# Complex numbers problem

• Sep 25th 2011, 09:41 AM
Jormirias
Complex numbers problem
Hello there, I'm having some trouble solving a problem. Would be really grateful if someone could lend a hand.

Prove the theorem: Given a,b,c,d belonging to Z, exist u,v belonging to Z so that (a^2+b^2)(c^2+d^2)=u^2+v^2

You're supposed to use complex numbers to solve the exercise.

I started as following: (a^2+b^2)(c^2+d^2)=U^2+v^2 <=>
<=> (a+ib)(a-ib)(c+id)(c-id)=(u+iv)(u-iv) <=>
[assuming Z1=a+ib, Z2=c+id and w=u+iv] Z1Z1Z2Z2=ww (also, assume the underline is actually an overline, meaning the conjugate)

I'm not sure of what to do next :/
• Sep 25th 2011, 01:15 PM
Deveno
Re: Complex numbers problem
note that a^2+b^2 = |z1|^2, and c^2+d^2 = |z2|^2 (this is the complex modulus, if z1 = a+ib, then |z1| = √(a^2 + b^2)).

in the complex numbers, is it true that |(z1)(z2)| = |z1||z2|?

if so (this is really what you must prove), then |(z1)(z2)|^2 = (|z1||z2|)^2 = |z1|^2|z2|^2, so we can take u and v to be the real and imaginary parts of z1z2.

after that, it is not hard to see that u and v must be integers.
• Sep 25th 2011, 10:23 PM
Jormirias
Re: Complex numbers problem
Thanks. That helped out a bunch ;)