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When a complex number $z=(a,b)=a+bi$ is interpreted as a vector, the absolute value of it is simply the length of the vector. Its definition is chosen in such a way that it is “compatible” with the Euclidean distance of two points in the complex plane.

Definition: Absolute Value of Complex Numbers

The absolute value \(|z|\) of a complex number \(z=a+bi\in\mathbb C\) is the (positive) square root of the dot product of \(z\) with itself, i.e. the non-negative real number

\[|z|:=\sqrt{\langle z, z\rangle}=\sqrt{\Re(z\cdot z^*)}=\sqrt{a^2+b^2}.\]

Geometrically, it is the real number, which equals the distance of the complex number from the origin:

| | | | | created: 2015-04-26 18:06:08 | modified: 2020-06-14 12:11:06 | by: bookofproofs | references: [696]

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Bibliography (further reading)

[696] Kramer Jürg, von Pippich, Anna-Maria: “Von den natürlichen Zahlen zu den Quaternionen”, Springer-Spektrum, 2013