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:

Chapters: 1 2 3
Definitions: 4 5 6 7 8 9 10 11 12
Examples: 13
Lemmas: 14
Proofs: 15 16 17 18 19 20 21 22 23
Propositions: 24 25 26 27 28 29
Theorems: 30

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  1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013