conman
Member
Can some one show that I I z1 I - I z2 I I =< I z1 + z2 I. Sate the condition when the equality occur????
Complex Number (1 Viewer)
- Thread starter conman
- Start date
One has the triangle inequality:
|A+B|<|A| + |B|
or |A+B| - |B| < |A|
Then if you let A=z1+z2, B=-z2 you get |z1| - |z2| < |z1 + z2|
and if you instead let B=-z1 you get |z2| - |z1| < |z1+z2|
combining the 2 you get
||z2| - |z1|| < |z1+z2|
equality occurs when z2 = -k*z1 where k is a non-negative real number
|A+B|<|A| + |B|
or |A+B| - |B| < |A|
Then if you let A=z1+z2, B=-z2 you get |z1| - |z2| < |z1 + z2|
and if you instead let B=-z1 you get |z2| - |z1| < |z1+z2|
combining the 2 you get
||z2| - |z1|| < |z1+z2|
equality occurs when z2 = -k*z1 where k is a non-negative real number