Properties of Modulus

Modulus is multiplicative — it converts products into products of real numbers. Also: |Re(z)| ≤ |z| and |Im(z)| ≤ |z|, since each component cannot exceed the full distance from origin.

The key to all modulus properties is the identity $z\bar{z} = |z|^2$. Once this is established, every property follows from algebra.

Property 1 — Modulus of a Product

Claim: $|z_1 z_2| = |z_1||z_2|$

Proof:

$$|z_1 z_2|^2 = (z_1 z_2)\overline{(z_1 z_2)} = (z_1 z_2)(\bar{z}_1 \bar{z}_2) = (z_1 \bar{z}_1)(z_2 \bar{z}_2) = |z_1|^2 |z_2|^2$$

Taking square roots (both sides non-negative):

$$|z_1 z_2| = |z_1||z_2| \qquad \blacksquare$$

Property 2 — Modulus of a Quotient

Claim: $\left|\dfrac{z_1}{z_2}\right| = \dfrac{|z_1|}{|z_2|}$, $z_2 \neq 0$

Apply Property 1 to $z_1 = \dfrac{z_1}{z_2} \cdot z_2$:

$$|z_1| = \left|\frac{z_1}{z_2}\right| \cdot |z_2| \implies \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|} \qquad \blacksquare$$

Property 3 — Modulus of Conjugate

Claim: $|\bar{z}| = |z|$

$$|\bar{z}|^2 = \bar{z}\,\overline{\bar{z}} = \bar{z} \cdot z = z\bar{z} = |z|^2$$

So $|\bar{z}| = |z|$. Geometrically: conjugation reflects across the real axis, preserving the distance from the origin. $\blacksquare$

Property 4 — Modulus of a Power

Claim: $|z^n| = |z|^n$

Apply Property 1 repeatedly:

$$|z^n| = |z \cdot z \cdots z| = |z||z|\cdots|z| = |z|^n \qquad \blacksquare$$

Component Bounds

Since $a^2 \leq a^2 + b^2$ and $b^2 \leq a^2 + b^2$:

$$|\operatorname{Re}(z)| = |a| \leq \sqrt{a^2 + b^2} = |z|$$ $$|\operatorname{Im}(z)| = |b| \leq \sqrt{a^2 + b^2} = |z|$$

Each component is at most as large as the full modulus. This is used constantly in bounding arguments: to show $|z|$ is small, it suffices to bound $|\operatorname{Re}(z)|$ and $|\operatorname{Im}(z)|$ separately.