What Algebra law is this called: $|\alpha \beta| \le |\alpha| |\beta|$ for complex numbers

Question

I'm trying to remember the name of this law in complex number algebra:

$$|\alpha \beta| \le |\alpha| |\beta|$$

Where:

$$\alpha = Re\{\alpha\}+ i \ Im\{\alpha\}$$ $$\beta = Re\{\beta\}+ i \ Im\{\beta\}$$

Answer 1

For complex this is an immediate consequence of $|z|^2=z\bar z$ and associative property of multiplication.

Indeed $|ab|^2=(ab)(\overline{ab})=ab\bar a\bar b=(a\bar a)(b\bar b)=|a|^2|b|^2$

And since $|\cdot|$ is positive then we can get rid of squares.

For a more general result, you may be interested in this post:

A name for the property $ \| x \star y \| = \| x \| \| y \| $.

Answer 2

You not only have $\vert \alpha \beta \vert \le \vert \alpha \vert \vert \beta \vert$ but even $\vert \alpha \beta \vert = \vert \alpha \vert \vert \beta \vert$.

The complex modulus is multiplicative