Complex numbers from cambridge (1 Viewer)

[hanod]

Can some one show me the solution for the following question from Cambridge book

let z1,z2 and z3 be three complex numbers with modulus 1. prove that |z1z2+z1z3+z2z3| = |z1+z2+z3|

 

[Lith_30]

using the identity




which will eventually simplify to which is

 

[hogzillaAnson]

Call them a,b,c instead for simplicity. A few things of note:

Also,

So,

To be honest, we don't need to use the fact that the conjugate is the reciprocal here, but it just made typesetting easier. The key is to use the fact that



then expand and equate

 

[hanod]

using the identity




which will eventually simplify to which is

Big thank you

 

[hanod]

Call them a,b,c instead for simplicity. A few things of note:

Also,

So,

To be honest, we don't need to use the fact that the conjugate is the reciprocal here, but it just made typesetting easier. The key is to use the fact that



then expand and equate

Big thank you

 

[idkkdi]

Can some one show me the solution for the following question from Cambridge book

let z1,z2 and z3 be three complex numbers with modulus 1. prove that |z1z2+z1z3+z2z3| = |z1+z2+z3|

|z1+z2+z3| = |conj(z1+z2+z3)|
Since mod 1, conj is 1/z_n, |z1+z2+z3| = |1/z1 + 1/z2 + 1/z3|
Multiply RHS by |z1||z2||z3| = 1,
|z1+z2+z3| = |z1z2+z1z3+z2z3|