Don't understand some parts of Complex Numbers (1 Viewer)

[someth1ng]

In the Patel textbook, I was going through the examples and got to this one:

Find the locus of w if w=(z-1)/z, given |z|=2

Can someone explain the steps because Patel doesn't explain it clearly.
- Why is |1-w|=|w-1|?
- Why is |z|=1/|1-w| if z=1/(1-w)?
[HR][/HR]Cheers

 

[Nooblet94]

Here's the answer to your first query:

You can also think about it using vectors, or you can simply factorise out a -1 and then you get the desired result.

Give me a few minutes and I'll do the other parts.

<a href="http://www.codecogs.com/eqnedit.php?latex=\\ \textrm{Let}~z=r_1cis\theta_1~\textrm{and}~w=r_2cis\theta_2\\ \frac{z}{w}\\ =\frac{r_1cis\theta_1}{r_2cis\theta_2}\\ =\frac{r_1}{r_2}cis (\theta_1-\theta_2)\\ ~\\ \textrm{Hence,} \left | \frac{z}{w}\right |=\frac{r_1}{r_2}=\frac{|z_1|}{|z_2|}\\ \therefore \textrm{If}~z=\frac{1}{1-w},~\textrm{then}~|z|=\left|\frac{1}{1-w}\right|=\frac{1}{|1-w|}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\\ \textrm{Let}~z=r_1cis\theta_1~\textrm{and}~w=r_2cis\theta_2\\ \frac{z}{w}\\ =\frac{r_1cis\theta_1}{r_2cis\theta_2}\\ =\frac{r_1}{r_2}cis (\theta_1-\theta_2)\\ ~\\ \textrm{Hence,} \left | \frac{z}{w}\right |=\frac{r_1}{r_2}=\frac{|z_1|}{|z_2|}\\ \therefore \textrm{If}~z=\frac{1}{1-w},~\textrm{then}~|z|=\left|\frac{1}{1-w}\right|=\frac{1}{|1-w|}" title="\\ \textrm{Let}~z=r_1cis\theta_1~\textrm{and}~w=r_2cis\theta_2\\ \frac{z}{w}\\ =\frac{r_1cis\theta_1}{r_2cis\theta_2}\\ =\frac{r_1}{r_2}cis (\theta_1-\theta_2)\\ ~\\ \textrm{Hence,} \left | \frac{z}{w}\right |=\frac{r_1}{r_2}=\frac{|z_1|}{|z_2|}\\ \therefore \textrm{If}~z=\frac{1}{1-w},~\textrm{then}~|z|=\left|\frac{1}{1-w}\right|=\frac{1}{|1-w|}" /></a>

 

[someth1ng]

Oh, that makes much more sense - using vectors, it would sort of be an inverted reflection!

 

[IamBread]

 

[someth1ng]

Yeah, I see now...I didn't realise that |1| was 1 so I got confused.

Thanks guys, I can now put this in my notes and make sense with it!

 

[IamBread]

For the first one, just remember modulus is basically a 2 dimensional absolute value, so the same way |1-6| = 5, |6-1| = 5, the modulus of complex numbers also works like that.

 

[someth1ng]

For the first one, just remember modulus is basically a 2 dimensional absolute value, so the same way |1-6| = 5, |6-1| = 5, the modulus of complex numbers also works like that.

You're saying that with modulus, in an equation, it could be treated like an absolute value?

 

[IamBread]

You're saying that with modulus, in an equation, it could be treated like an absolute value?

It's essentially what it is, distance from the origin.

 

[someth1ng]

It's essentially what it is, distance from the origin.

This makes me feel...much better.

Thanks for your help IamBread and Nooblet94!