## anonymous 5 years ago can u teach me, how i can solve this problem... Triangle inequality for complex numbers is |Z1 + Z2|<= |Z1| + |Z2| . Write down at least eight different conditions for which |Z1 + Z2|= |Z1| + |Z2| .

1. anonymous

Let the angle between the two complex position be @ So the length of the Z1+Z2 complex point or in other words |Z1+Z2| is = $\sqrt(Z _{1}^2\sin^2@+Z _{2}^2+Z _{1}^2\cos^2@+Z _{1}Z _{2}\cos@)$ Continued ........................

2. anonymous

Let name that expression as A

3. anonymous

Now when |Z1+Z2|=|Z1|+|Z2| Z1+Z2=A

4. anonymous

Squaring both sides and simplifying we get @=0

5. anonymous

So you can get the equation only when the amplitude of the complex number is the same. I hope you have understood Please note that here the symbol Z1 and Z2 is the magnitude of the actual Z1 and Z2

6. anonymous

i got what u had explained...thank you so much...this open stdy really halp me to think out of the box..

7. anonymous

As you asked on through chat how to calculate amplitude of a complex number, so here is the reply Lets take a complex number x+iy So its amplitude is y/x Lets take another example z=3+7i So its amplitude would be 7/3 According the reply I gave you, the equation |z1|+|z2|=|z1+z2| is valid only when the amplitude of two complex number is the same. So for an example The complex number z1= 2+i4 and the complex number z2=6+i12 holds the relationship |z1|+|z2|=|z1+z2| Hope, I have made myself clear

8. anonymous

So silly of me! All this time, I have been mentioning a wrong term. I am extremely sorry for that. Please note that in the above answer everywhere I have used the word ""amplitude"", it must be instead ""argument""