Understanding Inequality of Complex Numbers: |z+w|=|z-w|?

[mynameisfunk]

OK, in my book we have an inequality ||z|-|w||\leq|z+w|\leq|z|+|w| then from here it simply states, "Replacing w by -w here shows that ||z|-|w||\leq|z-w|\leq|z|+|w|.

How do we know that?
is |z+w|=|z-w|?? Note that z and w are complex numbers.

 

[Office_Shredder]

You can choose any number for w. In particular, -w works too. The key is that |-w|=|w|

 

[HallsofIvy]

OK, in my book we have an inequality ||z|-|w||\leq|z+w|\leq|z|+|w| then from here it simply states, "Replacing w by -w here shows that ||z|-|w||\leq|z-w|\leq|z|+|w|.

How do we know that?
is |z+w|=|z-w|?? Note that z and w are complex numbers.

No, |z+w| is not equal to |z- w| and it doesn't say that. "Replacing w by -w" changes |z+ w| to |z+ (-w)|= |z- w|. And, as Office_Shredder said, |-w|= |w| whether w is real or complex.

 

[mynameisfunk]

Thanks guys, I get it now. I very much appreciate the help, as always.