Complex Modulus
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Introduction, definition, and properties of the complex modulus
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Complex Modulus
AHL 1.12
The complex modulus ∣z∣ is a measure of the size of a complex number:
∣z∣=√a2+b2🚫
Modulus on the complex plane
AHL 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:
zz*=|z|²
AHL 1.12
Notice that
zz∗=(a+bi)(a−bi)=a2+b2=∣z∣2🚫
Properties of complex modulus
AHL 1.12
The following properties apply for the complex modulus:
∣z∗∣=∣z∣🚫
∣zw∣=∣z∥w∣🚫
∣∣∣wz∣∣∣=∣w∣∣z∣🚫
∣zn∣=∣z∣n🚫
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