Properties of Modulus of a Complex Number

Eight properties of the modulus (for complex numbers $z,z_1,z_2$ and integer $n$):

1. $|z|=|\overline z|$
2. $|z_1+z_2|\le|z_1|+|z_2|$ (Triangle Inequality)
3. $|z_1z_2|=|z_1||z_2|$
4. $\left|\dfrac{z_1}{z_2}\right|=\dfrac{|z_1|}{|z_2|},\ z_2\ne0$
5. $|z_1-z_2|\ge\big||z_1|-|z_2|\big|,\ z_2\ne0$
6. $|z^n|=|z|^n$
7. $\operatorname{Re}(z)\le|z|$
8. $\operatorname{Im}(z)\le|z|$

Proof — Triangle Inequality. Using $|w|^2=w\overline w$:

(since $z_1\overline{z_2}+\overline{z_1}z_2=z_1\overline{z_2}+\overline{z_1\overline{z_2}}=2\operatorname{Re}(z_1\overline{z_2})$). By property 7, $\operatorname{Re}(z_1\overline{z_2})\le|z_1\overline{z_2}|=|z_1||\overline{z_2}|=|z_1||z_2|$, so

It generalises to any finite number of terms: $|z_1+z_2+\cdots+z_n|\le|z_1|+|z_2|+\cdots+|z_n|$.

Geometric interpretation. In the triangle with vertices $O,z_1,z_1+z_2$, the side corresponding to the vector $z_1+z_2$ cannot be longer than the sum of the lengths of the other two sides — exactly the ordinary triangle inequality from geometry, which is where the property gets its name.

Distance property. If $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$, then

So $|z_1-z_2|$ is exactly the ordinary distance between the points $z_1$ and $z_2$ in the Argand plane. Considering $O,z_1,z_2$ as a triangle and applying the triangle inequality to its sides gives two useful bounds used repeatedly to estimate $|z|$ from a given constraint: $|z_1+z_2|\le|z_1|+|z_2|$ and $|z_1-z_2|\ge\big||z_1|-|z_2|\big|$. …