136k views
0 votes
We know for a fact that the equation:


|z+z'| \leqslant |z|+|z'|
holds for any 2 complex numbers

I came up with the conclusion that:

|z+z'|=|z|+|z'|only holds when z and z' are pure imaginary or pure real numbers of the SAME sign.
How can i prove this algebraically/geometrically?
Note: Irrelevant answers will be reported



User Gamote
by
8.0k points

1 Answer

6 votes

They don't need to be pure real or imaginary. Any "mixed" complex number works so long as
z=z'.

Let
z=z'=a+bi. Then


|z+z'| = |2a+2bi| = 2 √(a^2+b^2)


|z| + |z'| = 2|a+bi| = 2 √(a^2+b^2)

so
|z+z'|=|z|+|z'|.

The geometric interpretation is essentially identical.
|z+z'|=2|z| is a complex number twice the distance away from the origin in the complex plane as
z, which is exactly
|z|+|z|.

User Dorca
by
8.2k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories