sqrt{26}}{2}$

The argument would be found by solving

$tan(\theta) = \frac{-5/2}{-1/2} = 5$

$\theta = tan^{-1}(5) + 180\ deg$ (because the point would be in Quadrant III)

$\theta = 258.69\ deg$

01

6. Oh, ok thanks!