How to simplify complex conjugates and moduli

Question

I have a few questions regarding conjugates and moduli

1.) If $z=x+bi$, how would I simplify $\overline{z+a}$ or $\overline{z+i}$ Also, how would I simplify something like $\overline{2\overline{z}+5}$.

2.) Would $\overline{2} = 2$? In addition, would $\overline{3i}=-3i$?

3.) For the modulus, I have no idea how to simplify $|3z-i|$, or $|z+2|$.

Thank you so much!

Answer 1

Well, you don’t want to assume that any complicated-looking expression necessarily has a simplified version. As @ChrisCuster suggests in his answer, you can get pretty far by knowing that $\overline{z+w}=\overline z+\overline w$, and $\overline{zw}=\overline z\cdot\overline w$. And also, of course, that $\overline{(\overline z)}=z$.

At least one of your examples is truly jolly, namely $\overline{2\overline z +5}$, which you can simplify by writing it as $\overline 2\overline{(\overline z)}+\overline 5=2z+5$. But for something like $|z+2|$, you can expand it using the square-root expression that @ChrisCuster gave, but I prefer something else:

If somebody asks you to describe the set of all $z$ with (for example) $|z+2|=5$, then just remember this, that just as $|z|$ is the distance from the point $z$ in the Wessel/Argand/Gauss plane to the origin, so $|z-w|$ is the distance between the points $z$ and $w$. Thus $\mid z+2\mid$ is the distance from $z$ to $-2$, and asking for $\mid z+2\mid$ to be $5$ is asking $z$ to be on the circle of radius $5$ centered at the complex point $-2$.

Answer 2

Hint: $\overline {a+bi}= a-bi$.

So indeed $\overline 2=2$ and $\overline {3i}=-3i$.

And $\mid a+bi\mid=\sqrt{a^2+b^2}$.

So, say $z=a+bi$. Then $\mid 3z-i\mid=\mid3a+3bi-i\mid=\mid3a+(3b-1)i\mid=\sqrt{9a^2+(3b-1)^2}$.