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.
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