# Thread: Real and Complex Number Systems

1. ## Real and Complex Number Systems

Hi there....

This is what is given:

x = r(cos u + i sin u) and y = t(cos v + i sin v)

(1) Prove that the modulus of (xy) is the product of their moduli, showing each step.

(2) Prove that the amplitude of (xy) is the sum of their amplitudes, showing each step.

i need a starting point, desperately. This one is completely throwing me for a loop. Thanks!

2. Do you know of Eulers formula?

$e^{i\theta}=\cos{\theta}+i\sin{\theta}$

3. No, I do not....

4. Well applying it to your formulae:

$x=re^{iu},\quad y=te^{iv}$

5. (1) Prove that the modulus of (xy) is the product of their moduli, showing each step.
If you do not wish to resort to Euler's formula, you can multiply the terms out normally and then apply these formulas:
cos(u)cos(v)-sin(u)sin(v) = cos(u+v)
cos(u)sin(v)+sin(u)cos(v) = sin(u+v)

(2) Prove that the amplitude of (xy) is the sum of their amplitudes, showing each step.
...
What is amplitude?