Question

state and prove triangle inequality of complex no​

Answer 1

Answer:

Our eventual goal in this problem is to prove the triangle inequality involving complex numbers.

(a) Show that for every z∈C,

|Re(z)|≤|z| and |Im(z)|≤|z|.

(b) Given z, w∈C, show that

|z+w|2=|z|2+|w|2+2Re(zw′).

(c) Using parts (a) and (b), prove the triangle inequality

|z+w|≤|z|+|w|.

This is what I got.

(a) By definition for a complex number z=x+yi,

|z|2=x2+y2=Re(z)2+Im(z)2

From here,

|z|2≥Re(z)2 and |z|2≥Im(z)2

And, finally,

|z|≥|Re(z)| and |z|≥|Im(z)|

(b) |z+w|2=(z+w)⋅(z+w)′

=(z+w)⋅[z′+w′]

=zz′+[zw′+z′w]+ww′

=|z|2+2Re[zw′]+|w|2

≤|z|2+2|zw′|+|w|2

=|z|2+2|z||w|+|w|2

=(|z|+|w|)2.

(c) Since both |z+w| and |z|+|w| are non-negative,

|z+w|≤|z|+|w|