Complex Modulus

AHL HL 1.12

The complex modulus ∣z∣ is a measure of the size of a complex number:

∣z∣=√a2+b2🚫

AHL HL 1.12

On the complex plane, z=a+bi has coordinates (a,b). Therefore

∣z∣=√a2+b2🚫

represents the distance of z from the origin:

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AHL HL 1.12

Notice that

zz∗=(a+bi)(a−bi)=a2+b2=∣z∣2🚫

AHL HL 1.12

The following properties apply for the complex modulus:

∣z∗∣=∣z∣🚫

∣zw∣=∣z∥w∣🚫

∣∣∣wz∣∣∣=∣w∣∣z∣🚫

∣zn∣=∣z∣n🚫

Exercises