Complex Numbers Problem

ScummyMan

Can anyone help me with this problem? Its been driving me mad!!



"The complex number z satisfies the equation |z| = |z+2|
Show that the real part of z is -1."

I'd much appreciate any help, even a nudge in the right direction! There are other parts to the question, but I imagine that they will be fine once I get this part.

Indiego

Man.UnitedLad wrote: »

Can anyone help me with this problem? Its been driving me mad!!



"The complex number z satisfies the equation |z| = |z+2|
Show that the real part of z is -1."

I'd much appreciate any help, even a nudge in the right direction! There are other parts to the question, but I imagine that they will be fine once I get this part.

*disclaimer: Haven't done complex numbers yet so maybe im wrong*

Because its a modulus, you square both sides:
z^2 = z^2 + 4z + 4
so
4z + 4 = 0
4(z+1) = 0
therfore z is -1 because 4(-1+1)=0

ScummyMan

/facepalm

If that is right, which I guess it is, I will put my head through the wall. I have just spent the last half an hour trying ridiculously complicated calculations, and you just did it in 4 lines!!! :pac:

finality

Let z equal (a + bi) on both sides. Then square both sides. Remember that i^2 = -1.
Then ignore any term with i. Just let the real parts on either side equal to one another (as you're only concerned with the real parts here).

You can't really do it the way Indiego posted because that's treating z as if it's a real number, and not taking the imaginary part into account. (In this case the imaginary part of z is 0 but you can't assume that).

Indiego

Man.UnitedLad wrote: »

/facepalm

If that is right, which I guess it is, I will put my head through the wall. I have just spent the last half an hour trying ridiculously complicated calculations, and you just did it in 4 lines!!! :pac:

5th year naivity :cool:
I'm like 99.9999999% sure its right tbh, was just doubtful because I know nothing about complex numbers XD

finality

It's still not too complicated btw, I did it in 5 lines.

ScummyMan

finality wrote: »

Let z equal (a + bi) on both sides. Then square both sides. Remember that i^2 = -1.
Then ignore any term with i. Just let the real parts on either side equal to one another (as you're only concerned with the real parts here).

You can't really do it the way Indiego posted because that's treating z as if it's a real number, and not taking the imaginary part into account. (In this case the imaginary part of z is 0 but you can't assume that).

Yeah that was the problem I had with Indiegos answer, he/she just ignored the imaginary part which is what someone with no experience of complex numbers would do.

aarond280

another way of doing it would be to let z=a+bi
|a+bi|=|a+bi+2|
(a^2+b^2)^(1/2)= ((a+2)^2+b^2))^(1/2)
work this out and let real=real that is a and 4giving a=-1

MathsManiac

Another way is to think of it visually on the Argand diagram. The modulus of z is its distance from the origin (0+0i), and the modulus of z-w is the distance between z and w.

So, in this case, you've been told that z is the same distance from 0 as it is from -2. So it must be on the vertical line through -1. Voila!

finality

aarond280 wrote: »

another way of doing it would be to let z=a+bi
|a+bi|=|a+bi+2|
(a^2+b^2)^(1/2)= ((a+2)^2+b^2))^(1/2)
work this out and let real=real that is a and 4giving a=-1

So what I said then? :P

ScummyMan

MathsManiac wrote: »

Another way is to think of it visually on the Argand diagram. The modulus of z is its distance from the origin (0+0i), and the modulus of z-w is the distance between z and w.

So, in this case, you've been told that z is the same distance from 0 as it is from -2. So it must be on the vertical line through -1. Voila!