MHB Complex number geometrical problem

[amr21]

Show geometrically that if |z|=1 then, $Im[z/(z+1)^2]=0$

I am unsure how to begin this problem. I have sketched out |z|=1 but can't work out how to sketch the Imaginary part of the question.

 

[castor28]

Show geometrically that if |z|=1 then, $Im[z/(z+1)^2]=0$

I am unsure how to begin this problem. I have sketched out |z|=1 but can't work out how to sketch the Imaginary part of the question.

Hi amr21,

You should think in terms of arguments. In particular, for any complex numbers $a$ and $b$:

• $\mbox{Im}(a) = 0$ if $\arg(a)=0$ or $\arg(a)=\pi$.
• $\arg(ab) = \arg(a) + \arg(b) \pmod{2\pi}$

Try to use that to convert the question into a question about arguments.

You should then make a complete drawing. Pick a point $z$ such that $|z|=1$ and look at the triangle $(0,z,z+1)$.

Can you try that ? Write back if you need further help.

 

[amr21]

Hi amr21,

You should think in terms of arguments. In particular, for any complex numbers $a$ and $b$:

• $\mbox{Im}(a) = 0$ if $\arg(a)=0$ or $\arg(a)=\pi$.
• $\arg(ab) = \arg(a) + \arg(b) \pmod{2\pi}$

Try to use that to convert the question into a question about arguments.

You should then make a complete drawing. Pick a point $z$ such that $|z|=1$ and look at the triangle $(0,z,z+1)$.

Can you try that ? Write back if you need further help.

I don't think I understand, could you explain further? :)

 

[I like Serena]

I don't think I understand, could you explain further? :)

Let's take a look at a picture:
\begin{tikzpicture}[scale=3]
\coordinate (O) at (0,0);
\coordinate (Z) at ({sqrt(1/2)},{sqrt(1/2)});
\coordinate (Zp1) at ({sqrt(1/2)+1},{sqrt(1/2)});
\draw circle (1);
\draw (O) -- (Z) -- (Zp1) -- cycle;
\draw (O) -- (1,0) -- (Zp1);
\draw (1,0) -- (Z);
\path (O) node[below left] {0} -- (Z) node[above] {z} -- (Zp1) node[above right] {z+1} -- (1,0) node[below right] {1};
\path (O) -- node[below] {1} (1,0) -- node[below] {1} (Zp1) -- node[above] {1} (Z) -- node[above] {1} (O);
\end{tikzpicture}

What can we say about the angle (with the real axis) of $z$ in relation to the angle of $z+1$?

 

[castor28]

I don't think I understand, could you explain further? :)

Hi amr21,

Could you also confirm that you understand the first part, about converting the question to a question about arguments ? If you don't, could you pinpoint what you don't understand ?