Absolute value property of complex numbers

Question

Show that $ |z_1z_2|=|z_1||z_2| $

I have written out what $z_1$ and $z_2$ are and what $|z_1|$ and $|z_2|$ are but I still don't know how i would do it

edit: $z_1$ $=$ $x_1 + iy_1$ and $z_2 = x_2 +iy_2$ , so $z_1z_2$ $=$ $x_1x_2 +x_1iy_2+x_2iy_1+i^2y_1y_2$

Answer 1

Hint: Write your equation in the form $$|(x_1+y_1i)(x_2+y_2i)|=|x_1+y_1i||x_2+y_2i|$$

Answer 2

We have $z\overline{z}=|z|^2$ and hence $$ |z_1|^2|z_2|^2=z_1\overline{z_1}z_2\overline{z_2}=(z_1z_2)(\overline{z_1z_2})=|z_1z_2|^2, $$ and hence $|z_1||z_2|=|z_1z_2|$.

Answer 3

I like Dietrich Burde's answer. Here is an expansion of it.

As you have computed $$ (x_1+iy_1)(x_2+iy_2)=(x_1x_2-y_1y_2)+i(x_1y_2+x_2y_1) $$ Thus, $$ \begin{align} |z_1z_2|^2 &=(x_1x_2-y_1y_2)^2+(x_1y_2+x_2y_1)^2\\ &=x_1^2x_2^2+y_1^2y_2^2+x_1^2y_2^2+x_2^2y_1^2\\ &=\left(x_1^2+y_1^2\right)\left(x_2^2+y_2^2\right)\\ &=|z_1|^2|z_2|^2 \end{align} $$