A few complex number qs (1 Viewer)

5uckerberg · Apr 29, 2022

For d when you have $\arg\left(z^{2}\right)$ this implies that you need to double the argument to obtain the arg of z so in turn

$\arg\left(z^{2}\right)=2\arg\left(z\right)=\frac{2\pi}{3}$ .
Thus, $\arg\left(z\right)=\frac{\pi}{3}$

=)(= · Apr 29, 2022

I got that but in the solution for some reason they also drew in (-2pi/3)

yanujw · Apr 29, 2022

=)(= said:
I got that but in the solution for some reason they also drew in (-2pi/3)

This is because squaring complex nos with arg (-2pi/3) gives them an arg of -4pi/3, which is an equilvalent arg to 2pi/3 and therefore satisfies the condition.

=)(= · Apr 29, 2022

ohh okay I see but how would you get that working from pi/3

5uckerberg · Apr 29, 2022

De Moivre's Theorem that is what is happening squaring the angle doubles the angle of rotation and so on.

=)(= · Apr 29, 2022

I still can't see it for some reason

5uckerberg · Apr 29, 2022

=)(= said:
View attachment 35522
View attachment 35523

Part b well we need to know this fact $\omega\bar{\omega}=|\omega|^{2}=1$ and $\omega+\bar{\omega}=2Re\left(\omega\right)$ and $\omega-\bar{\omega}=2Im\left(\omega\right)$
Once this is known our next step is pretty straightforward.

$\frac{1+\omega}{1-\omega}=\frac{\left(1+\omega\right)\left(1-\bar{\omega}\right)}{\left(1-\omega\right)\left(1-\bar{\omega}\right)}$ .

$\frac{1+\omega-\bar{\omega}-\omega\bar{\omega}}{1-\omega-\bar{\omega}+\omega\bar{\omega}}$ .

$\frac{2Im\left(\omega\right)}{2-2Re\left(\omega\right)}$

$\frac{Im\left(\omega\right)}{1-Re\left(\omega\right)}$

$\frac{iy}{1-x}$

I would like @=)(= to confirm the answer for this one.

=)(= · Apr 29, 2022

5uckerberg said:
Part b well we need to know this fact $\omega\bar{\omega}=|\omega|^{2}=1$ and $\omega+\bar{\omega}=2Re\left(\omega\right)$ and $\omega-\bar{\omega}=2Im\left(\omega\right)$
Once this is known our next step is pretty straightforward.

$\frac{1+\omega}{1-\omega}=\frac{\left(1+\omega\right)\left(1-\bar{\omega}\right)}{\left(1-\omega\right)\left(1-\bar{\omega}\right)}$ .

$\frac{1+\omega-\bar{\omega}-\omega\bar{\omega}}{1-\omega-\bar{\omega}+\omega\bar{\omega}}$ .

$\frac{2Im\left(\omega\right)}{2-2Re\left(\omega\right)}$

$\frac{Im\left(\omega\right)}{1-Re\left(\omega\right)}$

$\frac{y}{1-x}$

I would like @=)(= to confirm the answer for this one.

the answer is the y-axis

A few complex number qs (1 Viewer)

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