# Homework Help: Complex numbers, basic problem

1. Aug 17, 2010

### jav

1. The problem statement, all variables and given/known data

Know: modulus(z) < 3
WTS: |Im(z2 - zbar + 6)| <12

where zbar is the complex conjugate

2. Relevant equations

z = x + iy

3. The attempt at a solution

|Im(z2 - zbar + 6)|
= |Im(x2 + 2i*x*y - y2 - x + iy + 6)|
= |2xy + y|

So I want to show |2xy + y|< 12

I already proved it using maximization and Lagrange multipliers, but it seems like overkill, and I think there is some kind of arithmetic trick I am missing. Anyone see it?

Thanks

Last edited: Aug 17, 2010
2. Aug 17, 2010

### ╔(σ_σ)╝

|Im(z)|$$\leq$$ |z|
Then use the triangle inequality to derive an upper bound.

3. Aug 17, 2010

### hunt_mat

One thing to be careful about is to use:
$$|z_{1}-z_{2}|\leqslant ||z_{1}|-|z_{2}||$$