Complex Modulus of Product of Complex Numbers

Theorem
Let $z_1, z_2 \in \C$ be complex numbers.

Let $\left\vert{z}\right\vert$ be the modulus of $z$.

Then $\left\vert{z_1 z_2}\right\vert = \left\vert{z_1}\right\vert \cdot \left\vert{z_2}\right\vert$.

Corollary
Let $x, y \in \R$ be real numbers.

Let $\left\vert{x}\right\vert$ be the absolute value of $x$.

Then $\left\vert{xy}\right\vert = \left\vert{x}\right\vert \cdot \left\vert{y}\right\vert$.

Proof 1
Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$, where $x_1, y_1, x_2, y_2 \in \R$.

Proof 2
Let $\overline z$ denote the complex conjugate of $z$.

By Product of Complex Conjugates, compute:

Proof of Corollary
Follows directly from the fact that the Multiplicative Group of Reals Subgroup of Complex.

Therefore, any result applying to all complex numbers will also hold for all reals.

Alternatively, a direct approach can be taken:

Alternatively, as Real Numbers form Ordered Integral Domain, we can go straight to Product of Absolute Values and use that directly. ‎