Topic explanation

Standard explanation

z + z-bar = 2x (real). z - z-bar = 2 i y. (z-bar)-bar = z. Product conjugate: (zw)-bar = z-bar w-bar. z * z-bar = |z|^2.

Key points

• flip Im only
• z-bar flips Im
• |z| = sqrt(x²+y²)
• modulus is real >= 0
• CONJUGATE RULES: z + z-bar = 2x (real). z - z-bar = 2 i y. (z-bar)-bar = z. Product conjugate: (zw)-bar = z-bar w-bar
• CONJUGATE RULES: z + z-bar = 2x (real).
• WHAT — CONJUGATE RULES: z + z-bar = 2x (real). z - z-bar = 2 i y. (z-bar)-bar = z. Product conjugate: (zw)-bar = z-bar w-bar
• WHY — Why it matters: This rule is a separate learning unit inside conjugate and modulus. It can be defined, applied, and

Labelled diagram