tricky complex numbers proof

lektor

I recently was confronted by this monstrosity of a question in one of my mock exams.

|Z1 + Z2| ≤ |Z1| + |Z2|

I made a few attempts at it before becoming demoralized with the lack of progress..
|Z^2| was equal to Z1(conjugate)Z1
Hence equaling X^2 + Y^2

However even when expanding into x+iy form etc no avail, help is appreciated.

1800bigk

square both side and see if you notice anything and remember
if A=A then, surely A is less than or equal to A

Galileo

Geometrically this is just the triangle inequality. It just says that the sum of two sides of a triangle is always greater or equal than the third.

Since you know |z|^2=z*z. Why not write the left side out in this form?
ie: |z+w|^2=(z*+w*)(z+w)

BerkMath

I'm sorry but I had to laugh when I read "I recently was confronted by this monstrosity of a question..." only to find the triangle inequality beneathe. lol.