Cue Column:

• What is |z|?
• Triangle inequality forms?
• When does equality hold?

Note Column: |z| = sqrt($a^2$ + $b^2$) = sqrt(z * z-bar). Key properties:

• |z1 * z2| = |z1| * |z2| (multiplicative)
• |z1/z2| = |z1|/|z2|
• |$z^n$| = |z|^n
• |z-bar| = |z|

Triangle inequality and its variants:

• |z1 + z2| <= |z1| + |z2| (equality iff arg(z1) = arg(z2), i.e., same direction)
• ||z1| - |z2|| <= |z1 + z2| (equality iff arg(z1) and arg(z2) differ by pi, i.e., opposite directions)
• |z1 - z2| >= ||z1| - |z2||

For finding max/min of |z|: if |z - z0| = r (z lies on circle), then:

• max |z| = |z0| + r (farthest point from origin)
• min |z| = ||z0| - r| (nearest point to origin)

Summary: Modulus is multiplicative. Triangle inequality gives bounds. Equality conditions depend on argument alignment.