Complex numbers...?

Question

x,y,z are complex numbers and theyr mudulus are equal(the three between them...they're equal) and x*y*z=x*y + x*z +y*z=1
Calc. or determine x+y+z=?...
I apreciate any ideas...

Answer 1

Represent the numbers in the form:
x = |r|*exp(i*phi_x),
y = |r|*exp(i*phi_y),
z = |r|*exp(i*phi_z).
Here, |r| is the modulus and
phi_x, phi_y, phi_z are phases.

The condition x*y*z= 1 means
|r|^3*exp(i*phi_x + i*phi_y +i*phi_z) = 1.
Take the modulus of both parts and get |r|^3=1, or
(A) |r|=1.
Consequently,
(B) exp(i*phi_x + i*phi_y +i*phi_z) = 1.

Since |r| = 1, the expression x*y + x*z +y*z is written as
x*y + x*z +y*z = exp(i*phi_x + i*phi_y)
+ exp(i*phi_x +i*phi_z) + exp(i*phi_y +i*phi_z).
The first term is transformed to:
exp(i*phi_x + i*phi_y + i*phi_z - i*phi_z) =
exp (-i*phi_z) (we have used (B)).
Other terms are transformed in the same way.
As the result,
x*y + x*z +y*z = exp(-i*phi_z) + exp(-i*phi_x) + exp(-i*phi_y).
The right part is the complex conjugated of
x+y+z = exp(i*phi_x) + exp(i*phi_y) + exp(i*phi_z).

Hence, x+y+z is equal to the complex conjugated of
(x*y + x*z +y*z).

x+y+z = 1.